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+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0015.png",
+      "answer": "144/5",
+      "fact_expr": "C: Ellipse;A: Point;P: Point;F: Point;Expression(C) = (x^2/25 + y^2/16 = 1);Coordinate(A) = (0, 36/5);RightFocus(C) = F;PointOnCurve(P, C);WhenMax(Perimeter(TriangleOf(A,P,F)))",
+      "coords": {
+        "A": [
+          0.0,
+          7.2
+        ]
+      }
+    },
+    {
+      "index": 16,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0016.png",
+      "answer": "Preserve all mathematical expressions, LaTeX formulas, symbols, and equations EXACTLY.  \nDo not add explanations, do not rewrite formulas, do not omit any content.  \n\nText:",
+      "fact_expr": "l: Line;C: Parabola;A: Point;Expression(C) = (y^2 = 6*x);Coordinate(A) = (0, 2);PointOnCurve(A, l);IsTangent(l, C)",
+      "coords": {
+        "A": [
+          0.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 17,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 18,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0018.png",
+      "answer": "5/4",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;Expression(OneOf(Asymptote(G))) = (y = (3/4)*x)",
+      "coords": {}
+    },
+    {
+      "index": 19,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 20,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0020.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;p: Number;H: Point;L:LineSegment;p>0;Expression(G) = (y^2 = 2*p*x);Coordinate(H) = (3, 1);IsChordOf(L,G);MidPoint(L)=H;Slope(L)=2",
+      "coords": {
+        "H": [
+          3.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 21,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          -1.0,
+          0.0
+        ],
+        "radius": 2.8284271247461903
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/circle/problem_0021.png",
+      "answer": "x^2/2+y^2=1",
+      "fact_expr": "C: Circle;B: Point;A: Point;P: Point;Expression(C) = (y^2 + (x - 1)^2 = 8);Coordinate(A) = (-1, 0);PointOnCurve(B, C);Intersection(PerpendicularBisector(LineSegmentOf(A,B)), LineSegmentOf(B, C)) = P",
+      "coords": {}
+    },
+    {
+      "index": 22,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.7320508075688772,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0022.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2/3 - y^2/3 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 23,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0023.png",
+      "answer": "1/2",
+      "fact_expr": "G: Parabola;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(A, G);PointOnCurve(B, G);PointOnCurve(F,LineOf(A,B));-Abs(LineSegmentOf(A, F)) + Abs(LineSegmentOf(B, F)) = 3/2",
+      "coords": {}
+    },
+    {
+      "index": 24,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0024.png",
+      "answer": "x = -1",
+      "fact_expr": "H: Circle;Expression(H) = (-6*x + x^2 + y^2 - 7 = 0);G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;IsTangent(H, Directrix(G)) = True",
+      "coords": {}
+    },
+    {
+      "index": 25,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0025.png",
+      "answer": "14",
+      "fact_expr": "G: Parabola;A: Point;B: Point;x1:Number;y1:Number;x2:Number;y2:Number;Expression(G) = (y^2 = 8*x);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);PointOnCurve(Focus(G),LineSegmentOf(A,B));IsChordOf(LineSegmentOf(A,B),G);x1 + x2 = 10",
+      "coords": {}
+    },
+    {
+      "index": 26,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 27,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0027.png",
+      "answer": "6",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;E: Point;Intersection(Directrix(C), xAxis) = E;G: Line;PointOnCurve(F, G);A: Point;B: Point;Intersection(G, C) = {A, B};M: Point;MidPoint(LineSegmentOf(A, B)) = M;Abs(LineSegmentOf(M, E)) = sqrt(11)",
+      "coords": {}
+    },
+    {
+      "index": 28,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0028.png",
+      "answer": "3*sqrt(5)/5",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;G: Line;Expression(G) = (-c + x = 0);c: Number;P: Point;OneOf(Intersection(G, C)) = P;Q: Point;Intersection(G, OneOf(Asymptote(C))) = Q;O: Origin;VectorOf(O, P) = (1/3)*VectorOf(O, F2) + (2/3)*VectorOf(O, Q)",
+      "coords": {}
+    },
+    {
+      "index": 29,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0029.png",
+      "answer": "(0,sqrt(3)/2]",
+      "fact_expr": "C: Ellipse;b: Number;G: Line;k: Number;b>0;Expression(C) = (x^2/4 + y^2/b^2 = 1);Expression(G) = (y = k*x - 1);PointOnCurve(Focus(C),xAxis);IsIntersect(G,C)",
+      "coords": {}
+    },
+    {
+      "index": 30,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0030.png",
+      "answer": "4",
+      "fact_expr": "M: Point;G: Parabola;Expression(G) = (y^2 = 4*x);PointOnCurve(M, G) = True;F: Point;Focus(G) = F;A: Point;C: Circle;Expression(C) = ((x - 4)^2 + (y - 1)^2 = 1);PointOnCurve(A, C) = True",
+      "coords": {}
+    },
+    {
+      "index": 31,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.8425120115280151,
+        "b": 1.6167607307434082,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 1.3765338735538535e-05,
+        "converged": true,
+        "iterations": 54
+      },
+      "output_path": "results/dev/hyperbola/problem_0031.png",
+      "answer": "3",
+      "fact_expr": "G: Hyperbola;a: Number;a>0;Expression(G) = (-y^2/7 + x^2/a^2 = 1);Eccentricity(G) = 4/3",
+      "coords": {}
+    },
+    {
+      "index": 32,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.23606797749979,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 5.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0032.png",
+      "answer": "11",
+      "fact_expr": "G: Ellipse;M: Point;N: Point;P: Point;Expression(G) = (x^2/9 + y^2/5 = 1);Coordinate(M) = (1, 4);Coordinate(N) = (2, 0);PointOnCurve(P, G)",
+      "coords": {
+        "M": [
+          1.0,
+          4.0
+        ],
+        "N": [
+          2.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 33,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.8284271247461903,
+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 8.0,
+        "y_coef": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0033.png",
+      "answer": "sqrt(107)/4",
+      "fact_expr": "E: Ellipse;Expression(E) = (x^2/8 + y^2/2 = 1);P: Point;PointOnCurve(P, E);A: Point;B: Point;Coordinate(A) = (-2, 1);Coordinate(B) = (2, -1);l: Line;Expression(l) = (x=4);M: Point;N: Point;Intersection(LineOf(A, P), l) = M;Intersection(LineOf(B, P), l) = N;Area(TriangleOf(A, B, P)) = Area(TriangleOf(M, N, P));O: Origin",
+      "coords": {
+        "A": [
+          -2.0,
+          1.0
+        ],
+        "B": [
+          2.0,
+          -1.0
+        ]
+      }
+    },
+    {
+      "index": 34,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0034.png",
+      "answer": "sqrt(7)",
+      "fact_expr": "G: Ellipse;b: Number;F: Point;P:Point;m:Number;M:Number;0 < b;b < 4;Expression(G) = (x^2/16 + y^2/b^2 = 1);LeftFocus(G) = F;PointOnCurve(P,G);Max(Length(LineSegmentOf(F,P)))=M;Min(Length(LineSegmentOf(F,P)))=m;M = 7*m",
+      "coords": {}
+    },
+    {
+      "index": 35,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0035.png",
+      "answer": "4*sqrt(6)",
+      "fact_expr": "E: Parabola;C: Circle;a: Number;Expression(E) = (x^2 = 4*y);Expression(C) = (a*y + 8*x + x^2 + y^2 - 5 = 0);PointOnCurve(Focus(E), C)",
+      "coords": {}
+    },
+    {
+      "index": 36,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0036.png",
+      "answer": "(sqrt(5)+1)/2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;B: Point;F: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(B) = (0, b);RightFocus(G) = F;IsPerpendicular(OverlappingLine(LineSegmentOf(B,F)),OneOf(Asymptote(G)))",
+      "coords": {}
+    },
+    {
+      "index": 37,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0037.png",
+      "answer": "1",
+      "fact_expr": "G: Hyperbola;H: Line;Expression(G) = (x^2 - 9*y^2 = 9);Expression(H) = (2*x - 6*y - 21 = 0)",
+      "coords": {}
+    },
+    {
+      "index": 38,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0038.png",
+      "answer": "sqrt(2)+1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;m: Number;m>0;Coordinate(OneOf(Focus(G))) = (m, 0);P: Point;Coordinate(P) = (m, 2*m);PointOnCurve(P, G)",
+      "coords": {}
+    },
+    {
+      "index": 39,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0039.png",
+      "answer": "6",
+      "fact_expr": "F: Point;C: Parabola;Expression(C) = (x^2 = 4*y);Focus(C) = F;PointOnCurve(F, l);l: Line;Intersection(l, C) = {A, B};A: Point;B: Point;l1: Line;l2: Line;TangentOnPoint(A, C) = l1;TangentOnPoint(B, C) = l2;Intersection(l1, l2)= P;P: Point;Negation(A=B)",
+      "coords": {}
+    },
+    {
+      "index": 40,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 41,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0041.png",
+      "answer": "3",
+      "fact_expr": "G: Ellipse;M: Point;F1: Point;O: Origin;N: Point;Expression(G) = (x^2/25 + y^2/16 = 1);PointOnCurve(M, G);LeftFocus(G) = F1;Distance(M, F1) = 4;MidPoint(LineSegmentOf(M, F1)) = N",
+      "coords": {}
+    },
+    {
+      "index": 42,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0042.png",
+      "answer": "12",
+      "fact_expr": "G: Ellipse;A: Point;B: Point;F2: Point;F1: Point;Expression(G) = (x^2/25 + y^2/16 = 1);LeftFocus(G) = F1;PointOnCurve(F1, LineSegmentOf(A, B));IsChordOf(LineSegmentOf(A,B),G);Abs(LineSegmentOf(A, F2)) + Abs(LineSegmentOf(B, F2)) = 8;RightFocus(G) = F2",
+      "coords": {}
+    },
+    {
+      "index": 43,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0043.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;A: Point;B: Point;P: Point;PointOnCurve(A, G);PointOnCurve(B, G);PointOnCurve(P, G);Negation(A=B);Negation(A=P);Negation(B=P);O: Origin;PointOnCurve(O, LineSegmentOf(A, B));k1: Number;k2: Number;Slope(LineOf(P, A)) = k1;Slope(LineOf(P, B)) = k2;k1*k2=3",
+      "coords": {}
+    },
+    {
+      "index": 44,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0044.png",
+      "answer": "2*sqrt(3)/3",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Inclination(OneOf(Asymptote(C))) = (2*pi)/3;Eccentricity(C)=e;e:Number",
+      "coords": {}
+    },
+    {
+      "index": 45,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 46,
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+      "conic_type": "hyperbola",
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+        "iterations": 93
+      },
+      "output_path": "results/dev/hyperbola/problem_0046.png",
+      "answer": "3, 15",
+      "fact_expr": "G: Hyperbola;F1: Point;F2: Point;P: Point;Expression(G) = (x^2/9 - y^2/16 = 1);Coordinate(F1) = (-5, 0);Coordinate(F2) = (5, 0);PointOnCurve(P, G);Distance(P, F1) = 9",
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+        "F1": [
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+          0.0
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+        "F2": [
+          5.0,
+          0.0
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+    {
+      "index": 47,
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+        "b": 1.0,
+        "major_axis": "y",
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+        "y_coef": 4.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0047.png",
+      "answer": "5",
+      "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2 + y^2/m = 1);FocalLength(G) = 4",
+      "coords": {}
+    },
+    {
+      "index": 48,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0048.png",
+      "answer": "sqrt(2)",
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+      "coords": {}
+    },
+    {
+      "index": 49,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.0,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0049.png",
+      "answer": "2+2*sqrt(3)",
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+      "coords": {}
+    },
+    {
+      "index": 50,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
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+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0050.png",
+      "answer": "25",
+      "fact_expr": "T: Point;Coordinate(T) = (-1, 1);C: Parabola;Expression(C) = (y^2 = 2*p*x);p: Number;L1: Line;L2: Line;TangentOfPoint(T, C) = {L1, L2};TangentPoint(L1, C) = A;TangentPoint(L2, C) = B;A: Point;B: Point;Focus(C) = F;F: Point;PointOnCurve(F, LineOf(A, B)) = True",
+      "coords": {
+        "T": [
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+          1.0
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+      }
+    },
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 52,
+      "success": true,
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+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0052.png",
+      "answer": "1/2",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;A: Point;M: Point;F2: Point;F1: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;UpperVertex(G)=A;VectorOf(A, M) = VectorOf(M, F2);DotProduct(VectorOf(M, F1), VectorOf(A, F2)) = 0",
+      "coords": {}
+    },
+    {
+      "index": 53,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 54,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 55,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0055.png",
+      "answer": "2",
+      "fact_expr": "G: Ellipse;O: Origin;M: Point;P: Point;F: Point;Expression(G) = (x^2/25 + y^2/16 = 1);PointOnCurve(P, G);Distance(P, LeftDirectrix(G)) = 10;LeftFocus(G) = F;VectorOf(O, M) = (VectorOf(O, F) + VectorOf(O, P))/2",
+      "coords": {}
+    },
+    {
+      "index": 56,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.4142135623730951,
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+        "x_coef": 4.0,
+        "y_coef": 2.0
+      },
+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0056.png",
+      "answer": "x+2*y-3=0",
+      "fact_expr": "G: Ellipse;H: LineSegment;Expression(G) = (x^2/4 + y^2/2 = 1);IsChordOf(H,G);Coordinate(MidPoint(H))=(1,1)",
+      "coords": {}
+    },
+    {
+      "index": 57,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 2.4270472526550293,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 4.453341261978494e-06,
+        "converged": true,
+        "iterations": 125
+      },
+      "output_path": "results/dev/hyperbola/problem_0057.png",
+      "answer": "y=pm*(1/2)*x\nx^2/4-y^2=1",
+      "fact_expr": "G: Hyperbola;a: Number;b: Number;F1: Point;P: Point;F2: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Focus(G) = {F1, F2};Eccentricity(G)=sqrt(5)/2;PointOnCurve(P, G);AngleOf(F1, P,F2) = ApplyUnit(90, degree);Area(TriangleOf(F1,P,F2))=1",
+      "coords": {}
+    },
+    {
+      "index": 58,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0058.png",
+      "answer": "x^2+y^2-2*x=0",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);H: Circle;Focus(G) =  Center(H);O: Origin;PointOnCurve(O, H)",
+      "coords": {}
+    },
+    {
+      "index": 59,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 60,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 2.23606797749979,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0060.png",
+      "answer": "11/21",
+      "fact_expr": "H: Ellipse;Expression(H) = (x^2/25 + y^2/16 = 1);G: Hyperbola;Expression(G) = (x^2/4 - y^2/5 = 1);F1: Point;F2: Point;Focus(H) = {F1, F2};Focus(G) = {F1, F2};P: Point;OneOf(Intersection(H, G)) = P",
+      "coords": {}
+    },
+    {
+      "index": 61,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0061.png",
+      "answer": "sqrt(10)/2",
+      "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;A: Point;F1: Point;F2: Point;B: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F2, l);Intersection(l, RightPart(C)) = {A, B};DotProduct(VectorOf(A, F1), VectorOf(A, F2)) = 0;3*VectorOf(A, F2) = VectorOf(F2, B)",
+      "coords": {}
+    },
+    {
+      "index": 62,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0062.png",
+      "answer": "sqrt(5)/7",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;P: Point;PointOnCurve(P, G);F1: Point;F2: Point;Focus(G) = {F1,F2};AngleOf(P, F1, F2) = alpha;AngleOf(P, F2, F1) = beta;alpha: Number;beta: Number;Cos(alpha) = sqrt(5)/5;Sin(alpha + beta) = 3/5",
+      "coords": {}
+    },
+    {
+      "index": 63,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 64,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 65,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0065.png",
+      "answer": "sqrt(7)/3",
+      "fact_expr": "F1: Point;F2: Point;Focus(E) = {F1, F2};E: Ellipse;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;A: Point;PointOnCurve(A, E) = True;AngleOf(F1, A, F2) = ApplyUnit(120, degree);Abs(LineSegmentOf(A,F1)) = 2*Abs(LineSegmentOf(A,F2))",
+      "coords": {}
+    },
+    {
+      "index": 66,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0066.png",
+      "answer": "pi/4",
+      "fact_expr": "C: Ellipse;H: Line;A: Point;B: Point;F2: Point;F1: Point;Expression(C) = (x^2/4 + y^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, H);Intersection(H, C) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 67,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 68,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0068.png",
+      "answer": "32",
+      "fact_expr": "T: Parabola;Expression(T) = (y^2 = 8*x);F: Point;Focus(T) = F;l1: Line;l2: Line;PointOnCurve(F, l1);PointOnCurve(F, l2);IsPerpendicular(l1, l2);A: Point;B: Point;C: Point;D: Point;Intersection(l1, T) = {A, B};Intersection(l2, T) = {C, D};P: Point;Q: Point;MidPoint(LineSegmentOf(A, B)) = P;MidPoint(LineSegmentOf(C, D)) = Q",
+      "coords": {}
+    },
+    {
+      "index": 69,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 70,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 0.5,
+        "major_axis": "y",
+        "x_coef": 0.25,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0070.png",
+      "answer": "4",
+      "fact_expr": "G: Ellipse;l: Line;F1: Point;A: Point;B: Point;F2:Point;Expression(G) = (4*x^2 + y^2 = 1);Focus(G)={F1,F2};PointOnCurve(F1,l);Intersection(l,G)={A,B}",
+      "coords": {}
+    },
+    {
+      "index": 71,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.830951894845301,
+        "b": 3.0,
+        "major_axis": "y",
+        "x_coef": 9.0,
+        "y_coef": 34.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0071.png",
+      "answer": "y^2/9 - x^2/16 = 1",
+      "fact_expr": "S: Hyperbola;G: Ellipse;Expression(G) = (x^2/9 + y^2/34 = 1);Focus(S)=Focus(G);Expression(OneOf(Asymptote(S)))=(y=(3/4)*x)",
+      "coords": {}
+    },
+    {
+      "index": 72,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 0.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0072.png",
+      "answer": "pm*x - 2*y = 0",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - 4*y^2 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 73,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 74,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0074.png",
+      "answer": "sqrt(15)",
+      "fact_expr": "C: Parabola;l: Line;Expression(C) = (y^2 = 4*x);Expression(l) = (2*x - y - 1 = 0)",
+      "coords": {}
+    },
+    {
+      "index": 75,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0075.png",
+      "answer": "3*sqrt(2) - 1",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);l: Line;Expression(l) = (x - y + 5 = 0);P: Point;PointOnCurve(P, G) = True;d1: Number;Distance(P, yAxis) = d1;d2: Number;Distance(P,l) = d2",
+      "coords": {}
+    },
+    {
+      "index": 76,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 77,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 78,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 79,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0079.png",
+      "answer": "4/3",
+      "fact_expr": "l: Line;C: Parabola;A: Point;D: Point;F: Point;B: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};Intersection(l, Directrix(C)) = D;MidPoint(LineSegmentOf(A, D)) = F",
+      "coords": {}
+    },
+    {
+      "index": 80,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0080.png",
+      "answer": "5/7",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;H: Line;PointOnCurve(F1, H);P: Point;Q: Point;Intersection(H, G) = {P, Q};LineSegmentOf(P, F2) = LineSegmentOf(F1, F2);3*LineSegmentOf(P, F1) = 4*LineSegmentOf(Q, F1)",
+      "coords": {}
+    },
+    {
+      "index": 81,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 3.4641016151377544,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0081.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/12 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 82,
+      "success": true,
+      "conic_type": "hyperbola",
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0082.png",
+      "answer": "1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2 = 1)",
+      "coords": {}
+    },
+    {
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+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0083.png",
+      "answer": "[-7,8]",
+      "fact_expr": "P: Point;G: Ellipse;Expression(G) = (x^2/45 + y^2/20 = 1);PointOnCurve(P, G) = True;Quadrant(P) = 3;J1: Point;J2: Point;Focus(G) = {J1, J2};IsPerpendicular(LineSegmentOf(P, J1), LineSegmentOf(P, J2)) = True;H: Line;Expression(H) = (-2*m + 4*x - 3*y + 1 = 0);Negation(Distance(P, H) > 3);m: Real",
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+    },
+    {
+      "index": 84,
+      "success": true,
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0084.png",
+      "answer": "2*sqrt(5)/5",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x - 2*y + 2 = 0);PointOnCurve(OneOf(Focus(G)),H);PointOnCurve(OneOf(Vertex(G)),H)",
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+    },
+    {
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+        "b": 1.5,
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0085.png",
+      "answer": "x^2/5-y^2/4=1",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;C: Circle;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(C) = (-6*x + x^2 + y^2 + 5 = 0);IsTangent(Asymptote(G),C);Center(C)=RightFocus(G)",
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+    },
+    {
+      "index": 86,
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+      "conic_type": "hyperbola",
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0086.png",
+      "answer": "y=pm*2*x",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P,C);Abs(VectorOf(P,F1)+VectorOf(P,F2))=Abs(VectorOf(F1,F2));Abs(VectorOf(P, F1)) = 2*Abs(VectorOf(P, F2))",
+      "coords": {}
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+    {
+      "index": 87,
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0087.png",
+      "answer": "(5*sqrt(5))/2",
+      "fact_expr": "H: Point;Coordinate(H) = (1, 0);l: Line;PointOnCurve(H, l);G: Parabola;Expression(G) = (y^2 = 4*x);A: Point;B: Point;Intersection(l, G) = {A, B};C: Point;Coordinate(C) = (-1, -1);IsPerpendicular(LineSegmentOf(C, A), LineSegmentOf(C, B))",
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+        "C": [
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0088.png",
+      "answer": "y=pm*2*x",
+      "fact_expr": "C: Parabola;N: Point;O: Origin;M: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;Intersection(Directrix(C), xAxis) = M;PointOnCurve(N, C);lambda:Number;Abs(VectorOf(M,N))=lambda*Abs(VectorOf(N,F));WhenMax(lambda)",
+      "coords": {}
+    },
+    {
+      "index": 89,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0089.png",
+      "answer": "{0,pm*1/2}",
+      "fact_expr": "l: Line;G: Parabola;k: Real;Expression(G) = (y^2 = x);Expression(l) = (y = k*(x + 1));NumIntersection(l,G)=1",
+      "coords": {}
+    },
+    {
+      "index": 90,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0090.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;H: Line;O: Origin;A: Point;B: Point;C: Point;Expression(G) = (x^2/12 - y^2/3 = 1);Expression(H) = (y = x*(sqrt(3)/3) - 2);Intersection(H, RightPart(G)) = {A, B};PointOnCurve(C, RightPart(G));VectorOf(O, A) + VectorOf(O, B) = t*VectorOf(O, C);t:Number",
+      "coords": {}
+    },
+    {
+      "index": 91,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 92,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0092.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;H: Parabola;p: Number;Expression(G) = (x^2/3 - y^2 = 1);Expression(H) = (y^2 = 2*(p*x));Focus(H) = RightFocus(G)",
+      "coords": {}
+    },
+    {
+      "index": 93,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 94,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0094.png",
+      "answer": "2",
+      "fact_expr": "C: Parabola;A: Point;B: Point;M: Point;F: Point;Expression(C) = (y^2 = 4*x);Coordinate(M) = (2, 2);Focus(C) = F;PointOnCurve(A, C);PointOnCurve(B, C);MidPoint(LineSegmentOf(A, B)) = M",
+      "coords": {
+        "M": [
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+          2.0
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+      }
+    },
+    {
+      "index": 95,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 96,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0096.png",
+      "answer": "32/3",
+      "fact_expr": "C: Parabola;G: Line;F: Point;A: Point;B: Point;l: Line;Expression(C) = (y^2 = 4*x);Focus(C) = F;Directrix(C) = l;PointOnCurve(F, G);Intersection(G, l) = A;OneOf(Intersection(G, C)) = B;VectorOf(F, A) = -3*VectorOf(F, B)",
+      "coords": {}
+    },
+    {
+      "index": 97,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 98,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0098.png",
+      "answer": "x^2+y^2/(2/3)=1",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;Eccentricity(C) = sqrt(3)/3;l: Line;PointOnCurve(F2, l);A: Point;B: Point;Intersection(l, C) = {A, B};Perimeter(TriangleOf(A, F1, B)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 99,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0099.png",
+      "answer": "3/2 - sqrt(2)",
+      "fact_expr": "G: Parabola;M: Point;O: Origin;A: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(M) = (0, 1);Focus(G) = F;Intersection(LineSegmentOf(F, M), G) = A",
+      "coords": {
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+          1.0
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+    },
+    {
+      "index": 100,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 101,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0101.png",
+      "answer": "4/5",
+      "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;I: Point;Expression(G) = (x^2/16 - y^2/9 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, RightPart(G));Incenter(TriangleOf(P, F1, F2)) = I;Area(TriangleOf(I,P,F2))=Area(TriangleOf(I,P,F1))-lambda*Area(TriangleOf(I,F1,F2));lambda:Number",
+      "coords": {}
+    },
+    {
+      "index": 102,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 3.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0102.png",
+      "answer": "48",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 12*y);H: Line;PointOnCurve(Focus(G), H);Inclination(H) = ApplyUnit(60, degree);A: Point;B: Point;Intersection(H, G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 103,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
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+          0.0
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+        "radius": 1.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/circle/problem_0103.png",
+      "answer": "(x - 3/2)^2 + y^2 = 1/4",
+      "fact_expr": "B: Point;Coordinate(B) = (3, 0);A: Point;G: Circle;Expression(G) = (x^2 + y^2 = 1);PointOnCurve(A, G) = True;M: Point;MidPoint(LineSegmentOf(A,B)) = M",
+      "coords": {}
+    },
+    {
+      "index": 104,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0104.png",
+      "answer": "x^2/4+y^2=1",
+      "fact_expr": "l: Line;E: Ellipse;b: Number;a: Number;A: Point;F: Point;B: Point;O:Origin;a > b;b > 0;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);RightFocus(E) = F;Eccentricity(E)=sqrt(3)/2;PointOnCurve(O,l);Inclination(l)=pi/3;Intersection(l,E)={A,B};Perimeter(TriangleOf(A, F, B))=4+(8*sqrt(13)/13)",
+      "coords": {}
+    },
+    {
+      "index": 105,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0105.png",
+      "answer": "5*sqrt(2)/2-1",
+      "fact_expr": "G: Parabola;H: Line;P: Point;A: Point;B: Point;L1:Line;L2:Line;Expression(G) = (y^2 = 4*x);Expression(H) = (x - y + 4 = 0);PointOnCurve(P, G);PointOnCurve(P,L1);PointOnCurve(P,L2);IsPerpendicular(L1,yAxis);IsPerpendicular(L2,H);FootPoint(L1,yAxis)=A;FootPoint(L2,H)=B",
+      "coords": {}
+    },
+    {
+      "index": 106,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0106.png",
+      "answer": "8",
+      "fact_expr": "G: Hyperbola;H: Line;A: Point;F2: Point;B: Point;F1: Point;Expression(G) = (x^2/4 - y^2/3 = 1);LeftFocus(G) =F1;RightFocus(G)=F2;PointOnCurve(F1, H);Intersection(H, LeftPart(G)) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 107,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 108,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 109,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
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+          2.0
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+        "radius": 3.872983346207417
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/circle/problem_0109.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/a = 1);a: Number;H: Circle;Expression(H) = (-4*y - 2*x + x^2 + y^2 - 10 = 0);PointOnCurve(Center(H), Asymptote(G))",
+      "coords": {}
+    },
+    {
+      "index": 110,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 3.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0110.png",
+      "answer": "4*sqrt(3)",
+      "fact_expr": "G: Ellipse;B: Point;C: Point;A: Point;Expression(G) = (x^2/3 + y^2 = 1);PointOnCurve(F2,LineSegmentOf(B,C));PointOnCurve(B, G);PointOnCurve(C, G);F1:Point;F2:Point;Focus(G)={F1,F2};A=F1",
+      "coords": {}
+    },
+    {
+      "index": 111,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0111.png",
+      "answer": "4",
+      "fact_expr": "l: Line;C: Parabola;B: Point;A: Point;F: Point;Expression(C) = (y = x^2/4);Focus(C) = F;PointOnCurve(F, l);Inclination(l)=pi/4;Intersection(l, C) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 112,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 113,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 114,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0114.png",
+      "answer": "sqrt(3)/3",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;F: Point;Focus(G) = F;A: Point;B: Point;PointOnCurve(A, G);PointOnCurve(B, G);AngleOf(A, F, B) = ApplyUnit(120, degree);IsChordOf(LineSegmentOf(A, B), G);MidPoint(LineSegmentOf(A, B)) = M;M: Point;L: Line;PointOnCurve(M, L);IsPerpendicular(L, l);Directrix(G) = l;l: Line;FootPoint(L, l) = M1;M1: Point",
+      "coords": {}
+    },
+    {
+      "index": 115,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0115.png",
+      "answer": "5",
+      "fact_expr": "C: Parabola;p: Number;F: Point;P: Point;A: Point;Expression(C) = (y^2 = 2*(p*x));Coordinate(A) = (4, 2);Coordinate(F) = (1, 0);PointOnCurve(P, C);Focus(C) = F",
+      "coords": {
+        "A": [
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+          2.0
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+        "F": [
+          1.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 116,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0116.png",
+      "answer": "3",
+      "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;P: Point;P0: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(P) = (0, 3*b);PointOnCurve(P, l);L1: Line;OneOf(Asymptote(C)) = L1;Slope(L1) > 0;IsParallel(l, L1);PointOnCurve(P0, RightPart(C));Distance(P0, l) > b",
+      "coords": {}
+    },
+    {
+      "index": 117,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0117.png",
+      "answer": "8",
+      "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;F:Point;Expression(G) = (y^2 = 4*x);Inclination(l)=ApplyUnit(45,degree);Focus(G)=F;PointOnCurve(F,l);Intersection(l,G)={A,B}",
+      "coords": {}
+    },
+    {
+      "index": 118,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0118.png",
+      "answer": "[sqrt(2)/2, 1)",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;Focus(G) = {F1, F2};Z1: Line;Z2: Line;Directrix(G) = {Z1, Z2};M: Point;N: Point;Intersection(Z1, xAxis) = M;Intersection(Z2, xAxis) = N;LineSegmentOf(M, N) <= 2*LineSegmentOf(F1, F2)",
+      "coords": {}
+    },
+    {
+      "index": 119,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0119.png",
+      "answer": "x = -4",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);H: Point;Coordinate(H) = (4, 0);PointOnCurve(H, G) = True;G: Line;Intersection(G, C) = {A, B};A: Point;B: Point",
+      "coords": {
+        "H": [
+          4.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 120,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          0.0,
+          0.0
+        ],
+        "radius": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/circle/problem_0120.png",
+      "answer": "{(1/3,+oo),(-oo,-1/3)}",
+      "fact_expr": "G: Hyperbola;k: Real;H: Circle;Expression(G) = (-1/(4*k^2)*y^2 + x^2/((9*k^2)) = 1);Expression(H) = (x^2 + y^2 = 1);NumIntersection(G, H)=0",
+      "coords": {}
+    },
+    {
+      "index": 121,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0121.png",
+      "answer": "5/7",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;G: Line;M: Point;F2: Point;F1: Point;N: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, G);Intersection(G, C) = {M, N};Abs(LineSegmentOf(M, F2)) = Abs(LineSegmentOf(F1, F2));7*Abs(LineSegmentOf(M, F1)) = 4*Abs(LineSegmentOf(M, N))",
+      "coords": {}
+    },
+    {
+      "index": 122,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0122.png",
+      "answer": "ApplyUnit(90, degree)",
+      "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 - y^2/16 = 1);Focus(G)={F1,F2};PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) - Abs(LineSegmentOf(P, F2)) = 32",
+      "coords": {}
+    },
+    {
+      "index": 123,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0123.png",
+      "answer": "5/3",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, RightPart(G));Abs(LineSegmentOf(P, F1)) = 4*Abs(LineSegmentOf(P, F2));e: Number;Eccentricity(G) = e",
+      "coords": {}
+    },
+    {
+      "index": 124,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0124.png",
+      "answer": "pm*sqrt(3)*a",
+      "fact_expr": "G: Parabola;a: Number;a>0;Expression(G) = (y^2 = 2*(a*x));H:Point;PointOnCurve(H,G);Distance(H,Focus(G))=2*a",
+      "coords": {}
+    },
+    {
+      "index": 125,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.7320508075688772,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 3.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0125.png",
+      "answer": "(0,pm*1)",
+      "fact_expr": "G: Ellipse;F1: Point;A: Point;F2: Point;B: Point;Expression(G) = (x^2/3 + y^2 = 1);LeftFocus(G) =F1;RightFocus(G)=F2;PointOnCurve(A,G);PointOnCurve(B,G);VectorOf(F1, A) = 5*VectorOf(F2, B)",
+      "coords": {}
+    },
+    {
+      "index": 126,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0126.png",
+      "answer": "-4",
+      "fact_expr": "l: Line;p: Number;A: Point;B: Point;p>0;x1: Number;y1: Number;x2: Number;y2: Number;Expression(G) = (y^2 = 2*(p*x));Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);PointOnCurve(Focus(G), l);Intersection(l, G) = {A, B};G: Parabola",
+      "coords": {}
+    },
+    {
+      "index": 127,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0127.png",
+      "answer": "10",
+      "fact_expr": "G: Parabola;M: Point;P: Point;F: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(P, G);L: Line;PointOnCurve(P, L);IsPerpendicular(L, Directrix(G));FootPoint(L, Directrix(G)) = M;Abs(LineSegmentOf(P, M)) = 5;Focus(G) = F",
+      "coords": {}
+    },
+    {
+      "index": 128,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 129,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0129.png",
+      "answer": "5",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;O: Origin;M: Point;PointOnCurve(M, LeftPart(G));Abs(LineSegmentOf(O, M)) = sqrt(a^2 + b^2);4*Abs(LineSegmentOf(M, F1)) = 3*Abs(LineSegmentOf(M, F2))",
+      "coords": {}
+    },
+    {
+      "index": 130,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 131,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 132,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 133,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 2.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0133.png",
+      "answer": "y=pm*2*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/4 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 134,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 135,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0135.png",
+      "answer": "x^2/5+y^2/4=1",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;H: Circle;B: Point;A: Point;J: Point;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2 + y^2 = 1);Coordinate(J) = (1, 1/2);PointOnCurve(Focus(G), xAxis);L1:Line;L2:Line;TangentOfPoint(J,H)={L1,L2};TangentPoint(L1,H)=A;TangentPoint(L2,H)=B;PointOnCurve(RightFocus(G),LineOf(A,B));PointOnCurve(UpperVertex(G),LineOf(A,B))",
+      "coords": {
+        "J": [
+          1.0,
+          0.5
+        ]
+      }
+    },
+    {
+      "index": 136,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0136.png",
+      "answer": "9",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;G: Line;PointOnCurve(F,G) = True;Intersection(G, C) = {A, B};A: Point;B: Point",
+      "coords": {}
+    },
+    {
+      "index": 137,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0137.png",
+      "answer": "x + 2*y - 2 = 0",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G);P: Point;Coordinate(P) = (1, 1/2);MidPoint(LineSegmentOf(A, B)) = P",
+      "coords": {
+        "P": [
+          1.0,
+          0.5
+        ]
+      }
+    },
+    {
+      "index": 138,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0138.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;F: Point;A: Point;B: Point;L: Line;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;PointOnCurve(F, L);IsPerpendicular(L, Asymptote(C));FootPoint(L, Asymptote(C)) = A;Intersection(L, yAxis) = B;VectorOf(F, A) = 2*VectorOf(A, B)",
+      "coords": {}
+    },
+    {
+      "index": 139,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0139.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "C: Parabola;p: Real;G: Circle;l:Line;A: Point;P: Point;F: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;Directrix(C)=l;Intersection(l,xAxis)=A;PointOnCurve(P,C);IsPerpendicular(LineSegmentOf(P,F),xAxis);IsDiameter(LineSegmentOf(A,F),G);Length(InterceptChord(LineOf(A,P),G))=2",
+      "coords": {}
+    },
+    {
+      "index": 140,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.6457513110645907,
+        "orientation": "vertical"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0140.png",
+      "answer": "8",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/7 - y^2/9 = -1)",
+      "coords": {}
+    },
+    {
+      "index": 141,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 142,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0142.png",
+      "answer": "8",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 8*x);F: Point;Focus(C) = F;l: Line;Directrix(C) = l;P: Point;PointOnCurve(P, C) = True;Quadrant(P) = 1;PointOnCurve(P, G) = True;IsPerpendicular(G, l) = True;G: Line;FootPoint(G, l) = M;M: Point;Slope(LineOf(P, F)) = sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 143,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0143.png",
+      "answer": "3",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 8*y);P: Point;PointOnCurve(P, G);Distance(P, Focus(G)) = 5",
+      "coords": {}
+    },
+    {
+      "index": 144,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0144.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;p: Number;p>0;Expression(G) = (y^2 = 2*(p*x));Expression(Directrix(G)) = (x = -1)",
+      "coords": {}
+    },
+    {
+      "index": 145,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 3.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0145.png",
+      "answer": "(3, 0)",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 12*x)",
+      "coords": {}
+    },
+    {
+      "index": 146,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0146.png",
+      "answer": "(sqrt(5)-1)/2",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;F1: Point;LeftFocus(G) = F1;O: Origin;P: Point;PointOnCurve(P, G);Q: Point;PointOnCurve(Q, RightDirectrix(G));VectorOf(P, Q) = 2*VectorOf(F1, O);lambda: Number;lambda>0;VectorOf(F1, Q) = lambda*(VectorOf(F1, O)/Abs(VectorOf(F1, O)) + VectorOf(F1, P)/Abs(VectorOf(F1, P)))",
+      "coords": {}
+    },
+    {
+      "index": 147,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0147.png",
+      "answer": "Preserve all mathematical expressions, LaTeX formulas, symbols, and equations EXACTLY.  \nDo not add explanations, do not rewrite formulas, do not omit any content.  \n\nText:",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;F: Point;Focus(G) = F;H: Line;PointOnCurve(F, H) = True;Intersection(H, G) = {P, Q};P: Point;Q: Point;IsPerpendicular(LineSegmentOf(P, H1), Directrix(G)) ;IsPerpendicular(LineSegmentOf(P, H2), Directrix(G)) ;FootPoint(LineSegmentOf(P, H1), Directrix(G)) = H1;FootPoint(LineSegmentOf(P, H2), Directrix(G)) = H2;PointOnCurve(P, LineSegmentOf(P, H1));PointOnCurve(Q, LineSegmentOf(P, H2));H1: Point;H2: Point;MidPoint(LineSegmentOf(H1, H2)) = M;M: Point;Abs(LineSegmentOf(P, F)) = a;a: Number;Abs(LineSegmentOf(Q, F)) = b;b: Number",
+      "coords": {}
+    },
+    {
+      "index": 148,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.366584062576294,
+        "b": 2.364173412322998,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.00017838670464698225,
+        "converged": true,
+        "iterations": 89
+      },
+      "output_path": "results/dev/hyperbola/problem_0148.png",
+      "answer": "4*sqrt(2)",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/6 + x^2/a^2 = 1);a: Number;a>0;Eccentricity(C) = 2",
+      "coords": {}
+    },
+    {
+      "index": 149,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0149.png",
+      "answer": "2*x+4*y-3=0",
+      "fact_expr": "G: Ellipse;H: LineSegment;Expression(G) = (x^2/2 + y^2 = 1);Coordinate(MidPoint(H)) = (1/2, 1/2);IsChordOf(H,G) = True",
+      "coords": {}
+    },
+    {
+      "index": 150,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0150.png",
+      "answer": "2",
+      "fact_expr": "G: Line;Expression(G) = (2*x + y - 3 = 0);Gamma: Parabola;Expression(Gamma) = (y^2 = 8*x);A: Point;B: Point;Intersection(G, Gamma) = {A, B};H: Circle;PointOnCurve(A, H);PointOnCurve(B, H);C: Point;D: Point;Intersection(H, Gamma) = {C, D};k: Number;Slope(LineOf(C, D)) = k",
+      "coords": {}
+    },
+    {
+      "index": 151,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0151.png",
+      "answer": "2*sqrt(3)/3",
+      "fact_expr": "O: Origin;F: Point;Vertex(G) = O;Focus(G) = F;G: Parabola;Expression(G) = (y^2 = 2*x);M: Point;PointOnCurve(M, G)",
+      "coords": {}
+    },
+    {
+      "index": 152,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0152.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;p: Number;H: Circle;p>0;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (2*x + x^2 + y^2 - 3 = 0);PointOnCurve(Focus(G), H)",
+      "coords": {}
+    },
+    {
+      "index": 153,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 154,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 6.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 36.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0154.png",
+      "answer": "9",
+      "fact_expr": "G: Ellipse;F1: Point;F2: Point;Expression(G) = (x^2/36 + y^2/9 = 1);Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 155,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0155.png",
+      "answer": "(1, sqrt(3))",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;H: Circle;Expression(H) = (y^2 + (x - sqrt(3))^2 = 2);IsIntersect(Asymptote(G), H)",
+      "coords": {}
+    },
+    {
+      "index": 156,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0156.png",
+      "answer": "y^2=4*x",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;l: Line;PointOnCurve(F, l) = True;Inclination(l) = pi/4;Intersection(l,G) = {A,B};A: Point;B: Point;IsDiameter(LineSegmentOf(A,B),H) = True;H: Circle;PointOnCurve(I,H) = True;I: Point;Coordinate(I) = (-p/2, 2)",
+      "coords": {}
+    },
+    {
+      "index": 157,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0157.png",
+      "answer": "(4/9)*x^2 - x^2/4 = 1",
+      "fact_expr": "G: Hyperbola;C:Hyperbola;A: Point;Expression(G) = (x^2/16 - y^2/9 = 1);Coordinate(A) = (2*sqrt(3), -3);PointOnCurve(A, C);Asymptote(C) = Asymptote(G)",
+      "coords": {
+        "A": [
+          3.4641016151377544,
+          -3.0
+        ]
+      }
+    },
+    {
+      "index": 158,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 159,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0159.png",
+      "answer": "sqrt(5)/2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(OneOf(Asymptote(G))) = (y = x/2)",
+      "coords": {}
+    },
+    {
+      "index": 160,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 161,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 162,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 163,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0163.png",
+      "answer": "[sqrt(3),+oo)",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;LeftFocus(C) = F1;RightFocus(C) = F2;F2: Point;F1: Point;PointOnCurve(P,C) = True;P: Point;DotProduct(VectorOf(P,F1),VectorOf(P,F2)) = -2*a^2",
+      "coords": {}
+    },
+    {
+      "index": 164,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 2.23606797749979,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0164.png",
+      "answer": "x^2/9+y^2/5=1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/5 = 1);Vertex(G) = Focus(H);Focus(G) = Vertex(H);H: Ellipse",
+      "coords": {}
+    },
+    {
+      "index": 165,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0165.png",
+      "answer": "5/2",
+      "fact_expr": "C: Ellipse;M: Point;F: Point;O: Point;N: Point;Expression(C) = (x^2/25 + y^2/9 = 1);LeftFocus(C) = F;PointOnCurve(M, C);MidPoint(LineSegmentOf(M, F)) = N;Center(C) = O;LineSegmentOf(O, N) = 4",
+      "coords": {}
+    },
+    {
+      "index": 166,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0166.png",
+      "answer": "{4, sqrt(34)}",
+      "fact_expr": "G: Ellipse;m: Real;m>0;Expression(G) = (x^2/25 + y^2/m^2 = 1);FocalLength(G)=6",
+      "coords": {}
+    },
+    {
+      "index": 167,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.4641016151377544,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 12.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0167.png",
+      "answer": "6",
+      "fact_expr": "G: Parabola;p: Number;H: Ellipse;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (x^2/12 + y^2/3 = 1);Focus(G) = RightFocus(H)",
+      "coords": {}
+    },
+    {
+      "index": 168,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0168.png",
+      "answer": "x^2/72+y^2/36=1",
+      "fact_expr": "M: Ellipse;b: Number;a: Number;G: Line;A: Point;B: Point;F: Point;D: Point;a > b;b > 0;Expression(M) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (-c, 0);Coordinate(D) = (-4, 2);Eccentricity(M) = sqrt(2)/2;LeftFocus(M)=F;PointOnCurve(F, G);Intersection(G, M) = {A, B};IsChordOf(LineSegmentOf(A, B),M);MidPoint(LineSegmentOf(A, B)) = D;c:Number",
+      "coords": {
+        "D": [
+          -4.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 169,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0169.png",
+      "answer": "8/5",
+      "fact_expr": "A: Point;Coordinate(A) = (4, 0);C: Parabola;Expression(C) = (y^2 = 2*p*x);p: Number;0 < p;p < 4;l: Line;Directrix(C) = l;P: Point;PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(P, H), l) = True;FootPoint(LineSegmentOf(P, H), l) = H;H: Point;Abs(LineSegmentOf(P, H)) = Abs(LineSegmentOf(P, A));AngleOf(A, P, H) = ApplyUnit(120, degree)",
+      "coords": {
+        "A": [
+          4.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 170,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0170.png",
+      "answer": "-7*sqrt(15)/17",
+      "fact_expr": "C1: Hyperbola;C2: Circle;A: Point;B: Point;F2: Point;F1: Point;m: Number;m > 0;Expression(C1) = (x^2/9 - y^2/m = 1);LeftFocus(C1) = F1;RightFocus(C1) = F2;Expression(Asymptote(C1)) = (sqrt(7)*x + pm*3*y = 0);PointOnCurve(A, C2);Expression(C2) = (x^2 + y^2 = 16);VectorOf(F1, A) = VectorOf(A, B);PointOnCurve(B, RightPart(C1))",
+      "coords": {}
+    },
+    {
+      "index": 171,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0171.png",
+      "answer": "sqrt(2) / 2",
+      "fact_expr": "F1: Point;F2: Point;C: Hyperbola;Expression(C) = (x^2 - y^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C) = True;AngleOf(F1, P, F2) = ApplyUnit(90, degree)",
+      "coords": {}
+    },
+    {
+      "index": 172,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0172.png",
+      "answer": "y^2=4*x",
+      "fact_expr": "G: Parabola;p: Number;A: Point;B: Point;F: Point;p>0;Expression(G) = (y^2 = 2*p*x);IsChordOf(LineSegmentOf(A,B),G);Focus(G) = F;PointOnCurve(F,LineSegmentOf(A,B));Abs(LineSegmentOf(A, B)) = 8;XCoordinate(MidPoint(LineSegmentOf(A, B))) = 3",
+      "coords": {}
+    },
+    {
+      "index": 173,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 174,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0174.png",
+      "answer": "sqrt(10)/5",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;Coordinate(F1) = (-1, 0);Coordinate(F2) = (1, 0);LeftFocus(C) = F1;RightFocus(C) = F2;G: Line;Expression(G) = (x + y - 2 = 0);A: Point;PointOnCurve(A, G);PointOnCurve(A, C)",
+      "coords": {
+        "F1": [
+          -1.0,
+          0.0
+        ],
+        "F2": [
+          1.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 175,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0175.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;M: Point;x0:Number;Expression(G) = (y^2 = 4*x);Coordinate(M) = (x0, 2*sqrt(3));PointOnCurve(M,G)",
+      "coords": {}
+    },
+    {
+      "index": 176,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0176.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "P: Point;F1: Point;F2: Point;Focus(G) = {F1, F2};G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;PointOnCurve(P, G) = True;IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2)) = True;Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 177,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 178,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0178.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;G: Circle;A1: Point;A2: Point;F: Point;P: Point;a > b;b > 0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;LeftVertex(C) = A1;RightVertex(C) = A2;PointOnCurve(F, l);L1: Line;L2: Line;OneOf(Asymptote(C)) = L1;OneOf(Asymptote(C)) = L2;Negation(L1=L2);IsParallel(l, L1);Intersection(l, L2) = P;IsDiameter(LineSegmentOf(A1, A2), G);PointOnCurve(P, G)",
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+    {
+      "index": 179,
+      "success": true,
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+        "direction": "up"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0179.png",
+      "answer": "y^2/4 - x^2/5 = 1",
+      "fact_expr": "C: Hyperbola;O: Origin;Center(C) = O;G: Parabola;Expression(G) = (y = x^2/12);OneOf(Focus(C)) = Focus(G);Expression(OneOf(Asymptote(C))) = (y = x*(2*sqrt(5)/5))",
+      "coords": {}
+    },
+    {
+      "index": 180,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+      "optimization": {
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+        "converged": true,
+        "iterations": 122
+      },
+      "output_path": "results/dev/hyperbola/problem_0180.png",
+      "answer": "x^2 - y^2/3 = 1",
+      "fact_expr": "H: Parabola;Expression(H) = (y^2 = 4*x);F: Point;Focus(H) = F;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;RightVertex(G) = F;Expression(Asymptote(G)) = (y = pm*sqrt(3)*x)",
+      "coords": {}
+    },
+    {
+      "index": 181,
+      "success": true,
+      "conic_type": "parabola",
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0181.png",
+      "answer": "pm*sqrt(3)",
+      "fact_expr": "G: Parabola;p: Number;H: Line;B: Point;A: Point;F: Point;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};VectorOf(A, F) = 3*VectorOf(F, B)",
+      "coords": {}
+    },
+    {
+      "index": 182,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 183,
+      "success": true,
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+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0183.png",
+      "answer": "(3-sqrt(5))/2",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;B: Point;A: Point;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftVertex(C) = A;UpperVertex(C) = B;LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P,LineSegmentOf(A,B));IsPerpendicular(LineSegmentOf(P,F1),LineSegmentOf(P,F2))",
+      "coords": {}
+    },
+    {
+      "index": 184,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0184.png",
+      "answer": "2",
+      "fact_expr": "G: Point;Coordinate(G) = (2, 3);PointOnCurve(G, C);C: Hyperbola;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;FocalLength(C) = 4",
+      "coords": {
+        "G": [
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+          3.0
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+      }
+    },
+    {
+      "index": 185,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0185.png",
+      "answer": "3",
+      "fact_expr": "C: Parabola;G: Line;F: Point;A: Point;B: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F, G);Slope(G) = sqrt(3);Intersection(G, C) = {A, B};Abs(LineSegmentOf(F, A)) > Abs(LineSegmentOf(F, B))",
+      "coords": {}
+    },
+    {
+      "index": 186,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0186.png",
+      "answer": "1",
+      "fact_expr": "C: Parabola;Expression(C) = (x^2 = 2*y);F: Point;Focus(C) = F;P: Point;y0: Number;Coordinate(P) = (1, y0);PointOnCurve(P, C);A: Point;l: Line;TangentOnPoint(P,C)=l;Intersection(l, yAxis) = A",
+      "coords": {}
+    },
+    {
+      "index": 187,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 188,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0188.png",
+      "answer": "(1/8,-1)",
+      "fact_expr": "P: Point;Coordinate(P) = (4, -1);G: Parabola;Expression(G) = (y^2 = 8*x);F: Point;Focus(G) = F;M: Point;PointOnCurve(M, G);WhenMin(Abs(LineSegmentOf(M, P)) + Abs(LineSegmentOf(M, F)))",
+      "coords": {
+        "P": [
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+          -1.0
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+    },
+    {
+      "index": 189,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0189.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "F: Point;C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;OneOf(Focus(C)) = F;P: Point;PointOnCurve(P, C);MidPoint(LineSegmentOf(P, F)) = OneOf(Endpoint(ImageinaryAxis(C)))",
+      "coords": {}
+    },
+    {
+      "index": 190,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 191,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0191.png",
+      "answer": "11/2",
+      "fact_expr": "C: Parabola;G: Line;M: Point;A: Point;F: Point;B: Point;m>0;Expression(C) = (y^2 = 4*x);Coordinate(M) = (m, 0);Focus(C)=F;PointOnCurve(F, G);Intersection(G, C) = {A, B};VectorOf(A, F) = 2*VectorOf(F, B);Abs(VectorOf(M, F)) = Abs(VectorOf(M, A));m:Number",
+      "coords": {}
+    },
+    {
+      "index": 192,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0192.png",
+      "answer": "15",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);A: Point;P: Point;F1: Point;Coordinate(A) = (-1, 3);LeftFocus(G) = F1;PointOnCurve(P, G)",
+      "coords": {
+        "A": [
+          -1.0,
+          3.0
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+      }
+    },
+    {
+      "index": 193,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 194,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0194.png",
+      "answer": "6",
+      "fact_expr": "G: Parabola;P: Point;A: Point;F: Point;Expression(G) = (y^2 = 8*x);Coordinate(A) = (4, 1);PointOnCurve(P, G);Focus(G) = F",
+      "coords": {
+        "A": [
+          4.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 195,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 196,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0196.png",
+      "answer": "6",
+      "fact_expr": "C: Ellipse;A: Point;B: Point;F: Point;Expression(C) = (x^2/4 + y^2/3 = 1);LeftVertex(C) = A;UpperVertex(C) = B;RightFocus(C) = F",
+      "coords": {}
+    },
+    {
+      "index": 197,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0197.png",
+      "answer": "x^2/5 - y^2 = 1",
+      "fact_expr": "C: Hyperbola;b: Number;A: Point;O: Origin;M: Point;N: Point;b>0;A0: Circle;Expression(C) = (x^2/5 - y^2/b^2 = 1);RightVertex(C) = A;Center(A0) = A;Radius(A0) = b;Intersection(A0, OneOf(Asymptote(C))) = {M, N};VectorOf(O, M) = (3/2)*VectorOf(O, N)",
+      "coords": {}
+    },
+    {
+      "index": 198,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_focus_mismatch",
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0198.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;F: Point;P: Point;A: Point;a > b;b > 0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (0, 2);Coordinate(F) = (-sqrt(5), 0);LeftFocus(G) = F;PointOnCurve(P, RightPart(G));Min(Perimeter(TriangleOf(P, A, F))) = 8",
+      "coords": {
+        "A": [
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+          2.0
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+        "F": [
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+          0.0
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+      },
+      "validation_reasons": [
+        "hyperbola_focus_mismatch"
+      ]
+    },
+    {
+      "index": 199,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 200,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0200.png",
+      "answer": "6",
+      "fact_expr": "G: Parabola;H: Line;B: Point;A: Point;C: Point;D: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(C) = (0, 3);Coordinate(D) = (4, 0);PointOnCurve(C,H);Intersection(H,G)={A,B};PointOnCurve(D,PerpendicularBisector(LineSegmentOf(A,B)));Focus(G)=F",
+      "coords": {
+        "C": [
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+          3.0
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+        "D": [
+          4.0,
+          0.0
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+    },
+    {
+      "index": 201,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0201.png",
+      "answer": "1/2",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;P: Point;F2: Point;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(60, degree)",
+      "coords": {}
+    },
+    {
+      "index": 202,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 203,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0203.png",
+      "answer": "y=pm*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/b^2 = 1);b: Number;e: Number;Eccentricity(G) = e;e = sqrt(2)",
+      "coords": {}
+    },
+    {
+      "index": 204,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 205,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 206,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0206.png",
+      "answer": "4",
+      "fact_expr": "l: Line;Slope(l) > 0;Sin(Inclination(l)) = sqrt(2)/2;Intercept(l, xAxis) = -2;Intersection(l, C) = {A, B};A: Point;B: Point;C: Parabola;Expression(C) = (x^2 = 2*p*y);p: Number;p>0;Abs(LineSegmentOf(A, B)) = 16",
+      "coords": {}
+    },
+    {
+      "index": 207,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.366584062576294,
+        "b": 2.364173412322998,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.00017838670464698225,
+        "converged": true,
+        "iterations": 89
+      },
+      "output_path": "results/dev/hyperbola/problem_0207.png",
+      "answer": "x^2-y^2/3=1",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Ellipse;m: Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2/(m^2 + 4) + y^2/m^2 = 1);Focus(H) = Focus(G);Eccentricity(G) = 2",
+      "coords": {}
+    },
+    {
+      "index": 208,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 209,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0209.png",
+      "answer": "{(1,4)+(4,oo)}",
+      "fact_expr": "G: Ellipse;n: Number;H: Line;k: Real;Expression(G) = (x^2/4 + y^2/n = 1);Expression(H) = (y = k*x + 1);NumIntersection(H, G) = 2",
+      "coords": {}
+    },
+    {
+      "index": 210,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0210.png",
+      "answer": "(1,sqrt(10)/2]",
+      "fact_expr": "Gamma: Ellipse;H: Circle;I: Line;N: Point;M: Point;O:Origin;Expression(I) = (x + y = 1);Expression(Gamma) = (y^2/b^2 + x^2/a^2 = 1);Intersection(I, Gamma) = {M, N};IsDiameter(LineSegmentOf(M,N),H);PointOnCurve(O,H);Negation(Eccentricity(Gamma)>sqrt(3)/2);a:Number;b:Number;a>b;b>0",
+      "coords": {}
+    },
+    {
+      "index": 211,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0211.png",
+      "answer": "2",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;M: Circle;e: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(M) = (y^2 + (x - 2)^2 = 3);IsTangent(Asymptote(C),M);Eccentricity(C) = e",
+      "coords": {}
+    },
+    {
+      "index": 212,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0212.png",
+      "answer": "(pm*sqrt(7), 0)",
+      "fact_expr": "C: Hyperbola;Expression(C) = (3*x^2 - 4*y^2 = 12)",
+      "coords": {}
+    },
+    {
+      "index": 213,
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+      "error": "Unsupported or unparseable expression"
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+    {
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+      "answer": "3",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-x^2/9 + y^2/16 = 1);F: Point;RightFocus(C) = F",
+      "coords": {}
+    },
+    {
+      "index": 215,
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+      "error": "Unsupported or unparseable expression"
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+      "index": 216,
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+      "answer": "4",
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+      },
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+      "answer": "1/4",
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+    {
+      "index": 218,
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+      "output_path": "results/dev/ellipse/problem_0218.png",
+      "answer": "4/5",
+      "fact_expr": "C: Ellipse;A: Point;P: Point;B: Point;Q: Point;M: Point;N: Point;Expression(C) = (x^2/4 + y^2 = 1);LeftVertex(C) = A;RightVertex(C) = B;PointOnCurve(M, xAxis);l:Line;PointOnCurve(M,l);Intersection(l, C) = {P, Q};l1:Line;PointOnCurve(M, l1);IsPerpendicular(l1,LineSegmentOf(A, P));Intersection(l1, LineSegmentOf(B, Q)) = N;IsPerpendicular(l,xAxis)",
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+    },
+    {
+      "index": 219,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 220,
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+      },
+      "output_path": "results/dev/hyperbola/problem_0220.png",
+      "answer": "3",
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+      "coords": {}
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+    {
+      "index": 221,
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+        "direction": "right"
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+      "output_path": "results/dev/parabola/problem_0221.png",
+      "answer": "y^2=8*x",
+      "fact_expr": "H: Line;Slope(H) = 2;G: Parabola;Expression(G) = (y^2 = a*x);a: Number;a>0;F: Point;Focus(G) = F;PointOnCurve(F, H) = True;Intersection(H, yAxis) = A;O: Origin;A: Point;Area(TriangleOf(O, A, F)) = 4",
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+    },
+    {
+      "index": 222,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 223,
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+        "note": "using explicit params"
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+      "answer": "-3",
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+    {
+      "index": 224,
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+      "output_path": "results/dev/hyperbola/problem_0224.png",
+      "answer": "3/2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/5 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 225,
+      "success": true,
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+        "y_coef": 4.0
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+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0225.png",
+      "answer": "[-sqrt(5),sqrt(5)]",
+      "fact_expr": "G: Ellipse;x: Real;Expression(G) = (x^2/5 + y^2/4 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 226,
+      "success": true,
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+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0226.png",
+      "answer": "1/4",
+      "fact_expr": "G: Hyperbola;B: Point;A: Point;Expression(G) = (x^2/4 - y^2 = 1);PointOnCurve(A,Asymptote(G));PointOnCurve(B,Asymptote(G));Coordinate(MidPoint(LineSegmentOf(A,B)))=(2,2)",
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+    },
+    {
+      "index": 227,
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0227.png",
+      "answer": "y^2=-12*x",
+      "fact_expr": "G: Hyperbola;H: Parabola;Expression(G) = (x^2/4 - y^2/5 = 1);LeftFocus(G)=Focus(H)",
+      "coords": {}
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+    {
+      "index": 228,
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+        "b": 4.0,
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+        "note": "using explicit params"
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+      "output_path": "results/dev/hyperbola/problem_0228.png",
+      "answer": "3",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);P: Point;PointOnCurve(P, RightPart(G));F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2",
+      "coords": {}
+    },
+    {
+      "index": 229,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 230,
+      "success": true,
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+        "x_coef": 2.0,
+        "y_coef": 1.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0230.png",
+      "answer": "{(0,-1),(-12/11,-7/11)}",
+      "fact_expr": "Gamma: Ellipse;H: Line;A: Point;C: Point;B: Point;Expression(H) = (x - 3*y + 1 = 0);Expression(Gamma)=(x^2/2+y^2=1);Intersection(H,Gamma)={A,B};PointOnCurve(C,Gamma);AngleOf(A, C, B) = ApplyUnit(90, degree)",
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+    },
+    {
+      "index": 231,
+      "success": true,
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+      "error": null,
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+        "direction": "up"
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+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0231.png",
+      "answer": "{x^2=-8*y,y^2=16*x}",
+      "fact_expr": "G: Parabola;H: Line;O: Origin;Expression(H) = (x - 2*y - 4 = 0);Vertex(G) = O;PointOnCurve(Focus(G), H);SymmetryAxis(G)=axis",
+      "coords": {}
+    },
+    {
+      "index": 232,
+      "success": true,
+      "conic_type": "ellipse",
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+        "y_coef": 5.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0232.png",
+      "answer": "x^2/18-y^2/2=1",
+      "fact_expr": "H: Ellipse;Expression(H) = (x^2/25 + y^2/5 = 1);G: Hyperbola;Focus(H) = Focus(G);I: Point;Coordinate(I) = (-3*sqrt(2), 0);PointOnCurve(I, G)",
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+    {
+      "index": 233,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 234,
+      "success": true,
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+      "error": null,
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+        "b": 1.0,
+        "major_axis": "y",
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+        "y_coef": 4.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0234.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "L: Line;Slope(L) = sqrt(2)/2;G: Ellipse;Expression(G) = (x^2 + y^2/b^2 = 1);b: Number;b>0;PointOnCurve(Focus(G), xAxis);P: Point;Q: Point;Negation(P=Q);Intersection(L, G) = {P, Q};F1: Point;F2: Point;Focus(G) = {F1, F2};Projection(P, xAxis) = F1;Projection(Q, xAxis) = F2",
+      "coords": {}
+    },
+    {
+      "index": 235,
+      "success": true,
+      "conic_type": "ellipse",
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+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0235.png",
+      "answer": "20",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/9 + x^2/a^2 = 1);a: Number;a>3;F2: Point;F1: Point;Focus(G) = {F1,F2};A: Point;B: Point;IsChordOf(LineSegmentOf(A,B),G) = True;Abs(LineSegmentOf(F1,F2)) = 8;PointOnCurve(F1, LineSegmentOf(A, B))",
+      "coords": {}
+    },
+    {
+      "index": 236,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 237,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "converged": true,
+        "iterations": 109
+      },
+      "output_path": "results/dev/hyperbola/problem_0237.png",
+      "answer": "y = pm*(4/3)*x",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = 5/3",
+      "coords": {}
+    },
+    {
+      "index": 238,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 239,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "y_coef": 6.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0239.png",
+      "answer": "3/2",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/8 + y^2/6 = 1);P: Point;Coordinate(P) = (-2, 1);l: Line;PointOnCurve(P, l);A: Point;B: Point;Intersection(l, C) = {A, B};MidPoint(LineSegmentOf(A, B)) = P",
+      "coords": {
+        "P": [
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+          1.0
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+    },
+    {
+      "index": 240,
+      "success": true,
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+      "error": null,
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+        "b": 1.7320508075688772,
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+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0240.png",
+      "answer": "5/7",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;F1: Point;F2: Point;Focus(C) = {F1, F2};P: Point;PointOnCurve(P, C);Abs(LineSegmentOf(P, F2)) = Abs(LineSegmentOf(F1, F2));Distance(F2, LineOf(P, F1)) = Length(MinorAxis(C))",
+      "coords": {}
+    },
+    {
+      "index": 241,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0241.png",
+      "answer": "20",
+      "fact_expr": "G: Ellipse;H: Line;A: Point;F2: Point;B: Point;F1: Point;Expression(G) = (x^2/25 + y^2/16 = 1);LeftFocus(G)=F1;PointOnCurve(F1, H);Intersection(H, G)={A,B};RightFocus(G) = F2",
+      "coords": {}
+    },
+    {
+      "index": 242,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 243,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 2.23606797749979,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 5.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0243.png",
+      "answer": "5",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/5 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 244,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0244.png",
+      "answer": "5",
+      "fact_expr": "C: Ellipse;a: Number;P: Point;F1: Point;F2: Point;a>0;Expression(C) = (y^2/16 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);Perimeter(TriangleOf(P, F1, F2)) = 16",
+      "coords": {}
+    },
+    {
+      "index": 245,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0245.png",
+      "answer": "sqrt(3)/3",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;M: Point;Intersection(Directrix(G), xAxis) = M;P: Point;PointOnCurve(P, G);Q: Point;MidPoint(LineSegmentOf(P, F)) = Q",
+      "coords": {}
+    },
+    {
+      "index": 246,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0246.png",
+      "answer": "x^2/4+y^2=1",
+      "fact_expr": "G: Parabola;H: Ellipse;b: Number;a: Number;Expression(G) = (y^2 = 4*(sqrt(3)*x));a>0;b>0;Expression(H) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(OneOf(Focus(H)), Directrix(G));Length(MajorAxis(H)) = 2*Length(MinorAxis(H))",
+      "coords": {}
+    },
+    {
+      "index": 247,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 248,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0248.png",
+      "answer": "x^2/16+y^2/4=1",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Eccentricity(G) = sqrt(3)/2;H: Line;Slope(H) = 1;M: Point;Coordinate(M) = (b, 0);PointOnCurve(M, H);A: Point;B: Point;Intersection(H, G) = {A, B};O: Origin;DotProduct(VectorOf(O, A), VectorOf(O, B)) = 32/(5*Tan(AngleOf(A, O, B)))",
+      "coords": {}
+    },
+    {
+      "index": 249,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 250,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0250.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;p: Number;H: Circle;p > 0;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (-6*x + x^2 + y^2 - 7 = 0);IsTangent(H, Directrix(G))",
+      "coords": {}
+    },
+    {
+      "index": 251,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0251.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "l: Line;G: Parabola;p: Number;A: Point;B: Point;C: Point;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;PointOnCurve(F, l);Slope(l) > 0;Intersection(l, G) = {A, B};Quadrant(A) = 1;Intersection(l,Directrix(G)) = C;Abs(LineSegmentOf(B, C)) = 3*Abs(LineSegmentOf(B, F))",
+      "coords": {}
+    },
+    {
+      "index": 252,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0252.png",
+      "answer": "y = x - 1",
+      "fact_expr": "l: Line;C: Parabola;A: Point;B: Point;F: Point;k: Number;k > 0;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F, l);Slope(l) = k;Intersection(l, C) = {A, B};LineSegmentOf(A, B) = 8",
+      "coords": {}
+    },
+    {
+      "index": 253,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0253.png",
+      "answer": "6",
+      "fact_expr": "C: Parabola;G: Circle;M: Point;A: Point;B: Point;F: Point;a: Number;b: Number;Expression(C) = (y^2 = 4*x);Coordinate(M) = (a, b);PointOnCurve(M, C);Focus(C) = F;Distance(M, F) = 5;Center(G) = M;PointOnCurve(F, G);Intersection(G, yAxis) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 254,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0254.png",
+      "answer": "4",
+      "fact_expr": "H: Ellipse;Expression(H) = (x^2/5 + (16*y^2)/p^2 = 1);p: Number;G: Parabola;Expression(G) = (y^2 = 2*p*x);PointOnCurve(LeftFocus(H), Directrix(G)) = True",
+      "coords": {}
+    },
+    {
+      "index": 255,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0255.png",
+      "answer": "[sqrt(10), +oo)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/b^2 = 1);b: Number;b >= 9",
+      "coords": {}
+    },
+    {
+      "index": 256,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0256.png",
+      "answer": "9*sqrt(3)/4",
+      "fact_expr": "C: Parabola;A: Point;F: Point;O: Origin;N: Point;l: Line;Expression(C) = (y^2 = 6*x);Focus(C) = F;Directrix(C) = l;PointOnCurve(A, C);Distance(A, l) = 6;Intersection(PerpendicularBisector(LineSegmentOf(A, F)), l) = N",
+      "coords": {}
+    },
+    {
+      "index": 257,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.23606797749979,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 5.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0257.png",
+      "answer": "-10",
+      "fact_expr": "l: Line;C: Ellipse;F: Point;M: Point;B: Point;A: Point;Expression(C) = (x^2/5 + y^2 = 1);RightFocus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};Intersection(l, yAxis) =M;PointOnCurve(B, LineSegmentOf(F,M))",
+      "coords": {}
+    },
+    {
+      "index": 258,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          0.0,
+          0.0
+        ],
+        "radius": 6.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/circle/problem_0258.png",
+      "answer": "((x-2)^2 + y^2 = 4)&Negation(x = 4)",
+      "fact_expr": "L: Line;O: Circle;M: Point;N: Point;G: Point;Expression(O) = (x^2 + y^2 = 36);Expression(L) = (y = k*(x - 4));Intersection(L, O) = {M, N};MidPoint(LineSegmentOf(M, N)) = G",
+      "coords": {}
+    },
+    {
+      "index": 259,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0259.png",
+      "answer": "[2, 2*sqrt(2))",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/2 + y^2 = 1);F1: Point;F2: Point;Focus(C) = {F1, F2};P: Point;x0: Number;y0: Number;Coordinate(P) = (x0, y0);0<x0^2/2+y0^2;x0^2/2+y0^2<1",
+      "coords": {}
+    },
+    {
+      "index": 260,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 6.928203230275509,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0260.png",
+      "answer": "6",
+      "fact_expr": "G: Hyperbola;H: Circle;E: Circle;P: Point;M: Point;N: Point;Expression(G) = (x^2 - y^2/48 = 1);Expression(H) = (y^2 + (x + 7)^2 = 9);Expression(E) = ((x - 7)^2 + y^2 = 1);PointOnCurve(P, RightPart(G));PointOnCurve(M, H);PointOnCurve(N, E)",
+      "coords": {}
+    },
+    {
+      "index": 261,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 2.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0261.png",
+      "answer": "2*sqrt(5)/5",
+      "fact_expr": "G: Hyperbola;H: Line;a: Number;Expression(G) = (x^2 - y^2/4 = 1);Expression(H) = (a*x + y + 2 = 0);l1: Line;OneOf(Asymptote(G)) = l1;IsParallel(H,l1)",
+      "coords": {}
+    },
+    {
+      "index": 262,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 10.0,
+        "b": 6.0,
+        "major_axis": "x",
+        "x_coef": 100.0,
+        "y_coef": 36.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0262.png",
+      "answer": "12",
+      "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/100 + y^2/36 = 1);PointOnCurve(P, G);Distance(P,RightDirectrix(G)) = 10",
+      "coords": {}
+    },
+    {
+      "index": 263,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0263.png",
+      "answer": "8",
+      "fact_expr": "F: Point;Focus(C) = F;C: Parabola;Expression(C) = (y^2 = 2*p*x);p: Number;p>0;G: Line;PointOnCurve(F, G);Inclination(G) = ApplyUnit(60, degree);Intersection(G, C) = {A, B};A: Point;B: Point;Abs(LineSegmentOf(A, F)) - Abs(LineSegmentOf(B, F)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 264,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0264.png",
+      "answer": "40",
+      "fact_expr": "C: Parabola;p: Number;G: Line;p>0;Expression(C) = (y^2 = 2*(p*x));Expression(G) = (x + 2*y - 2 = 0);PointOnCurve(Focus(C), G)",
+      "coords": {}
+    },
+    {
+      "index": 265,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 266,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 267,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0267.png",
+      "answer": "x+2*y-3=0",
+      "fact_expr": "G: Ellipse;H: LineSegment;P: Point;Expression(G) = (x^2/4 + y^2/2 = 1);Coordinate(P) = (1, 1);IsChordOf(H, G);PointOnCurve(P, H);MidPoint(H) = P",
+      "coords": {
+        "P": [
+          1.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 268,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 269,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0269.png",
+      "answer": "x^2+y^2=4",
+      "fact_expr": "G: Parabola;H: Circle;Expression(G) = (y^2 = 4*x);Center(H) = Vertex(G);Radius(H) = Distance(Focus(G), Directrix(G))",
+      "coords": {}
+    },
+    {
+      "index": 270,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0270.png",
+      "answer": "x^2/4-y^2=1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;l: Line;Expression(l) = (x - 2*y - sqrt(5) = 0);IsParallel(OneOf(Asymptote(G)), l) = True;PointOnCurve(OneOf(Focus(G)), l)",
+      "coords": {}
+    },
+    {
+      "index": 271,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0271.png",
+      "answer": "sqrt(6)/2",
+      "fact_expr": "Gamma: Hyperbola;Expression(Gamma) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;M: Point;Coordinate(M) = (1, 1);H: Line;Slope(H) = 1/2;PointOnCurve(M, H);A: Point;B: Point;Intersection(H, Gamma) = {A, B};MidPoint(LineSegmentOf(A, B)) = M",
+      "coords": {
+        "M": [
+          1.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 272,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0272.png",
+      "answer": "sqrt(10)/2",
+      "fact_expr": "E: Hyperbola;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(E) = F1;RightFocus(E) = F2;O: Origin;A: Point;PointOnCurve(A, E);AngleOf(F1, A, F2) = ApplyUnit(120, degree);Abs(LineSegmentOf(O, A)) = b",
+      "coords": {}
+    },
+    {
+      "index": 273,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 274,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0274.png",
+      "answer": "8",
+      "fact_expr": "G: Parabola;F: Point;A: Point;P: Point;l: Line;Expression(G) = (y^2 = 8*x);Focus(G) = F;Directrix(G) = l;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, A), l);FootPoint(LineSegmentOf(P, A), l) = A;Slope(LineOf(A, F)) = -sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 275,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0275.png",
+      "answer": "sqrt(7)/2",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;k: Number;A: Point;F1: Point;B: Point;F2: Point;a > b;b > 0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y = k*x);LeftFocus(C) = F1;RightFocus(C) = F2;Intersection(G, C) = {A, B};Abs(LineSegmentOf(A, F1)) = 3*Abs(LineSegmentOf(B, F1));AngleOf(F1, A, F2) = ApplyUnit(60, degree)",
+      "coords": {}
+    },
+    {
+      "index": 276,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 277,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0277.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;A: Point;B: Point;O: Origin;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(A, G);PointOnCurve(B, G);PointOnCurve(P, G);Negation(A = B);Negation(A = P);Negation(B = P);PointOnCurve(O, LineSegmentOf(A, B));kPA: Number;kPB: Number;Slope(LineOf(P, A)) = kPA;Slope(LineOf(P, B)) = kPB;kPA*kPB = 3",
+      "coords": {}
+    },
+    {
+      "index": 278,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0278.png",
+      "answer": "(3/2,17/8)",
+      "fact_expr": "M: Point;Coordinate(M) = (3/2, -1);l:Line;C:Parabola;Expression(C)=(x^2=4*y);A: Point;B: Point;PointOnCurve(Focus(C),l);Intersection(l,C)={A,B};IsTangent(LineSegmentOf(A,M),C)",
+      "coords": {
+        "M": [
+          1.5,
+          -1.0
+        ]
+      }
+    },
+    {
+      "index": 279,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0279.png",
+      "answer": "-3",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2=4*x);H: Point;Coordinate(H) = (1, 0);L: Line;PointOnCurve(H,L) = True;Intersection(L,G) = {P,Q};P: Point;Coordinate(P) = (x1, y1);x1: Number;y1: Number;Q: Point;Coordinate(Q) = (x2,y2);x2: Number;y2: Number",
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+          0.0
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+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0280.png",
+      "answer": "3",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);C: Circle;Expression(C) = ((x + 3)^2 + (y - 3)^2 = 1);P: Point;Q: Point;R: Point;PointOnCurve(P, G);PointOnCurve(Q, C);Projection(P, yAxis) = R",
+      "coords": {}
+    },
+    {
+      "index": 281,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 282,
+      "success": true,
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0282.png",
+      "answer": "1",
+      "fact_expr": "C: Ellipse;M: Circle;r: Number;P: Point;A: Point;B: Point;Expression(C) = (x^2/8 + y^2/2 = 1);0 < r;r < sqrt(2);Expression(M) = (-r^2 - 4*sqrt(2)*y + 2*(sqrt(2)*x) + x^2 + y^2 + 10 = 0);UpperVertex(C)=P;L1:Line;L2:Line;TangentOfPoint(P,M)={L1,L2};Intersection(L1,C)=A;Intersection(L2,C)=B;Negation(A=P);Negation(B=P)",
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+    },
+    {
+      "index": 283,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 284,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 2.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0284.png",
+      "answer": "y=pm*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 4)",
+      "coords": {}
+    },
+    {
+      "index": 285,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0285.png",
+      "answer": "x - y - 3 = 0",
+      "fact_expr": "G: Hyperbola;H: LineSegment;P: Point;Expression(G) = (x^2 - 4*y^2 = 4);Coordinate(P) = (4, 1);MidPoint(H) = P;IsChordOf(H, G)",
+      "coords": {
+        "P": [
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+          1.0
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+    },
+    {
+      "index": 286,
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+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 287,
+      "success": true,
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+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0287.png",
+      "answer": "x^2/9+y^2/8=1",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;C1: Circle;Expression(C1) = (x^2 + y^2 = 9);NumIntersection(C,C1) = 2;C2: Circle;Expression(C2) = (x^2 + y^2 = 8);NumIntersection(C,C2) = 2",
+      "coords": {}
+    },
+    {
+      "index": 288,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 289,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0289.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;M: Circle;A: Point;B: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(M) = (y^2 + (x - 3)^2 = 8);Intersection(OneOf(Asymptote(C)),M) = {A, B};Abs(LineSegmentOf(A, B)) = 2*sqrt(2)",
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+    },
+    {
+      "index": 290,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0290.png",
+      "answer": "y^2=16*x",
+      "fact_expr": "G: Hyperbola;H: Parabola;Expression(G) = (x^2/16 - y^2/9 = 1);RightVertex(G)=Focus(H)",
+      "coords": {}
+    },
+    {
+      "index": 291,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0291.png",
+      "answer": "10;y = pm*(4/3)*x;4",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 292,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0292.png",
+      "answer": "(x^2+y^2=1)&(Negation(y=0))",
+      "fact_expr": "P: Point;PointOnCurve(P, RightPart(G));G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;Q: Point;lambda: Number;DotProduct(VectorOf(Q, F2), (VectorOf(P, F1)/Abs(VectorOf(P, F1)) + VectorOf(P, F2)/Abs(VectorOf(P, F2)))) = 0;VectorOf(Q, P) + lambda*(VectorOf(P, F1)/Abs(VectorOf(P, F1)) + VectorOf(P, F2)/Abs(VectorOf(P, F2))) = 0",
+      "coords": {}
+    },
+    {
+      "index": 293,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0293.png",
+      "answer": "1/2",
+      "fact_expr": "G: Parabola;H: Line;A: Point;Q: Point;B: Point;P: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (-1, 0);Coordinate(B) = (1, 0);PointOnCurve(A, H);Intersection(H, G) = {P, Q};MidPoint(LineSegmentOf(A, Q)) = P",
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+          0.0
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+        "B": [
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+          0.0
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+    {
+      "index": 294,
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0294.png",
+      "answer": "[2, 2*sqrt(2)]",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/2 + y^2 = 1);F1: Point;F2: Point;Focus(C) = {F1, F2};P: Point;x0: Number;y0: Number;Coordinate(P) = (x0, y0);0 < x0^2/2+y0^2;x0^2/2+y0^2 < 1",
+      "coords": {}
+    },
+    {
+      "index": 295,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 296,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 297,
+      "success": true,
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+        "x_coef": 16.0,
+        "y_coef": 7.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0297.png",
+      "answer": "6",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/7 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0",
+      "coords": {}
+    },
+    {
+      "index": 298,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0298.png",
+      "answer": "(1,2]",
+      "fact_expr": "G: Hyperbola;a: Number;b: Number;P: Point;F1: Point;F2: Point;e: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G)=F2;PointOnCurve(P, RightPart(G));Abs(LineSegmentOf(P, F1)) = 3*Abs(LineSegmentOf(P, F2));Eccentricity(G) = e",
+      "coords": {}
+    },
+    {
+      "index": 299,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0299.png",
+      "answer": "y^2 - x^2 / 3 = 1",
+      "fact_expr": "G: Hyperbola;a: Number;Expression(G) = (y^2 + x^2/a = 1);Expression(OneOf(Asymptote(G))) =  (x + sqrt(3)*y = 0)",
+      "coords": {}
+    },
+    {
+      "index": 300,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 3.1622776601683795,
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0300.png",
+      "answer": "y=pm*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 10)",
+      "coords": {}
+    },
+    {
+      "index": 301,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 302,
+      "success": false,
+      "error": "Circle from diameter: missing point coordinates"
+    },
+    {
+      "index": 303,
+      "success": true,
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+      "error": null,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "results/dev/ellipse/problem_0303.png",
+      "answer": "1/3",
+      "fact_expr": "G: Ellipse;a: Number;b: Number;c: Number;F1: Point;F2: Point;P: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);Focus(G) = {F1, F2};PointOnCurve(P, G);Area(TriangleOf(P, F1, F2)) = b^2*(sqrt(2)/2)",
+      "coords": {}
+    },
+    {
+      "index": 304,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.0,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0304.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "G: Hyperbola;H: Line;P: Point;Q: Point;Expression(G) = (x^2 - y^2 = 1);PointOnCurve(OneOf(Focus(G)), H);IsPerpendicular(H, xAxis);L1:Line;L2:Line;Asymptote(G)={L1,L2};Intersection(H,L1)=P;Intersection(H,L2)=Q",
+      "coords": {}
+    },
+    {
+      "index": 305,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0305.png",
+      "answer": "5/2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;H: Line;PointOnCurve(F, H);A: Point;B: Point;Intersection(H, G) = {A, B};O: Origin;Abs(LineSegmentOf(A, F)) = 5",
+      "coords": {}
+    },
+    {
+      "index": 306,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 2.364173412322998,
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+      "optimization": {
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+        "converged": true,
+        "iterations": 89
+      },
+      "output_path": "results/dev/hyperbola/problem_0306.png",
+      "answer": "(pm*2,0)",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Length(RealAxis(C))=2;Eccentricity(C) = 2",
+      "coords": {}
+    },
+    {
+      "index": 307,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0307.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;A: Point;B: Point;PointOnCurve(A, G);PointOnCurve(B, G);AngleOf(A, F, B) = pi/2;IsChordOf(LineSegmentOf(A, B), G);M: Point;MidPoint(LineSegmentOf(A, B)) = M;N: Point;Projection(M, Directrix(G)) = N",
+      "coords": {}
+    },
+    {
+      "index": 308,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "x_coef": 3.0,
+        "y_coef": 2.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0308.png",
+      "answer": "4*sqrt(3)",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/3 + y^2/2 = 1);l: Line;P: Point;Q: Point;F2: Point;F1: Point;Expression(l) = (x - 2*y + 1 = 0);LeftFocus(C) = F1;RightFocus(C) = F2;Intersection(l, C) = {P, Q}",
+      "coords": {}
+    },
+    {
+      "index": 309,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 8.0,
+        "y_coef": 4.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0309.png",
+      "answer": "[0, sqrt(2)]",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/8 + y^2/4 = 1);P: Point;PointOnCurve(P, G);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;O: Origin",
+      "coords": {}
+    },
+    {
+      "index": 310,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0310.png",
+      "answer": "1",
+      "fact_expr": "l: Line;G: Ellipse;P: Point;Q: Point;A: Point;B: Point;O: Origin;m: Number;n: Number;Expression(G) = (x^2/4 + y^2/3 = 1);Coordinate(P) = (1, 1);Coordinate(Q) = (m, n);PointOnCurve(P, l);Intersection(l, G) = {A, B};lambda: Number;VectorOf(A, P) = lambda*VectorOf(P, B);VectorOf(A, Q) = -lambda*VectorOf(Q, B)",
+      "coords": {
+        "P": [
+          1.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 311,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0311.png",
+      "answer": "9",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);P: Point;PointOnCurve(P, G);Distance(P, xAxis) = 8",
+      "coords": {}
+    },
+    {
+      "index": 312,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 313,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 314,
+      "success": true,
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+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0314.png",
+      "answer": "2/a",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G) = F2",
+      "coords": {}
+    },
+    {
+      "index": 315,
+      "success": true,
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+        "y_coef": 3.0
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+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0315.png",
+      "answer": "3*x-4*y+7=0",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);P: Point;Coordinate(P) = (-1, 1);H: LineSegment;IsChordOf(H, G);MidPoint(H) = P",
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+          1.0
+        ]
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+    {
+      "index": 316,
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+        "y_coef": 9.0
+      },
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+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0316.png",
+      "answer": "16",
+      "fact_expr": "G: Ellipse;H: Line;M: Point;N: Point;F2: Point;F1: Point;Expression(G) = (x^2/16 + y^2/9 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, H);Intersection(H, G) = {M, N}",
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+    },
+    {
+      "index": 317,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 318,
+      "success": true,
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+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0318.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;F2: Point;A: Point;F1: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = -(b/a)*x);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(A, RightPart(G));IsParallel(LineOf(A,F2),H);Perimeter(TriangleOf(A, F1, F2)) = 9*a",
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+    },
+    {
+      "index": 319,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 320,
+      "success": true,
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+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0320.png",
+      "answer": "8*sqrt(3)/3",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/8 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 321,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 2.0,
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0321.png",
+      "answer": "8*sqrt(3)/3",
+      "fact_expr": "G: Hyperbola;H: Line;F: Point;P: Point;Q: Point;Expression(G) = (x^2 - y^2 = 4);RightFocus(G)=F;PointOnCurve(F,H);Inclination(H)=ApplyUnit(105,degree);Intersection(H,G)={P,Q}",
+      "coords": {}
+    },
+    {
+      "index": 322,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 323,
+      "success": true,
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0323.png",
+      "answer": "12/13",
+      "fact_expr": "C: Hyperbola;P: Point;F2: Point;F1: Point;Expression(C) = (x^2/16 - y^2/20 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);LineSegmentOf(P, F2) = 5",
+      "coords": {}
+    },
+    {
+      "index": 324,
+      "success": true,
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0324.png",
+      "answer": "64",
+      "fact_expr": "G: Parabola;H: Line;A: Point;O: Origin;B: Point;F: Point;Expression(G) = (y^2 = 16*x);Focus(G) = F;PointOnCurve(F, H);Inclination(H) = ApplyUnit(30, degree);Intersection(H, G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 325,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 326,
+      "success": true,
+      "conic_type": "ellipse",
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+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0326.png",
+      "answer": "1/3",
+      "fact_expr": "H: Circle;Expression(H) = (y^2 + (x - 2)^2 = 1);G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;PointOnCurve(OneOf(Vertex(G)), H);PointOnCurve(OneOf(Focus(G)), H);e: Number;Eccentricity(G) = e",
+      "coords": {}
+    },
+    {
+      "index": 327,
+      "success": true,
+      "conic_type": "parabola",
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0327.png",
+      "answer": "x=-4",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;G: Line;Expression(G) = (x + 2*y - 4 = 0);PointOnCurve(Focus(C), G)",
+      "coords": {}
+    },
+    {
+      "index": 328,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 2.070901870727539,
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+        "converged": true,
+        "iterations": 106
+      },
+      "output_path": "results/dev/hyperbola/problem_0328.png",
+      "answer": "x^2/3-y^2/6=1",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;Focus(C) = {F1, F2};Eccentricity(C) = sqrt(3);P: Point;PointOnCurve(P, C);Abs(LineSegmentOf(P, F1)) - Abs(LineSegmentOf(P, F2)) = 2*sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 329,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0329.png",
+      "answer": "4*sqrt(7)/7",
+      "fact_expr": "G: Hyperbola;a: Number;H: Parabola;O: Origin;Expression(G) = (-y^2/9 + x^2/a^2 = 1);Expression(H) = (y^2 = 16*x);RightFocus(G) = Focus(H);Center(G) = O",
+      "coords": {}
+    },
+    {
+      "index": 330,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
+      },
+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0330.png",
+      "answer": "5",
+      "fact_expr": "P: Point;G: Parabola;Expression(G) = (y^2=4*x);PointOnCurve(P,G) = True;F: Point;Focus(G) = F;A: Point;Coordinate(A) = (4,a);a: Number;Abs(a) < 4",
+      "coords": {}
+    },
+    {
+      "index": 331,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 332,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0332.png",
+      "answer": "4*sqrt(2)",
+      "fact_expr": "l: Line;C: Parabola;A: Point;O: Origin;B: Point;k1: Number;k2: Number;Expression(C) = (y^2 = 4*x);Intersection(C, l) = {A, B};Slope(LineOf(O, A)) = k1;Slope(LineOf(O, B)) = k2;k1*k2 = -2",
+      "coords": {}
+    },
+    {
+      "index": 333,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "down"
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0333.png",
+      "answer": "(0,-2)",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = -8*y)",
+      "coords": {}
+    },
+    {
+      "index": 334,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 335,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0335.png",
+      "answer": "(4, 4)",
+      "fact_expr": "G: Parabola;M: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(M, G);Distance(M, Focus(G)) = 5;Quadrant(M)=1",
+      "coords": {}
+    },
+    {
+      "index": 336,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0336.png",
+      "answer": "(x - 1)^2 + (y + pm*2*sqrt(2))^2 = 9",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 8*x);A: Point;PointOnCurve(A, C) ;G: Circle;Center(G) = A;PointOnCurve(Vertex(C), G) ;PointOnCurve(Focus(C), G)",
+      "coords": {}
+    },
+    {
+      "index": 337,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0337.png",
+      "answer": "5/3\n4",
+      "fact_expr": "G: Hyperbola;e: Number;Expression(G) = (x^2/9 - y^2/16 = 1);Eccentricity(G) = e",
+      "coords": {}
+    },
+    {
+      "index": 338,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0338.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;A: Point;B: Point;F: Point;a > 0;b > 0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftVertex(G) = A;Slope(H) = 1;Intersection(H, RightPart(G)) = B;Projection(B, xAxis) = F;RightFocus(G) = F;PointOnCurve(A, H)",
+      "coords": {}
+    },
+    {
+      "index": 339,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0339.png",
+      "answer": "5",
+      "fact_expr": "G: Parabola;A: Point;Expression(G) = (x^2 = 4*y);Coordinate(A) = (m, 4);PointOnCurve(A,G);m:Number",
+      "coords": {}
+    },
+    {
+      "index": 340,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+      "optimization": {
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+        "converged": true,
+        "iterations": 89
+      },
+      "output_path": "results/dev/hyperbola/problem_0340.png",
+      "answer": "2*sqrt(3)/3",
+      "fact_expr": "C1: Hyperbola;b: Number;a: Number;C2: Hyperbola;a>0;b>0;Expression(C1) = (x^2/a^2 - y^2/b^2 = 1);Expression(C2) = (y^2/b^2 - x^2/a^2 = 1);Eccentricity(C1) = 2",
+      "coords": {}
+    },
+    {
+      "index": 341,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 4.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0341.png",
+      "answer": "8",
+      "fact_expr": "G: Parabola;B: Point;A: Point;P:Point;Expression(G) = (y = x^2/16);PointOnCurve(A,G);PointOnCurve(B,G);Abs(LineSegmentOf(A, B)) = 24;MidPoint(LineSegmentOf(A,B))=P;WhenMin(Distance(P,xAxis))",
+      "coords": {}
+    },
+    {
+      "index": 342,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0342.png",
+      "answer": "-1/3",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/2 + y^2 = 1);l: Line;PointOnCurve(OneOf(Focus(G)), l);Inclination(l) = ApplyUnit(45, degree);A: Point;B: Point;Intersection(l, G) = {A, B};O: Origin",
+      "coords": {}
+    },
+    {
+      "index": 343,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "y",
+        "x_coef": 1.0,
+        "y_coef": 4.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0343.png",
+      "answer": "7",
+      "fact_expr": "C: Circle;Expression(C) = (y^2 + (x - 4)^2 = 4);G: Ellipse;Expression(G) = (x^2 + y^2/4 = 1);P: Point;Q: Point;PointOnCurve(P,C) = True;PointOnCurve(Q,G) = True",
+      "coords": {}
+    },
+    {
+      "index": 344,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0344.png",
+      "answer": "{y^2=8*x, y^2=-16*x}",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = m*x);m: Number;H: Line;Expression(H) = (x = 1);Distance(Directrix(G), H) = 3",
+      "coords": {}
+    },
+    {
+      "index": 345,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0345.png",
+      "answer": "12",
+      "fact_expr": "G: Hyperbola;m: Number;m>0;Expression(G) = (-y^2/3 + x^2/m = 1);Expression(OneOf(Asymptote(G))) = (y = x/2)",
+      "coords": {}
+    },
+    {
+      "index": 346,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 347,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.4641016151377544,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 12.0,
+        "y_coef": 9.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0347.png",
+      "answer": "3*sqrt(3)",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/12 + y^2/9 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C) = True;AngleOf(F1, P, F2) = ApplyUnit(60, degree)",
+      "coords": {}
+    },
+    {
+      "index": 348,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0348.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(G) = F;L: Line;PointOnCurve(F,L)=True;IsPerpendicular(Asymptote(G),L)=True;FootPoint(Asymptote(G),L) = P;P: Point;PointOnCurve(MidPoint(LineSegmentOf(P,F)),G) = True",
+      "coords": {}
+    },
+    {
+      "index": 349,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 350,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0350.png",
+      "answer": "sqrt(10)/5",
+      "fact_expr": "l: Line;C: Parabola;E:Circle;E1: Point;O: Origin;A: Point;B: Point;Expression(C) = (y^2 = 4*x);Coordinate(E1) = (1, 1);Slope(l)=1/3;Intersection(l,C)={A,B};Center(E)=E1;InscribedCircle(TriangleOf(O,A,B))=E",
+      "coords": {
+        "E1": [
+          1.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 351,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0351.png",
+      "answer": "x+2*y-2=0",
+      "fact_expr": "P: Point;Coordinate(P) = (1, 1);H: LineSegment;G: Ellipse;Expression(G) = (x^2/4 + y^2/2 = 1);IsChordOf(H, G) = True;MidPoint(H) = P",
+      "coords": {
+        "P": [
+          1.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 352,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0352.png",
+      "answer": "2*sqrt(3)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 353,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0353.png",
+      "answer": "y=pm*x",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;O: Origin;P: Point;F2: Point;F1: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P,C);DotProduct((VectorOf(O,P)+VectorOf(O,F2)),VectorOf(F2,P))=0;Abs(VectorOf(P, F1))*Abs(VectorOf(P, F2)) = 2*a^2",
+      "coords": {}
+    },
+    {
+      "index": 354,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 355,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0355.png",
+      "answer": "pm*(1/2)",
+      "fact_expr": "G: Line;Expression(G) = (y = k * (x + 1));C: Parabola;Expression(C) = (y^2 = 4*x);A: Point;Coordinate(A) = (x1, y1);x1: Number;y1: Number;B: Point;Coordinate(B) = (x2, y2);x2: Number;y2: Number;Intersection(G, C) = {A, B};Focus(C) = F;F: Point;Abs(LineSegmentOf(F, A)) + Abs(LineSegmentOf(F, B)) = 16;k: Number",
+      "coords": {}
+    },
+    {
+      "index": 356,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0356.png",
+      "answer": "pm*sqrt(15)",
+      "fact_expr": "l: Line;C: Parabola;p: Number;Q: Point;F: Point;P: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = P;Intersection(l, Directrix(C)) = Q;VectorOf(Q, F) = 3*VectorOf(F, P)",
+      "coords": {}
+    },
+    {
+      "index": 357,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0357.png",
+      "answer": "8/3",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/25 + y^2/16 = 1);PointOnCurve(P, G);Focus(G) = {F1, F2};Radius(InscribedCircle(TriangleOf(P, F1, F2))) = 1;Quadrant(P) = 1",
+      "coords": {}
+    },
+    {
+      "index": 358,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0358.png",
+      "answer": "2",
+      "fact_expr": "C: Parabola;G: Line;A: Point;F: Point;B: Point;k:Number;Expression(C) = (y^2 = 4*x);k>0;Expression(G) = (y = k*(x - 2));Focus(C) = F;Intersection(G, C) = {A, B};Abs(LineSegmentOf(A, F))/Abs(LineSegmentOf(B, F)) = 2/5;Negation(A=B)",
+      "coords": {}
+    },
+    {
+      "index": 359,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0359.png",
+      "answer": "2",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;LeftFocus(C) = F;H: Line;PointOnCurve(F, H);Z1: Line;Z2: Line;Asymptote(C) = {Z1, Z2};Q: Point;Intersection(H, Z1) = Q;P: Point;Intersection(H, Z2) = P;2*VectorOf(F, Q) = VectorOf(F, P);O: Origin;G: Circle;Center(G) = O;IsTangent(G, LineOf(F, P));TangentPoint(G, LineOf(F, P)) = Q",
+      "coords": {}
+    },
+    {
+      "index": 360,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 361,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0361.png",
+      "answer": "y=3*x-2",
+      "fact_expr": "G: Curve;Expression(G) = (y = x^2);H: Point;Coordinate(H) = (1, 1)",
+      "coords": {
+        "H": [
+          1.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 362,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0362.png",
+      "answer": "9/2",
+      "fact_expr": "G: Parabola;A: Point;P: Point;M: Point;Expression(G) = (y^2 = 2*x);Coordinate(A) = (7/2, 4);PointOnCurve(P, G);Projection(P, yAxis) = M",
+      "coords": {
+        "A": [
+          3.5,
+          4.0
+        ]
+      }
+    },
+    {
+      "index": 363,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 6.0,
+        "y_coef": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0363.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;p: Number;H: Ellipse;p>0;Expression(G) = (y^2 = 2*p*x);Expression(H) = (x^2/6 + y^2/2 = 1);Focus(G) = RightFocus(H)",
+      "coords": {}
+    },
+    {
+      "index": 364,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.7320508075688772,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 3.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0364.png",
+      "answer": "3*sqrt(2)/2",
+      "fact_expr": "l: Line;C: Curve;P: Point;Expression(C) = (x^2/3 + y^2 = 1);Expression(l) = (x + y + 1 = 0);PointOnCurve(P, C)",
+      "coords": {}
+    },
+    {
+      "index": 365,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0365.png",
+      "answer": "(1, 3*sqrt(2)/4]",
+      "fact_expr": "E: Hyperbola;a: Number;b: Number;C: Parabola;A: Point;P: Point;F: Point;a>0;b>0;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);Expression(C) = (y^2 = 8*(a*x));RightVertex(E) = A;Focus(C) = F;PointOnCurve(P, Asymptote(E));IsPerpendicular(VectorOf(A,P),VectorOf(F,P))",
+      "coords": {}
+    },
+    {
+      "index": 366,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.8284271247461903,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 8.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0366.png",
+      "answer": "y^2=-8*x",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/8 + y^2/4 = 1);P: Point;Distance(P, LeftFocus(G)) = Distance(P, H);H: Line;Expression(H) = (x = 2)",
+      "coords": {}
+    },
+    {
+      "index": 367,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0367.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;A: Point;B: Point;O: Origin;a > b;b > 0;LeftVertex(G)=A;RightVertex(G)=B;P:Point;PointOnCurve(P, G);Negation(P=A);Negation(P=B);Slope(LineOf(P, A)) *Slope(LineOf(P, B)) = -1/2",
+      "coords": {}
+    },
+    {
+      "index": 368,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 369,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0369.png",
+      "answer": "sqrt(55)",
+      "fact_expr": "G: Parabola;M: Point;P0: Point;Expression(G) = (y^2 = 10*x);Coordinate(M) = (8, 0);PointOnCurve(P0, G)",
+      "coords": {
+        "M": [
+          8.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 370,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0370.png",
+      "answer": "y^2=20*x",
+      "fact_expr": "H: Parabola;Expression(H) = (y^2 = 2*m*x);m: Number;m>0;G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1);Distance(Focus(H),OneOf(Asymptote(G))) = 3",
+      "coords": {}
+    },
+    {
+      "index": 371,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0371.png",
+      "answer": "2",
+      "fact_expr": "C: Parabola;G: Line;M: Point;A: Point;B: Point;k:Number;Expression(C) = (y^2 = 8*x);Coordinate(M) = (-2, 2);PointOnCurve(Focus(C), G);Slope(G) = k;Intersection(G, C) = {A, B};DotProduct(VectorOf(M, A), VectorOf(M, B)) = 0",
+      "coords": {
+        "M": [
+          -2.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 372,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.06247315928339958,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0014067133888602257,
+        "converged": true,
+        "iterations": 338
+      },
+      "output_path": "results/dev/parabola/problem_0372.png",
+      "answer": "y=-1/16",
+      "fact_expr": "C: Parabola;G: Point;O: Origin;Coordinate(G) = (-1, 4);Vertex(C) = O;PointOnCurve(Focus(C), yAxis);PointOnCurve(G,C)",
+      "coords": {
+        "G": [
+          -1.0,
+          4.0
+        ]
+      }
+    },
+    {
+      "index": 373,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0373.png",
+      "answer": "(1,sqrt(7)-1)",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;P: Point;PointOnCurve(P,RightPart(C)) = True;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;M: Circle;IsInscribedCircle(M,TriangleOf(P,F1,F2)) = True;A: Point;TangentPoint(LineSegmentOf(P, F1), M) = A;Abs(LineSegmentOf(P,A)) = c^2/2*a;c: Number;HalfFocalLength(C) = c;Abs(LineSegmentOf(P, F1))/Abs(LineSegmentOf(P, F2)) > 2",
+      "coords": {}
+    },
+    {
+      "index": 374,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0374.png",
+      "answer": "y^2=4*x",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(C) = F;H: Line;PointOnCurve(F, H);A: Point;B: Point;Intersection(H, C) = {A, B};G: Circle;IsDiameter(LineSegmentOf(A, B), G);M: Point;Coordinate(M) = (-p/2, 3);TangentPoint(Directrix(C), G) = M;O: Origin;Area(TriangleOf(A, O, B)) = sqrt(13)",
+      "coords": {}
+    },
+    {
+      "index": 375,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0375.png",
+      "answer": "1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1);H: Parabola;Expression(H) = (y^2 = 2*p*x);p: Number;A: Point;F: Point;B: Point;p>0;Focus(H)=F;l:Line;PointOnCurve(F,l);IsParallel(l,OneOf(Asymptote(G)));Intersection(l,H)={A,B};Abs(LineSegmentOf(A, F)) > Abs(LineSegmentOf(B, F));Abs(LineSegmentOf(A, F)) = 2",
+      "coords": {}
+    },
+    {
+      "index": 376,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0376.png",
+      "answer": "(sqrt(5)+1)/2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;L: Line;A: Point;B: Point;F: Point;O:Origin;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F;PointOnCurve(F, L);IsPerpendicular(L,xAxis);Intersection(L, G) = {A, B};IsDiameter(LineSegmentOf(A,B),H);PointOnCurve(O,H)",
+      "coords": {}
+    },
+    {
+      "index": 377,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+      },
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+      "answer": "5/2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = x);A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G) = True;Distance(MidPoint(LineSegmentOf(A, B)), yAxis) = 1",
+      "coords": {}
+    },
+    {
+      "index": 378,
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0378.png",
+      "answer": "sqrt(5)/2",
+      "fact_expr": "C: Hyperbola;b: Number;G: Parabola;p: Number;M: Point;A: Point;b>0;Expression(C) = (x^2 - y^2/b^2 = 1);p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(M) = (1, y0);y0:Number;PointOnCurve(M, G);Distance(M, Focus(G)) = 5;LeftVertex(C) = A;IsPerpendicular(OneOf(Asymptote(C)),LineOf(A,M))",
+      "coords": {}
+    },
+    {
+      "index": 379,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 380,
+      "success": false,
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+      "error": "validation: hyperbola_focus_mismatch",
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0380.png",
+      "answer": "6",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;F: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (5, 0);RightFocus(C)=F;Distance(F,Asymptote(C))=4",
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+        "F": [
+          5.0,
+          0.0
+        ]
+      },
+      "validation_reasons": [
+        "hyperbola_focus_mismatch"
+      ]
+    },
+    {
+      "index": 381,
+      "success": true,
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+        "direction": "up"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0381.png",
+      "answer": "x^2+(y-1)^2=10",
+      "fact_expr": "C: Circle;G: Parabola;Expression(G) = (x^2 = 4*y);Center(C) = Focus(G);H: Line;Expression(H) = (4*x - 3*y - 2 = 0);A: Point;B: Point;Intersection(H, C) = {A, B};Abs(LineSegmentOf(A, B)) = 6",
+      "coords": {}
+    },
+    {
+      "index": 382,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 2.5275303393962245,
+        "orientation": "horizontal"
+      },
+      "optimization": {
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+        "converged": true,
+        "iterations": 95
+      },
+      "output_path": "results/dev/hyperbola/problem_0382.png",
+      "answer": "x^2 - y^2/4 = 1",
+      "fact_expr": "C: Hyperbola;l: Line;OneOf(Asymptote(C)) = l;Expression(l) = (y = 2*x);Length(RealAxis(C)) < 4",
+      "coords": {}
+    },
+    {
+      "index": 383,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0383.png",
+      "answer": "sqrt(5)-1",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C) = True;IsDiameter(LineSegmentOf(F1, F2), G) = True;G: Circle;Intersection(G, Asymptote(C)) = Q;Q: Point;Quadrant(P) = 1;Quadrant(Q) = 1;MidPoint(LineSegmentOf(Q, F2)) = P",
+      "coords": {}
+    },
+    {
+      "index": 384,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 2.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0384.png",
+      "answer": "y^2 = 12*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/5 - y^2/4 = 1);RightFocus(G) = Focus(H);H: Parabola",
+      "coords": {}
+    },
+    {
+      "index": 385,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0385.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;D: Point;OneOf(Vertex(ImageinaryAxis(C))) = D;G: Line;Expression(G) = (x = 2*a);A: Point;B: Point;Intersection(G, C) = {A, B};PointOnCurve(Orthocenter(TriangleOf(A, B, D)), OneOf(Asymptote(C)))",
+      "coords": {}
+    },
+    {
+      "index": 386,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0386.png",
+      "answer": "-1",
+      "fact_expr": "C1: Hyperbola;Expression(C1) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;l: Line;LeftDirectrix(C1) = l;F1: Point;F2: Point;LeftFocus(C1) = F1;RightFocus(C1) = F2;C2: Parabola;Directrix(C2) = l;Focus(C2) = F2;M: Point;Intersection(C1, C2) = M",
+      "coords": {}
+    },
+    {
+      "index": 387,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 6.0,
+        "b": 6.708203932499369,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0387.png",
+      "answer": "y=pm*x*sqrt(5)/2\n3/2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/36 - y^2/45 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 388,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0388.png",
+      "answer": "sqrt(10)/5",
+      "fact_expr": "C: Ellipse;a: Number;b: Number;G: Line;A: Point;F1: Point;B: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, G);Intersection(G, C) = {A, B};Abs(LineSegmentOf(A, F1)) = 3*Abs(LineSegmentOf(B, F1));Abs(LineSegmentOf(A, B)) = Abs(LineSegmentOf(B, F2))",
+      "coords": {}
+    },
+    {
+      "index": 389,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0389.png",
+      "answer": "x+y-1=0",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);L: Line;PointOnCurve(Focus(G), L);H: Circle;Expression(H) = (2*y - 4*x + x^2 + y^2 = 0);WhenMax(Length(InterceptChord(L, H)))",
+      "coords": {}
+    },
+    {
+      "index": 390,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 8.0,
+        "b": 6.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0390.png",
+      "answer": "32/5",
+      "fact_expr": "G: Hyperbola;P: Point;Expression(G) = (x^2/64 - y^2/36 = 1);PointOnCurve(P, G);Distance(P,RightFocus(G)) = 8",
+      "coords": {}
+    },
+    {
+      "index": 391,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 392,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0392.png",
+      "answer": "{1, 3}",
+      "fact_expr": "L: Line;Expression(L) = (y = -t + x);t: Number;C: Parabola;Expression(C) = (y^2 = 4*x);A: Point;B: Point;Intersection(L, C) = {A, B};Negation(A=B);O: Origin;DotProduct(VectorOf(O, A), VectorOf(O, B)) = -3",
+      "coords": {}
+    },
+    {
+      "index": 393,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 394,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0394.png",
+      "answer": "{(x^2/8+y^2/6=1),(y^2/25/3+x^2/25/4=1)}",
+      "fact_expr": "G1: Ellipse;Expression(G1) = (x^2/4 + y^2/3 = 1);G2: Ellipse;Eccentricity(G1) = Eccentricity(G2);H: Point;Coordinate(H) = (2, -sqrt(3));PointOnCurve(H,G2) = True",
+      "coords": {
+        "H": [
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+          -1.7320508075688772
+        ]
+      }
+    },
+    {
+      "index": 395,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 396,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0396.png",
+      "answer": "(4*sqrt(2),4*sqrt(6))",
+      "fact_expr": "l: Line;G: Parabola;O: Point;A: Point;P: Point;Q: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (2, 0);Vertex(G)=O;Inclination(l)=pi/4;PointOnCurve(O,l);PointOnCurve(A,l);IsIntersect(LineSegmentOf(O,A),l);Intersection(l,LineSegmentOf(O,A))={P,Q}",
+      "coords": {
+        "A": [
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+          0.0
+        ]
+      }
+    },
+    {
+      "index": 397,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0397.png",
+      "answer": "3*sqrt(5)+13",
+      "fact_expr": "F: Point;C: Hyperbola;Expression(C) = (x^2/16 - y^2/9 = 1);RightFocus(C) = F;P: Point;PointOnCurve(P, LeftPart(C)) = True;A: Point;Coordinate(A) = (1, 3)",
+      "coords": {
+        "A": [
+          1.0,
+          3.0
+        ]
+      }
+    },
+    {
+      "index": 398,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0398.png",
+      "answer": "2*sqrt(3)",
+      "fact_expr": "l: Line;C: Parabola;G: Line;B: Point;A: Point;O: Origin;Expression(C) = (y^2 = x);Expression(G) = (y = 1);Intersection(l, C) = {A, B};PointOnCurve(MidPoint(LineSegmentOf(A, B)), G);DotProduct(VectorOf(O, A), VectorOf(O, B)) = 2",
+      "coords": {}
+    },
+    {
+      "index": 399,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0399.png",
+      "answer": "(-1/2, pm*3/8)",
+      "fact_expr": "H: Parabola;Expression(H) = (y^2 = 2*x);G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 400,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0400.png",
+      "answer": "17/4",
+      "fact_expr": "A: Point;Coordinate(A) = (4, 0);B: Point;Coordinate(B) = (2, 2);M: Point;PointOnCurve(M, G) = True;G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1)",
+      "coords": {
+        "A": [
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+          0.0
+        ],
+        "B": [
+          2.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 401,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0401.png",
+      "answer": "sqrt(7)/2",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2/4 - y^2/b^2 = 1);b: Number;b>0;F1: Point;F2: Point;Focus(C) = {F1, F2};P: Point;x0: Number;y0: Number;Coordinate(P) = (x0, y0);Quadrant(P) = 1;PointOnCurve(P, C);l: Line;TangentOnPoint(P, C) = l;Distance(F1, l)*Distance(F2, l) = 3",
+      "coords": {}
+    },
+    {
+      "index": 402,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 3.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0402.png",
+      "answer": "y=-3",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 12*y)",
+      "coords": {}
+    },
+    {
+      "index": 403,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.7320508075688772,
+        "b": 1.4142135623730951,
+        "major_axis": "y",
+        "x_coef": 2.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0403.png",
+      "answer": "(0, pm*1)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/2 + y^2/3 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 404,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0404.png",
+      "answer": "(1,3]",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F2: Point;F1: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P,LeftPart(G));Abs(LineSegmentOf(P,F2))^2/Abs(LineSegmentOf(P,F1))=8*a",
+      "coords": {}
+    },
+    {
+      "index": 405,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0405.png",
+      "answer": "[0,b^4/c^2]",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;c:Number;M: Point;O: Origin;x0:Number;y0:Number;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(M) = (x0, y0);PointOnCurve(M, C);HalfFocalLength(C) = c;Abs(LineSegmentOf(M, O)) <= c",
+      "coords": {}
+    },
+    {
+      "index": 406,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0406.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;A: Point;B: Point;C: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightVertex(G)=A;Slope(H)=-1;PointOnCurve(A,H);L1:Line;L2:Line;Asymptote(G)={L1,L2};Intersection(H,L1)=B;Intersection(H,L2)=C;VectorOf(A, B) = VectorOf(B, C)/2",
+      "coords": {}
+    },
+    {
+      "index": 407,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 408,
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+      "error": "Unsupported or unparseable expression"
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+        "note": "using explicit params"
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+      "answer": "pi/2",
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+      "answer": "sqrt(5)/5",
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+        "note": "using explicit params"
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+      "answer": "2-sqrt(3)",
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+    {
+      "index": 418,
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+        "note": "using explicit params"
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+      "output_path": "results/dev/parabola/problem_0418.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;A: Point;B: Point;Expression(G) = (y^2 = 4*x);IsChordOf(LineSegmentOf(A,B),G);IsPerpendicular(LineSegmentOf(A,B),xAxis);Abs(LineSegmentOf(A, B)) = 4*sqrt(3)",
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+    {
+      "index": 419,
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+        "note": "using explicit params"
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+      "output_path": "results/dev/hyperbola/problem_0419.png",
+      "answer": "3",
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+        "converged": true
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+      "output_path": "results/dev/parabola/problem_0420.png",
+      "answer": "25/4",
+      "fact_expr": "l: Line;G: Parabola;p: Number;B: Point;A: Point;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(F) = (1, 0);Slope(l)=4/3;Focus(G) = F;PointOnCurve(F,l);Intersection(l, G) = {A, B}",
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+    {
+      "index": 421,
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+        "b": 1.5,
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0421.png",
+      "answer": "y=pm*(sqrt(10)/4)*x",
+      "fact_expr": "H: Parabola;Expression(H) = (y^2 = 2*p*x);p: Number;p>0;F: Point;Focus(H) = F;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;RightFocus(G) = F;I: Point;Coordinate(I) = ((3*a^2)/p, (2*b^2)/p);PointOnCurve(I, G)",
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+    {
+      "index": 422,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 423,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 424,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 425,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 426,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_focus_mismatch",
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+        "b": 1.5,
+        "orientation": "vertical"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "results/dev/hyperbola/problem_0426.png",
+      "answer": "16",
+      "fact_expr": "m: Number;F: Point;Coordinate(F) = (0, 5);G: Hyperbola;Expression(G) = (-x^2/9 + y^2/m = 1);OneOf(Focus(G)) = F",
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+    {
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
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+      "output_path": "results/dev/circle/problem_0427.png",
+      "answer": "x^2/25+y^2/16=1",
+      "fact_expr": "M: Circle;A: Point;Coordinate(A) = (-3, 0);PointOnCurve(A, M);Expression(B)=((x-3)^2+y^2=100);M1:Point;Center(M)=M1;IsInTangent(M,B);B:Circle",
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+    {
+      "index": 428,
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+        "note": "using explicit params"
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+      "output_path": "results/dev/parabola/problem_0428.png",
+      "answer": "5",
+      "fact_expr": "G: Parabola;M: Point;Expression(G) = (y^2 = 4*x);Coordinate(M) = (4, y0);y0:Number;PointOnCurve(M, G);Focus(G)=F;F: Point",
+      "coords": {}
+    },
+    {
+      "index": 429,
+      "success": true,
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "results/dev/parabola/problem_0429.png",
+      "answer": "(1,2)",
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+          2.0
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+    {
+      "index": 430,
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+        "b": 2.0,
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+        "x_coef": 9.0,
+        "y_coef": 4.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0430.png",
+      "answer": "x^2/3-y^2/2=1",
+      "fact_expr": "G: Hyperbola;H: Ellipse;A: Point;Expression(H) = (x^2/9 + y^2/4 = 1);Coordinate(A) = (3,-2);PointOnCurve(A, G);Focus(G)=Focus(H)",
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+    {
+      "index": 431,
+      "success": true,
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+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "results/dev/hyperbola/problem_0431.png",
+      "answer": "1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/3 + x^2/a^2 = 1);a: Number;a>0;H: Parabola;Expression(H) = (y^2 = 8*x);OneOf(Focus(G)) = Focus(H)",
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+    },
+    {
+      "index": 432,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 3.4641016151377544,
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0432.png",
+      "answer": "4 + 3*sqrt(5)",
+      "fact_expr": "G: Hyperbola;A: Point;P: Point;F: Point;Expression(G) = (x^2/4 - y^2/12 = 1);Coordinate(A) = (1, 6);LeftFocus(G) = F;PointOnCurve(P, RightPart(G))",
+      "coords": {
+        "A": [
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+          6.0
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+    {
+      "index": 433,
+      "success": true,
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+      "error": null,
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+        "b": 4.0,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0433.png",
+      "answer": "y=pm*2*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/16 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 434,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "results/dev/hyperbola/problem_0434.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;P: Point;PointOnCurve(P, RightPart(G));F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;I: Point;Incenter(TriangleOf(P, F1, F2)) = I;TriangleOf(I, P, F1) = (1/2)*TriangleOf(I, F1, F2) + TriangleOf(I, P, F2)",
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+    },
+    {
+      "index": 435,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 436,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "iterations": 89
+      },
+      "output_path": "results/dev/hyperbola/problem_0436.png",
+      "answer": "3/16",
+      "fact_expr": "G: Hyperbola;m: Number;n: Number;H: Parabola;Negation(m*n=0);Expression(G) = (-y^2/n + x^2/m = 1);Expression(H) = (y^2 = 4*x);Eccentricity(G) = 2;OneOf(Focus(G)) = Focus(H)",
+      "coords": {}
+    },
+    {
+      "index": 437,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 438,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0438.png",
+      "answer": "12*sqrt(2)/7",
+      "fact_expr": "G: Ellipse;H: Line;k: Number;F: Point;A: Point;B: Point;Expression(G) = (x^2/4 + y^2/3 = 1);Expression(H) = (y = k*x - 1);LeftFocus(G) = F;Intersection(H, G) = {A, B};WhenMax(Perimeter(TriangleOf(F,A,B)))",
+      "coords": {}
+    },
+    {
+      "index": 439,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "vertical"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0439.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/3 + y^2/4 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 440,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 441,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          -3.0,
+          0.0
+        ],
+        "radius": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/circle/problem_0441.png",
+      "answer": "x^2/25+y^2/16=1",
+      "fact_expr": "G: Circle;IsOutTangent(G,Q1) = True;Q1: Circle;Expression(Q1) = ((x+3)^2+y^2=1);Q2: Circle;Expression(Q2) = ((x-3)^2+y^2=81);IsInTangent(G,Q2) = True",
+      "coords": {}
+    },
+    {
+      "index": 442,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 443,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0443.png",
+      "answer": "pm*(4/3)",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, C) = {A, B};Abs(LineSegmentOf(A, F)) = 4*Abs(LineSegmentOf(B, F))",
+      "coords": {}
+    },
+    {
+      "index": 444,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0444.png",
+      "answer": "sqrt(6)/3",
+      "fact_expr": "C: Ellipse;A: Point;P: Point;Expression(C) = (x^2/4 + y^2 = 1);Coordinate(A) = (1, 0);PointOnCurve(P,C)",
+      "coords": {
+        "A": [
+          1.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 445,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0445.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;F: Point;A: Point;B: Point;e: Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(Focus(G),xAxis);L1: Line;L2: Line;Asymptote(G) = {L1, L2};Intersection(RightDirectrix(G), L1) = A;Intersection(RightDirectrix(G), L2) = B;RightFocus(G) = F;DotProduct(VectorOf(F, A), VectorOf(F, B)) = 0;Eccentricity(G) = e",
+      "coords": {}
+    },
+    {
+      "index": 446,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0446.png",
+      "answer": "{2, (2*sqrt(15))/5}",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1);F: Point;RightFocus(G) = F;H: Line;PointOnCurve(F, H);Slope(H)>0;A: Point;B: Point;Intersection(H, Asymptote(G)) = {A, B};O: Origin;Area(TriangleOf(O, A, B)) = 16*sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 447,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0447.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;c: Number;F: Point;RightFocus(C) = F;Coordinate(F) = (c, 0);P: Point;Q: Point;PointOnCurve(P, C);PointOnCurve(Q, C);M: Point;Coordinate(M) = (-c/2, 0);Distance(M, LineOf(F, P)) = c/2;Incenter(TriangleOf(P, Q, F)) = M;e: Number;Eccentricity(C) = e",
+      "coords": {}
+    },
+    {
+      "index": 448,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0448.png",
+      "answer": "[(sqrt(17)-3)/4, 1)",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C);e: Number;Eccentricity(C) = e;LineSegmentOf(P, F1) = (2*e)*LineSegmentOf(P, F2)",
+      "coords": {}
+    },
+    {
+      "index": 449,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 450,
+      "success": false,
+      "error": "Circle from diameter: missing point coordinates"
+    },
+    {
+      "index": 451,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0451.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;P: Point;F2: Point;F1: Point;Expression(G) = (x^2 - y^2/3 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);Sin(AngleOf(P, F2, F1))/Sin(AngleOf(P, F1, F2)) = e;e:Number;Eccentricity(G)=e",
+      "coords": {}
+    },
+    {
+      "index": 452,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0452.png",
+      "answer": "pm*sqrt(2)",
+      "fact_expr": "G: Hyperbola;H: Circle;a: Number;Expression(G) = (x^2 - y^2 = 1);Expression(H) = (y^2 + (-a + x)^2 = 1);IsTangent(H,Asymptote(G))",
+      "coords": {}
+    },
+    {
+      "index": 453,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0453.png",
+      "answer": "4*sqrt(7)/7",
+      "fact_expr": "G: Ellipse;H: Line;A: Point;B: Point;F2: Point;F1: Point;Expression(G) = (x^2/16 + y^2/9 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, H);Intersection(H, G) = {A,B}",
+      "coords": {}
+    },
+    {
+      "index": 454,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 455,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0455.png",
+      "answer": "8",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);H: Line;Slope(H) = 1;PointOnCurve(Focus(G), H);A: Point;B: Point;Intersection(H, G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 456,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0456.png",
+      "answer": "5",
+      "fact_expr": "G: Parabola;H: Line;A: Point;P: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (x = -1);Coordinate(A) = (0, -1);PointOnCurve(P, G)",
+      "coords": {
+        "A": [
+          0.0,
+          -1.0
+        ]
+      }
+    },
+    {
+      "index": 457,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0457.png",
+      "answer": "{(x-1)^2+(y-2)^2=4,(x-1)^2+(y+2)^2=4}",
+      "fact_expr": "G: Parabola;C: Circle;Expression(G) = (y^2 = 4*x);PointOnCurve(Center(C),G);IsTangent(C,xAxis);IsTangent(C,Directrix(G))",
+      "coords": {}
+    },
+    {
+      "index": 458,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0458.png",
+      "answer": "4*a+2*m",
+      "fact_expr": "G: Hyperbola;a: Number;b: Number;A: Point;B: Point;F1:Point;F2: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F2;Focus(G)={F1,F2};Abs(LineSegmentOf(A, B)) = m;IsChordOf(LineSegmentOf(A, B),LeftPart(G));PointOnCurve(F1,LineSegmentOf(A, B));m:Number",
+      "coords": {}
+    },
+    {
+      "index": 459,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0459.png",
+      "answer": "(2*sqrt(5))/5",
+      "fact_expr": "H: Line;Expression(H) = (x + 2*y - 2 = 0);G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;PointOnCurve(OneOf(Focus(G)), H);PointOnCurve(OneOf(Vertex(G)), H)",
+      "coords": {}
+    },
+    {
+      "index": 460,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0460.png",
+      "answer": "{14, 2}",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/4 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 8",
+      "coords": {}
+    },
+    {
+      "index": 461,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0461.png",
+      "answer": "10",
+      "fact_expr": "l: Line;Slope(l) = 2;G: Parabola;Expression(G) = (y^2 = 8*x);F: Point;Focus(G) = F;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 462,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0462.png",
+      "answer": "16/5",
+      "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 - y^2/16 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0",
+      "coords": {}
+    },
+    {
+      "index": 463,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.7320508075688772,
+        "b": 1.4142135623730951,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0463.png",
+      "answer": "(pm*sqrt(5), 0)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2/2 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 464,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0464.png",
+      "answer": "x=-3/2",
+      "fact_expr": "l: Line;C: Parabola;P: Number;P>0;Expression(l)=(y = x + 3/2);Expression(C) = (y^2 = 2*(P*x));NumIntersection(l,C)=1",
+      "coords": {}
+    },
+    {
+      "index": 465,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.0,
+        "major_axis": "y",
+        "x_coef": 4.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0465.png",
+      "answer": "sqrt(5)/3",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/9 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 466,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0466.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;H: Circle;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(Focus(G),H);PointOnCurve(Endpoint(MinorAxis(G)),H)",
+      "coords": {}
+    },
+    {
+      "index": 467,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0467.png",
+      "answer": "2*sqrt(5)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;p: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);p>0;Expression(H) = (y^2 = 2*(p*x));Distance(LeftVertex(G), Focus(H)) = 4;Coordinate(Intersection(OneOf(Asymptote(G)),Directrix(H)))=(-2,-1)",
+      "coords": {}
+    },
+    {
+      "index": 468,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.015625,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0468.png",
+      "answer": "(0, 1/64)",
+      "fact_expr": "G: Parabola;Expression(G) = (y = 16*x^2)",
+      "coords": {}
+    },
+    {
+      "index": 469,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.7320508075688772,
+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 3.0,
+        "y_coef": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0469.png",
+      "answer": "4*sqrt(3)",
+      "fact_expr": "C: Ellipse;P: Point;Q: Point;F2: Point;Expression(C) = (x^2/3 + y^2/2 = 1);RightFocus(C) = F2;PointOnCurve(P, C);PointOnCurve(Q, C);LineSegmentOf(P, Q) = 2*sqrt(2);WhenMax(LineSegmentOf(P,F2)+LineSegmentOf(Q,F2))",
+      "coords": {}
+    },
+    {
+      "index": 470,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0470.png",
+      "answer": "16/3",
+      "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, H);Inclination(H) = pi/3;Intersection(H, G) = {A, B};IsChordOf(LineSegmentOf(A,B),G)",
+      "coords": {}
+    },
+    {
+      "index": 471,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0471.png",
+      "answer": "8",
+      "fact_expr": "l: Line;C: Parabola;M: Point;A: Point;B: Point;F: Point;N: Point;Expression(C) = (y^2 = 4*x);Coordinate(M) = (-1, 0);PointOnCurve(M, l);Intersection(l, C) = {A, B};Between(A, M, B);Focus(C) = F;VectorOf(N, F) = 6*VectorOf(A, F)",
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+        "M": [
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+          0.0
+        ]
+      }
+    },
+    {
+      "index": 472,
+      "success": true,
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+        "b": 6.0,
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+        "y_coef": 36.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0472.png",
+      "answer": "14",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/100 + y^2/36 = 1);PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G)=F2;Abs(LineSegmentOf(P, F1)) = 6",
+      "coords": {}
+    },
+    {
+      "index": 473,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 474,
+      "success": true,
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+      "error": null,
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+        "b": 1.5,
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0474.png",
+      "answer": "y=pm*(4/3)*x",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, RightPart(G));Abs(LineSegmentOf(P, F1)) - Abs(LineSegmentOf(P, F2)) = (3/5)*Abs(LineSegmentOf(F1, F2))",
+      "coords": {}
+    },
+    {
+      "index": 475,
+      "success": true,
+      "conic_type": "ellipse",
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+        "b": 2.0,
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+        "y_coef": 4.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0475.png",
+      "answer": "16/3",
+      "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/9 + y^2/4 = 1);PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G) = F2;AngleOf(F1, P, F2) = ApplyUnit(60, degree)",
+      "coords": {}
+    },
+    {
+      "index": 476,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0476.png",
+      "answer": "3",
+      "fact_expr": "C: Parabola;F: Point;P: Point;Q: Point;l: Line;Expression(C) = (y^2 = 8*x);Focus(C) = F;Directrix(C) = l;PointOnCurve(P, l);OneOf(Intersection(LineOf(P,F),C))=Q;VectorOf(F, P) = 4*VectorOf(F, Q)",
+      "coords": {}
+    },
+    {
+      "index": 477,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 478,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 479,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0479.png",
+      "answer": "4/5",
+      "fact_expr": "l: Line;G: Hyperbola;Q: Point;P: Point;A: Point;Expression(G) = (x^2/5 - y^2/4 = 1);Coordinate(A) = (1, 1);Intersection(l, G) = {P, Q};MidPoint(LineSegmentOf(P,Q))=A",
+      "coords": {
+        "A": [
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+          1.0
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+      }
+    },
+    {
+      "index": 480,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 2.434081792831421,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "iterations": 124
+      },
+      "output_path": "results/dev/hyperbola/problem_0480.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(OneOf(Asymptote(G)))=(y=2*x)",
+      "coords": {}
+    },
+    {
+      "index": 481,
+      "success": true,
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+      "error": null,
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+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0481.png",
+      "answer": "16",
+      "fact_expr": "m: Line;E: Parabola;A: Point;M: Point;F: Point;N: Point;B: Point;l: Line;Expression(E) = (y^2 = 12*x);Focus(E) = F;Directrix(E) = l;PointOnCurve(F,m);Intersection(m, E) = {A, B};PointOnCurve(A,LineSegmentOf(A,M));IsPerpendicular(LineSegmentOf(A,M),l);FootPoint(LineSegmentOf(A,M),l)=M;MidPoint(LineSegmentOf(A, M)) = N;IsPerpendicular(LineSegmentOf(A, M), LineSegmentOf(F, N))",
+      "coords": {}
+    },
+    {
+      "index": 482,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          2.0,
+          0.0
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+        "radius": 6.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/circle/problem_0482.png",
+      "answer": "x^2/9+y^2/5=1",
+      "fact_expr": "G: Circle;A: Point;N: Point;M: Point;P: Point;Expression(G) = (y^2 + (x + 2)^2 = 36);Coordinate(N) = (2, 0);Center(G)=M;PointOnCurve(A, G);Intersection(PerpendicularBisector(LineSegmentOf(A,N)), LineSegmentOf(M, A)) = P",
+      "coords": {}
+    },
+    {
+      "index": 483,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 4.898979485566356,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0483.png",
+      "answer": "24",
+      "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2 - y^2/24 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);3*LineSegmentOf(P, F1) = 4*LineSegmentOf(P, F2)",
+      "coords": {}
+    },
+    {
+      "index": 484,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 2.5275303393962245,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "iterations": 95
+      },
+      "output_path": "results/dev/hyperbola/problem_0484.png",
+      "answer": "x^2/2 - y^2/8 = 1",
+      "fact_expr": "C: Hyperbola;Expression(OneOf(Asymptote(C))) = (y = 2*x)",
+      "coords": {}
+    },
+    {
+      "index": 485,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 2.23606797749979,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 5.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0485.png",
+      "answer": "x^2-y^2=2",
+      "fact_expr": "H: Ellipse;Expression(H) = (x^2/9 + y^2/5 = 1);G: Hyperbola;Focus(H) = Focus(G);L1: Line;L2: Line;Asymptote(G) = {L1, L2};IsPerpendicular(L1, L2)",
+      "coords": {}
+    },
+    {
+      "index": 486,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 487,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0487.png",
+      "answer": "9",
+      "fact_expr": "G: Hyperbola;H: Circle;P: Point;M: Point;N: Point;Expression(G) = (x^2/9-y^2/16 = 1);Expression(C) = (y^2 + (x + 5)^2 = 4);PointOnCurve(P,RightPart(G));PointOnCurve(M,C);PointOnCurve(N,H);C:Circle;Expression(H)=(y^2 + (x-5)^2 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 488,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0488.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) =F1;RightFocus(C)=F2;PointOnCurve(P, C);Abs(VectorOf(P, F1) + VectorOf(P, F2)) = sqrt(Abs(VectorOf(P, F1))^2 + Abs(VectorOf(P, F2))^2);Abs(VectorOf(P, F1)) = 2*Abs(VectorOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 489,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 490,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "major_axis": "x",
+        "x_coef": 8.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0490.png",
+      "answer": "y=-x+3",
+      "fact_expr": "G: Ellipse;H: LineSegment;Expression(G) = (x^2/8 + y^2/4 = 1);Coordinate(MidPoint(H)) = (2, 1);IsChordOf(H,G)",
+      "coords": {}
+    },
+    {
+      "index": 491,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 9.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0491.png",
+      "answer": "16",
+      "fact_expr": "G: Ellipse;H: Line;A: Point;F2: Point;B: Point;F1: Point;Expression(G) = (x^2/16 + y^2/9 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, H);Intersection(H, G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 492,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0492.png",
+      "answer": "x-y+1=0",
+      "fact_expr": "G: Parabola;P: Point;A: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (-1, 0);PointOnCurve(P, G);Quadrant(P)=1;Focus(G) = F;WhenMin(Abs(LineSegmentOf(P, F))/Abs(LineSegmentOf(P, A)))",
+      "coords": {
+        "A": [
+          -1.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 493,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 494,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0494.png",
+      "answer": "4",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, C) = {A, B};Abs(LineSegmentOf(A, F)) = 4/3",
+      "coords": {}
+    },
+    {
+      "index": 495,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 496,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 2.070901870727539,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 1.4661259228887502e-06,
+        "converged": true,
+        "iterations": 106
+      },
+      "output_path": "results/dev/hyperbola/problem_0496.png",
+      "answer": "y=pm*sqrt(2)*x",
+      "fact_expr": "G: Hyperbola;a: Number;a>0;Expression(G) = (-y^2/2 + x^2/a^2 = 1);Eccentricity(G) = sqrt(3)*a",
+      "coords": {}
+    },
+    {
+      "index": 497,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0497.png",
+      "answer": "1/2",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 2*y);F: Point;Focus(G) = F;l: Line;Directrix(G) = l",
+      "coords": {}
+    },
+    {
+      "index": 498,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 499,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0499.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 4*a);a: Real;Negation(a=0);H: Ellipse;Expression(H) = (x^2/16 + y^2/9 = 1);RightFocus(G) = OneOf(Vertex(H))",
+      "coords": {}
+    },
+    {
+      "index": 500,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 501,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 502,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0502.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;p: Number;H: Circle;p>0;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (-6*x + x^2 + y^2 - 7 = 0);IsTangent(H, Directrix(G))",
+      "coords": {}
+    },
+    {
+      "index": 503,
+      "success": false,
+      "conic_type": "parabola",
+      "error": "validation: parabola_point_off_curve",
+      "params": {
+        "p": 0.9927864074707031,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.00015326803259085864,
+        "converged": true,
+        "iterations": 44
+      },
+      "output_path": "results/dev/parabola/problem_0503.png",
+      "answer": "5",
+      "fact_expr": "C: Parabola;p: Number;P: Point;F: Point;p>0;Expression(C) = (x^2 = 2*(p*y));Coordinate(P) = (4, 4);Focus(C) = F;PointOnCurve(P, C)",
+      "coords": {
+        "P": [
+          4.0,
+          4.0
+        ]
+      },
+      "validation_reasons": [
+        "parabola_point_off_curve"
+      ]
+    },
+    {
+      "index": 504,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 505,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.7320508075688772,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 3.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0505.png",
+      "answer": "x^2+x+3*y^2=0",
+      "fact_expr": "C: Ellipse;G: Point;A: Point;B: Point;Expression(C) = (x^2/3 + y^2 = 1);Coordinate(G) = (-1, 0);IsChordOf(LineSegmentOf(A,B),C);PointOnCurve(G,LineSegmentOf(A,B))",
+      "coords": {
+        "G": [
+          -1.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 506,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.4142135623730951,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0506.png",
+      "answer": "y=pm*(sqrt(2)/2)*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/2 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 507,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "y",
+        "x_coef": 16.0,
+        "y_coef": 25.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0507.png",
+      "answer": "3*sqrt(7)",
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+          -3.0
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+        "note": "using explicit params"
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+      "output_path": "results/dev/parabola/problem_0508.png",
+      "answer": "2",
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+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0509.png",
+      "answer": "(-oo,-1]+[1,+oo)",
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+    },
+    {
+      "index": 510,
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+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 511,
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+        "note": "using explicit params"
+      },
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+      "answer": "2*sqrt(2)-1",
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+      },
+      "output_path": "results/dev/parabola/problem_0512.png",
+      "answer": "12",
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+    },
+    {
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+      "error": "Unsupported or unparseable expression"
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+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0514.png",
+      "answer": "y = pm*sqrt(2)*x",
+      "fact_expr": "G: Hyperbola;a: Number;a>0;Expression(G) = (-y^2/2 + x^2/a = 1);Eccentricity(G) = sqrt(3*a)",
+      "coords": {}
+    },
+    {
+      "index": 515,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 516,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 517,
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+        "iterations": 2
+      },
+      "output_path": "results/dev/ellipse/problem_0517.png",
+      "answer": "{9/2, 8}",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/6 + x^2/m = 1);m: Number;Eccentricity(G) = 1/2",
+      "coords": {},
+      "validation_reasons": [
+        "ellipse_ecc_mismatch"
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+    {
+      "index": 518,
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+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0518.png",
+      "answer": "16/3",
+      "fact_expr": "C: Parabola;G: Line;A: Point;B: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F, G);Inclination(G) = ApplyUnit(60, degree);Intersection(G, C) = {A, B}",
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+    },
+    {
+      "index": 519,
+      "success": true,
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+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0519.png",
+      "answer": "4",
+      "fact_expr": "F1: Point;F2: Point;Focus(G) = {F1, F2};G: Ellipse;Expression(G) = (x^2/9 + y^2/3 = 1);H: Line;PointOnCurve(F1, H);Intersection(H, G) = {A, B};A: Point;B: Point;Abs(LineSegmentOf(A, F2)) + Abs(LineSegmentOf(B, F2)) = 8",
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+    },
+    {
+      "index": 520,
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+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0520.png",
+      "answer": "sqrt(5), sqrt(5)/2",
+      "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;O: Origin;A: Point;B: Point;a>0;b>0;L1:Line;L2:Line;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(l)=(x=4*a);Asymptote(C)={L1,L2};Intersection(l,L1)=A;Intersection(l,L2)=B;Area(TriangleOf(O,A,B))=32;FocalLength(C) = 2*sqrt(5)",
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+    },
+    {
+      "index": 521,
+      "success": true,
+      "conic_type": "ellipse",
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+        "x_coef": 4.0,
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+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0521.png",
+      "answer": "sqrt(3)/2",
+      "fact_expr": "G: Ellipse;n: Number;m: Number;m>0;n>0;2/n + 1/m = 1;WhenMin(m*n);Expression(G) = (y^2/n^2 + x^2/m^2 = 1)",
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+    },
+    {
+      "index": 522,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0522.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Distance(Focus(G),Asymptote(G))= a^2/b",
+      "coords": {}
+    },
+    {
+      "index": 523,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 524,
+      "success": true,
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+        "b": 3.0,
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+        "x_coef": 16.0,
+        "y_coef": 9.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0524.png",
+      "answer": "12*(2+sqrt(2))/5\n12*(2-sqrt(2))/5",
+      "fact_expr": "G: Ellipse;H: Line;P: Point;Expression(G) = (x^2/16 + y^2/9 = 1);Expression(H) = (3*x - 4*y = 24);PointOnCurve(P, G)",
+      "coords": {}
+    },
+    {
+      "index": 525,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0525.png",
+      "answer": "(1, 3/2)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;H: Circle;Expression(H) = (-6*y + x^2 + y^2 + 5 = 0);NumIntersection(Asymptote(G), H) = 0",
+      "coords": {}
+    },
+    {
+      "index": 526,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 527,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
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+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0527.png",
+      "answer": "2",
+      "fact_expr": "A: Point;B: Point;G: Parabola;p: Number;H: Line;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G)=F;PointOnCurve(F, H);Inclination(H)=ApplyUnit(45, degree);Intersection(H,G) = {B, A};Length(LineSegmentOf(A, B)) = 8;F: Point",
+      "coords": {}
+    },
+    {
+      "index": 528,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "vertical"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0528.png",
+      "answer": "y=pm*(sqrt(3)/3)*x",
+      "fact_expr": "G: Hyperbola;b: Number;b>0;Expression(G) = (y^2/4 - x^2/b^2 = 1);FocalLength(G) = 8",
+      "coords": {}
+    },
+    {
+      "index": 529,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0529.png",
+      "answer": "sqrt(6)",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;M: Point;PointOnCurve(M, G);AngleOf(F1, M, F2) = ApplyUnit(60, degree);DotProduct(VectorOf(M, F1), VectorOf(M, F2)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 530,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0530.png",
+      "answer": "32/5",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);F2: Point;RightFocus(G) = F2;H: LineSegment;IsChordOf(H, G);PointOnCurve(F2, H)",
+      "coords": {}
+    },
+    {
+      "index": 531,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_focus_mismatch",
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0531.png",
+      "answer": "y=pm*(2/3)*x",
+      "fact_expr": "G: Hyperbola;b: Number;H: Point;Expression(G) = (x^2/9 - y^2/b^2 = 1);Coordinate(H) = (sqrt(13), 0);RightFocus(G) = H",
+      "coords": {
+        "H": [
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+          0.0
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+      "validation_reasons": [
+        "hyperbola_focus_mismatch"
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+    },
+    {
+      "index": 532,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "left"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0532.png",
+      "answer": "x^2-y^2/3=1",
+      "fact_expr": "O: Origin;G:Hyperbola;H: Parabola;Expression(H) = (y^2 = -8*x);Eccentricity(G)=2;Center(G)=O;OneOf(Vertex(G))=Focus(H)",
+      "coords": {}
+    },
+    {
+      "index": 533,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0533.png",
+      "answer": "(2,+oo)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;P:Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P,G);XCoordinate(P)=3*a/2;Distance(P,RightFocus(G))>Distance(P,LeftDirectrix(G))",
+      "coords": {}
+    },
+    {
+      "index": 534,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0534.png",
+      "answer": "2",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;FocalLength(C) = 4*sqrt(5);Slope(OneOf(Asymptote(C))) = 2",
+      "coords": {}
+    },
+    {
+      "index": 535,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 536,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0536.png",
+      "answer": "9/2",
+      "fact_expr": "G: Parabola;A: Point;P: Point;M: Point;Expression(G) = (y^2 = 2*x);Coordinate(A) = (7/2, 4);PointOnCurve(P, G);Projection(P, yAxis) = M",
+      "coords": {
+        "A": [
+          3.5,
+          4.0
+        ]
+      }
+    },
+    {
+      "index": 537,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0537.png",
+      "answer": "-4",
+      "fact_expr": "C: Hyperbola;A: Point;B: Point;P: Point;Expression(C) = (x^2/16 - y^2/9 = 1);Coordinate(A) = (-5, 0);Coordinate(B) = (5, 4);PointOnCurve(P, RightPart(C))",
+      "coords": {
+        "A": [
+          -5.0,
+          0.0
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+        "B": [
+          5.0,
+          4.0
+        ]
+      }
+    },
+    {
+      "index": 538,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0538.png",
+      "answer": "(-oo, -2*sqrt(2)] + (0, +oo)",
+      "fact_expr": "l: Line;G: Parabola;p: Number;O: Origin;A: Point;F: Point;B: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B};PointOnCurve(A, xAxis);Abs(VectorOf(O, A)) >= Abs(VectorOf(F, A))",
+      "coords": {}
+    },
+    {
+      "index": 539,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0539.png",
+      "answer": "x^2/(9/4)-y^2/4=1",
+      "fact_expr": "G: Hyperbola;C:Hyperbola;A: Point;Expression(G) = (x^2/9 - y^2/16 = 1);Coordinate(A) = (3, -2*sqrt(3));Asymptote(C) = Asymptote(G);PointOnCurve(A, C)",
+      "coords": {
+        "A": [
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+          -3.4641016151377544
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+      }
+    },
+    {
+      "index": 540,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.2152259349823,
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+      "output_path": "results/dev/ellipse/problem_0540.png",
+      "answer": "x^2/16+y^2/7=1",
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+      "output_path": "results/dev/parabola/problem_0541.png",
+      "answer": "4",
+      "fact_expr": "F: Point;G: Parabola;Expression(G) = (y^2 = 4*x);Focus(G) = F;M: Point;PointOnCurve(M, G);P: Point;Coordinate(P) = (3, 1)",
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+          1.0
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+      }
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+    {
+      "index": 542,
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+      },
+      "output_path": "results/dev/hyperbola/problem_0542.png",
+      "answer": "x^2/4 - y^2/12 = 1",
+      "fact_expr": "G: Hyperbola;Expression(OneOf(Asymptote(G))) = (y = sqrt(3)*x);Coordinate(Focus(G))=(4,0)",
+      "coords": {}
+    },
+    {
+      "index": 543,
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+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0543.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 8*y)",
+      "coords": {}
+    },
+    {
+      "index": 544,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 545,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0545.png",
+      "answer": "3",
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+      "coords": {}
+    },
+    {
+      "index": 546,
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+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0546.png",
+      "answer": "sqrt(3)-1",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;G: Line;O: Origin;A: Point;B:Point;F: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y = sqrt(3)*x);RightFocus(C) = F;Intersection(G,C)={A,B};Quadrant(A)=1;LineSegmentOf(O, A) = LineSegmentOf(O, F)",
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+    },
+    {
+      "index": 547,
+      "success": true,
+      "conic_type": "ellipse",
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+        "y_coef": 16.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0547.png",
+      "answer": "4",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/25 + y^2/16 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 6",
+      "coords": {}
+    },
+    {
+      "index": 548,
+      "success": true,
+      "conic_type": "ellipse",
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+        "y_coef": 6.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0548.png",
+      "answer": "2*sqrt(3)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/6 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);WhenMin(1/Abs(LineSegmentOf(P, F1))+4/Abs(LineSegmentOf(P, F2)))",
+      "coords": {}
+    },
+    {
+      "index": 549,
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+      "conic_type": "hyperbola",
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+        "converged": true,
+        "iterations": 94
+      },
+      "output_path": "results/dev/hyperbola/problem_0549.png",
+      "answer": "(3*x^2/2-y^2/2 = 1)",
+      "fact_expr": "C: Hyperbola;C1: Point;Coordinate(C1) = (1,1);Expression(OneOf(Asymptote(C))) = (y = sqrt(3)*x)",
+      "coords": {
+        "C1": [
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+          1.0
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+      }
+    },
+    {
+      "index": 550,
+      "success": true,
+      "conic_type": "ellipse",
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+      "params": {
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+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
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+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0550.png",
+      "answer": "x^2/9 - y^2/27 = 1",
+      "fact_expr": "H: Ellipse;Expression(H) = (x^2/25 + y^2/16 = 1);G: Hyperbola;Focus(H) = Vertex(G);Eccentricity(G) = 2",
+      "coords": {}
+    },
+    {
+      "index": 551,
+      "success": true,
+      "conic_type": "parabola",
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "results/dev/parabola/problem_0551.png",
+      "answer": "4",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);Focus(C) = F;F: Point;A: Point;Coordinate(A) = (x0, y0);x0: Number;y0: Number;PointOnCurve(A, C) = True;Abs(LineSegmentOf(A, F)) = (5/4)*x0",
+      "coords": {}
+    },
+    {
+      "index": 552,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0552.png",
+      "answer": "(4,pm*4*sqrt(2))",
+      "fact_expr": "G: Parabola;P: Point;Expression(G) = (y^2 = 8*x);PointOnCurve(P, G);Distance(P, Focus(G)) = 6",
+      "coords": {}
+    },
+    {
+      "index": 553,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "y_coef": 3.0
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+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0553.png",
+      "answer": "sqrt(3)/2",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;P: Point;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Focus(G)={F1,F2};PointOnCurve(P,G);Max(AngleOf(F1, P, F2)) = (2*pi)/3",
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+    },
+    {
+      "index": 554,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0554.png",
+      "answer": "sqrt(3)+1",
+      "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;A: Point;B: Point;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;Expression(l)=(y = sqrt(3)*x) ;Intersection(l, C) = {A, B};Abs(LineSegmentOf(A, B)) = Abs(LineSegmentOf(F1, F2))",
+      "coords": {}
+    },
+    {
+      "index": 555,
+      "success": true,
+      "conic_type": "ellipse",
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+        "y_coef": 8.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0555.png",
+      "answer": "(pm*2*sqrt(2), 0)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/8 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 556,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 5.0,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0556.png",
+      "answer": "y=pm*(5/3)*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/25 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 557,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 558,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 3.4641016151377544,
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+      "optimization": {
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+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0558.png",
+      "answer": "3/2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/12 = 1);H: Parabola;Expression(H) = (y^2 = 2*p*x);p: Real;PointOnCurve(Intersection(RightDirectrix(G), Asymptote(G)), H)",
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+    },
+    {
+      "index": 559,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0559.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "C: Parabola;E: Ellipse;A: Point;B: Point;O:Origin;Center(E)=O;Eccentricity(E)=sqrt(3)/2;Expression(C) = (y^2 = 12*x);RightFocus(E) = Focus(C);Intersection(Directrix(C),E)={A,B}",
+      "coords": {}
+    },
+    {
+      "index": 560,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0560.png",
+      "answer": "y^2=8*x",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;A: Point;PointOnCurve(A, G);XCoordinate(A) = 3;Distance(A, Focus(G)) = 5",
+      "coords": {}
+    },
+    {
+      "index": 561,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_ecc_mismatch",
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+        "converged": true,
+        "iterations": 2
+      },
+      "output_path": "results/dev/ellipse/problem_0561.png",
+      "answer": "{3, 16/3}",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/m = 1);m: Number;Eccentricity(G) = 1/2",
+      "coords": {},
+      "validation_reasons": [
+        "ellipse_ecc_mismatch"
+      ]
+    },
+    {
+      "index": 562,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 563,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0563.png",
+      "answer": "{10,2}",
+      "fact_expr": "G: Hyperbola;P: Point;Expression(G) = (x^2 - 4*y^2 = 4);PointOnCurve(P, G);F1:Point;F2:Point;OneOf(Focus(G))=F1;OneOf(Focus(G)) = F2;Negation(F1=F2);Distance(P, F1) = 6",
+      "coords": {}
+    },
+    {
+      "index": 564,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0564.png",
+      "answer": "x - 2*y + 4 = 0",
+      "fact_expr": "E: Ellipse;b: Number;a: Number;M: Circle;r: Number;B: Point;A: Point;c: Number;a > b;b > 0;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);Expression(M) = ((x + 2)^2 + (y - 1)^2 = r^2);HalfFocalLength(E) = c;c = sqrt(3)*b;PointOnCurve(A, E);PointOnCurve(B, E);OneOf(Diameter(LineSegmentOf(A, B), M))",
+      "coords": {}
+    },
+    {
+      "index": 565,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_ecc_mismatch",
+      "params": {
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+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 2.5576271215184017e-15,
+        "converged": true,
+        "iterations": 2
+      },
+      "output_path": "results/dev/ellipse/problem_0565.png",
+      "answer": "x^2/4 + y^2/3 = 1",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Eccentricity(C) = 1/2;F2: Point;RightFocus(C) = F2;M: Point;G: Circle;Expression(G) = (x^2 + y^2 = b^2);PointOnCurve(M, G) = True;Quadrant(M) = 1;P: Point;Q: Point;l: Line;TangentOnPoint(M,G) = l;Perimeter(TriangleOf(P, F2, Q)) = 4;Intersection(l,C) = {P,Q}",
+      "coords": {},
+      "validation_reasons": [
+        "ellipse_ecc_mismatch"
+      ]
+    },
+    {
+      "index": 566,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0566.png",
+      "answer": "[sqrt(2)/2, 1)",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 567,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 568,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 4.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0568.png",
+      "answer": "5",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 + y^2/4 = 1);PointOnCurve(P, G);Focus(G)={F1,F2}",
+      "coords": {}
+    },
+    {
+      "index": 569,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0569.png",
+      "answer": "4",
+      "fact_expr": "C: Parabola;Expression(C) = (x^2 = 2*p*y);p: Number;p > 0;F: Point;Focus(C) = F;P: Point;Coordinate(P) = (4, y0);y0: Number;PointOnCurve(P, C) = True;Abs(LineSegmentOf(F, P)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 570,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "answer": "(pm*1, 0)",
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+    },
+    {
+      "index": 571,
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+    {
+      "index": 572,
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+      },
+      "output_path": "results/dev/ellipse/problem_0572.png",
+      "answer": "4*sqrt(5)/5",
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+    {
+      "index": 573,
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+      "error": "Unsupported or unparseable expression"
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+      "index": 574,
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+        "note": "using explicit params"
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+      "output_path": "results/dev/ellipse/problem_0574.png",
+      "answer": "21",
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+    },
+    {
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+        "note": "using explicit params"
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+      "output_path": "results/dev/ellipse/problem_0575.png",
+      "answer": "3/4",
+      "fact_expr": "l: Line;G: Ellipse;Expression(G) = (x^2/8 + y^2/6 = 1);A: Point;B: Point;Intersection(l, G) = {A, B};Coordinate(MidPoint(LineSegmentOf(A,B)))=(-1,1)",
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+    },
+    {
+      "index": 576,
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+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0576.png",
+      "answer": "sqrt(2)",
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+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0577.png",
+      "answer": "sqrt(2)/4",
+      "fact_expr": "C: Parabola;P: Point;O: Origin;F: Point;x0: Number;y0: Number;Expression(C) = (y^2 = 2*x);Coordinate(P) = (x0, y0);Focus(C) = F;PointOnCurve(P, C);Abs(LineSegmentOf(P, F)) = (3/2)*x0",
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+    {
+      "index": 578,
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+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0578.png",
+      "answer": "{(x^2/3-y^2/3=1),(x^2/8+y^2/2=1)}",
+      "fact_expr": "G: Hyperbola;H: ConicSection;P:Point;Expression(G) = (x^2/4 - y^2/2 = 1);Coordinate(P) = (2, 1);Focus(G)=Focus(H);PointOnCurve(P,H)",
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+    {
+      "index": 579,
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+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0579.png",
+      "answer": "2*(sqrt(2)-1)",
+      "fact_expr": "G: Parabola;H: Line;M: Point;F: Point;N: Point;A: Point;B: Point;l: Line;Expression(G) = (y^2 = 4*x);Focus(G) = F;Directrix(G) = l;PointOnCurve(F,H);Intersection(H,G) = {A,B};VectorOf(A, F) = VectorOf(F, B);Projection(A,l)=M;Projection(B,l)=N",
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+    {
+      "index": 580,
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+      "optimization": {
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+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0580.png",
+      "answer": "y=pm*(2/3)*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/4 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 581,
+      "success": true,
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0581.png",
+      "answer": "0",
+      "fact_expr": "C: Parabola;Expression(C) = (x^2 = 4*y);P: Point;Coordinate(P) = (0, -1);TangentOfPoint(P, C) = {LineOf(P, A), LineOf(P, B)};TangentPoint(LineOf(P, A), C) = A;TangentPoint(LineOf(P, B), C) = B;A: Point;B: Point",
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+    {
+      "index": 582,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 583,
+      "success": true,
+      "conic_type": "parabola",
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0583.png",
+      "answer": "4+2*sqrt(2)",
+      "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = 8;Abs(LineSegmentOf(A, F)) < Abs(LineSegmentOf(B, F))",
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+    {
+      "index": 584,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "a": 3.0,
+        "b": 1.0,
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0584.png",
+      "answer": "sqrt(10)/3\nx+pm*3*y=0",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2 = 1);e: Number;Eccentricity(G) = e",
+      "coords": {}
+    },
+    {
+      "index": 585,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 586,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0586.png",
+      "answer": "3",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/6 + x^2/m = 1);m: Number;FocalLength(G) = 6",
+      "coords": {}
+    },
+    {
+      "index": 587,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 10.0,
+        "b": 5.0,
+        "major_axis": "x",
+        "x_coef": 100.0,
+        "y_coef": 25.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0587.png",
+      "answer": "20",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2 + 4*y^2 = 100)",
+      "coords": {}
+    },
+    {
+      "index": 588,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0588.png",
+      "answer": "2*sqrt(3)/3",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;G: Circle;Expression(G) = (y^2 + (x - 2)^2 = 2);IsIntersect(Asymptote(C), G);L1: Line;L2: Line;Asymptote(C) = {L1, L2};Length(InterceptChord(L1, G)) = 2;Length(InterceptChord(L2, G)) = 2",
+      "coords": {}
+    },
+    {
+      "index": 589,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.449489742783178,
+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 6.0,
+        "y_coef": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0589.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;H: Ellipse;Expression(H) = (x^2/6 + y^2/2 = 1);Focus(G) = RightFocus(H)",
+      "coords": {}
+    },
+    {
+      "index": 590,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0590.png",
+      "answer": "1/8",
+      "fact_expr": "P: Point;PointOnCurve(P,RightPart(G)) = True;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;H: Circle;Center(H) = O;O: Origin;Radius(H) = a;TangentPoint(LineOf(P,F1),H) = A;A: Point;PointOnCurve(F2,PerpendicularBisector(LineSegmentOf(P,F1))) = True;IsTangent(LineOf(P,F1),H) = True",
+      "coords": {}
+    },
+    {
+      "index": 591,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0591.png",
+      "answer": "(-oo,2]",
+      "fact_expr": "G: Parabola;a: Real;Expression(G) = (x^2 = 4*y);A:Point;Coordinate(A)=(0,a);WhenMin(Distance(Vertex(G),A))",
+      "coords": {}
+    },
+    {
+      "index": 592,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0592.png",
+      "answer": "0",
+      "fact_expr": "G: Hyperbola;b: Number;P: Point;F1: Point;F2: Point;b>0;Expression(G) = (x^2/2 - y^2/b^2 = 1);y0:Number;Coordinate(P) = (sqrt(3), y0);LeftFocus(G) = F1;RightFocus(G) = F2;Expression(OneOf(Asymptote(G))) = (y = x);PointOnCurve(P, G)",
+      "coords": {}
+    },
+    {
+      "index": 593,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 594,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0594.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "C: Parabola;p: Number;F: Point;M: Point;A: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Coordinate(F) = (1, 0);Focus(C) = F;Intersection(Directrix(C),axis)=A;PointOnCurve(M, C);WhenMin(Abs(LineSegmentOf(M,F))/Abs(LineSegmentOf(M,A)))",
+      "coords": {
+        "F": [
+          1.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 595,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          0.0,
+          0.0
+        ],
+        "radius": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/circle/problem_0595.png",
+      "answer": "(x^2/16+y^2/12=1)&Negation(y=0)",
+      "fact_expr": "O: Circle;Expression(O) = (x^2 + y^2 = 16);A: Point;Coordinate(A) = (-2, 0);B: Point;Coordinate(B) = (2, 0);IsTangent(l, O) = True;l: Line;PointOnCurve(A, G) = True;PointOnCurve(B, G) = True;G: Parabola;Directrix(G) = l",
+      "coords": {}
+    },
+    {
+      "index": 596,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0596.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;A: Point;PointOnCurve(A, G);F: Point;Focus(G) = F;O: Origin;Area(TriangleOf(O, A, F)) = p;Abs(LineSegmentOf(A, F)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 597,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0597.png",
+      "answer": "[1,5)",
+      "fact_expr": "G: Ellipse;t: Number;H: Line;k: Real;Expression(G) = (x^2/5 + y^2/t = 1);Expression(H) = (y = k*x + 1);PointOnCurve(Focus(G),xAxis);IsIntersect(G,H)",
+      "coords": {}
+    },
+    {
+      "index": 598,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0598.png",
+      "answer": "4",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/16 + y^2/b^2 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;Eccentricity(C) = sqrt(3)/2;PointOnCurve(P,C) = True;P: Point;AngleOf(F1, P, F2) = ApplyUnit(90, degree);b:Number;4>b;b>0",
+      "coords": {}
+    },
+    {
+      "index": 599,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          0.0,
+          0.0
+        ],
+        "radius": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/circle/problem_0599.png",
+      "answer": "Preserve all mathematical expressions, LaTeX formulas, symbols, and equations EXACTLY.  \nDo not add explanations, do not rewrite formulas, do not omit any content.  \n\nText:",
+      "fact_expr": "G: Circle;B: Point;A: Point;Endpoint(LineSegmentOf(A,B))={A,B};Expression(G) = (x^2 + y^2 = 1);Coordinate(B)=(4,0);PointOnCurve(A, G)",
+      "coords": {}
+    },
+    {
+      "index": 600,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 0.5,
+        "b": 0.7071067811865476,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0600.png",
+      "answer": "sqrt(3)/2",
+      "fact_expr": "G: Hyperbola;F1: Point;P: Point;F2: Point;Expression(G) = (4*x^2 - 2*y^2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(60, degree)",
+      "coords": {}
+    },
+    {
+      "index": 601,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0601.png",
+      "answer": "(1, 2)",
+      "fact_expr": "G: Hyperbola;a: Number;b: Number;H: Circle;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2 + (y - 2)^2 = 1);NumIntersection(Asymptote(G), H) = 0",
+      "coords": {}
+    },
+    {
+      "index": 602,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 603,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0603.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Z: Line;PointOnCurve(OneOf(Focus(G)), Z);IsPerpendicular(Z, OneOf(Asymptote(G)));H: Curve;Expression(H) = (x^2/b^2 + y^2/a^2 = 1);PointOnCurve(FootPoint(Z, OneOf(Asymptote(G))), H)",
+      "coords": {}
+    },
+    {
+      "index": 604,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0604.png",
+      "answer": "(1/2, 1)",
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+      "coords": {}
+    },
+    {
+      "index": 605,
+      "success": true,
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+      "error": null,
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+      },
+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0605.png",
+      "answer": "16/3",
+      "fact_expr": "m: Line;E: Parabola;A: Point;F: Point;P: Point;Q: Point;B: Point;l: Line;Expression(E) = (y^2 = 4*x);Focus(E) = F;Directrix(E) = l;PointOnCurve(F, m);Intersection(m, E) = {A, B};Intersection(PerpendicularBisector(LineSegmentOf(A,F)),l)=P;Intersection(PerpendicularBisector(LineSegmentOf(A,F)),xAxis)=Q;AngleOf(A,F,P)=AngleOf(A,F,Q)",
+      "coords": {}
+    },
+    {
+      "index": 606,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0606.png",
+      "answer": "a^2/b^2",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;O: Circle;B: Point;A: Point;N: Point;M: Point;P:Point;a > b;b > 0;L1:Line;L2:Line;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(O)=(x^2+y^2=b^2);PointOnCurve(P,G);Negation(P=Vertex(G));TangentOfPoint(P,O)={L1,L2};TangentPoint(L1,O)=A;TangentPoint(L2,O)=B;Intersection(LineOf(A,B), xAxis) = M;Intersection(LineOf(A,B), yAxis) = N",
+      "coords": {}
+    },
+    {
+      "index": 607,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0607.png",
+      "answer": "4",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(C) = F;M: Point;PointOnCurve(M, C);N: Point;Intersection(OverlappingLine(LineSegmentOf(F, M)), yAxis) = N;MidPoint(LineSegmentOf(F, N)) = M;Abs(LineSegmentOf(F, N)) = 6",
+      "coords": {}
+    },
+    {
+      "index": 608,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 609,
+      "success": true,
+      "conic_type": "ellipse",
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+        "b": 4.0,
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+        "y_coef": 16.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0609.png",
+      "answer": "x^2/9 - y^2/16 = 1",
+      "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/25 + y^2/16 = 1);Vertex(H)=Focus(G);Focus(H)=Vertex(G)",
+      "coords": {}
+    },
+    {
+      "index": 610,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 611,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0611.png",
+      "answer": "3/5",
+      "fact_expr": "G: Ellipse;a: Number;b: Number;m: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(P) = (m, 4);PointOnCurve(P, G);Focus(G) = {F1, F2};Radius(InscribedCircle(TriangleOf(P, F1, F2))) = 3/2",
+      "coords": {}
+    },
+    {
+      "index": 612,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_point_off_curve",
+      "params": {
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+        "b": 1.6304504558913928,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 6.201316546158653e-06,
+        "converged": true,
+        "iterations": 81
+      },
+      "output_path": "results/dev/ellipse/problem_0612.png",
+      "answer": "x^2/4+y^2/3=1",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Coordinate(P) = (1, 3/2);Focus(C) = {F1, F2};PointOnCurve(P, C);Abs(LineSegmentOf(P, F1)) + Abs(LineSegmentOf(P, F2)) = 4",
+      "coords": {
+        "P": [
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+          1.5
+        ]
+      },
+      "validation_reasons": [
+        "ellipse_point_off_curve"
+      ]
+    },
+    {
+      "index": 613,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 614,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0614.png",
+      "answer": "7",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(Asymptote(G))=(4*x+pm*3*y=0);PointOnCurve(P,G);LeftFocus(G) = F1;RightFocus(G) = F2;Area(TriangleOf(P, F1, F2)) = 16;DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0",
+      "coords": {}
+    },
+    {
+      "index": 615,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 616,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0616.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "Slope(l) = sqrt(2)/2;l: Line;G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Intersection(l, G) = {A, B};A: Point;B: Point;Focus(G) = {Projection(A, xAxis), Projection(B, xAxis)}",
+      "coords": {}
+    },
+    {
+      "index": 617,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0617.png",
+      "answer": "x^2/8 - y^2/18 = 1",
+      "fact_expr": "G: Hyperbola;H: Hyperbola;M: Point;Expression(G) = (x^2/4 - y^2/9 = 1);Coordinate(M) = (2*sqrt(6), 6);PointOnCurve(M,H);Asymptote(G) = Asymptote(H)",
+      "coords": {
+        "M": [
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+          6.0
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+      }
+    },
+    {
+      "index": 618,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0618.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "E: Hyperbola;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;M: Point;Coordinate(M) = (1, 8);PointOnCurve(M, G);G: Line;Intersection(G, E) = {A, B};B: Point;A: Point;MidPoint(LineSegmentOf(A, B)) = M;IsPerpendicular(LineOf(A, B), OneOf(Asymptote(E))) = True",
+      "coords": {
+        "M": [
+          1.0,
+          8.0
+        ]
+      }
+    },
+    {
+      "index": 619,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0619.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;P: Point;Expression(G) = (y^2 = 4*x);Coordinate(P) = (3, y1);PointOnCurve(P,G);y1:Number",
+      "coords": {}
+    },
+    {
+      "index": 620,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.03125,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0620.png",
+      "answer": "65/16",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = y/8);H: Line;PointOnCurve(Focus(G), H);A: Point;B: Point;Intersection(H, G) = {A, B};M: Point;MidPoint(LineSegmentOf(A, B)) = M;YCoordinate(M) = 2",
+      "coords": {}
+    },
+    {
+      "index": 621,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0621.png",
+      "answer": "8",
+      "fact_expr": "l: Line;G: Parabola;Expression(G) = (y^2 = 8*x);F: Point;Focus(G) = F;PointOnCurve(F, l);A: Point;B: Point;Intersection(l,G) = {A,B};Distance(MidPoint(LineSegmentOf(A,B)), yAxis) = 2",
+      "coords": {}
+    },
+    {
+      "index": 622,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0622.png",
+      "answer": "sqrt(6)/2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;P: Point;PointOnCurve(P, G) ;L: Line;PointOnCurve(P,L);IsParallel(L,RealAxis(G)) ;L1: Line;L2: Line;Asymptote(G) = {L1,L2};Intersection(L,L1) = M;M: Point;Intersection(L,L2) = N;N: Point;DotProduct(VectorOf(P, M), VectorOf(P, N)) = 2*b^2",
+      "coords": {}
+    },
+    {
+      "index": 623,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 624,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0624.png",
+      "answer": "(0,sqrt(2)]",
+      "fact_expr": "E: Ellipse;A: Point;B: Point;F: Point;P: Point;t:Number;Expression(E) = (x^2/2 + y^2 = 1);Coordinate(P)=(2,t);LeftFocus(E) = F;TangentOfPoint(P,E)={LineOf(P,A),LineOf(P,B)};TangentPoint(LineOf(P,B),E)=A;TangentPoint(LineOf(P,A),E)=B",
+      "coords": {}
+    },
+    {
+      "index": 625,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0625.png",
+      "answer": "1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 626,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0626.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;l: Line;PointOnCurve(F1, l);Inclination(l) = pi/6;A: Point;Intersection(l, LeftPart(G)) = A;B: Point;Intersection(l, RightPart(G)) = B;Abs(LineSegmentOf(A, F2)) = Abs(LineSegmentOf(B, F2))",
+      "coords": {}
+    },
+    {
+      "index": 627,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.4641016151377544,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 12.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0627.png",
+      "answer": "9",
+      "fact_expr": "G: Ellipse;A: Point;B: Point;P: Point;Expression(G) = (x^2/16 + y^2/12 = 1);Coordinate(A) = (-2, 0);Coordinate(B) = (2, 0);PointOnCurve(P, G);Abs(LineSegmentOf(P, A)) - Abs(LineSegmentOf(P, B)) = 2",
+      "coords": {
+        "A": [
+          -2.0,
+          0.0
+        ],
+        "B": [
+          2.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 628,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0628.png",
+      "answer": "x^2/2+y^2=1",
+      "fact_expr": "C: Ellipse;G:Line;a: Number;b: Number;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x - y + 2*sqrt(2) = 0);Eccentricity(C) = sqrt(2)/2;Distance(RightVertex(C),G)=3",
+      "coords": {}
+    },
+    {
+      "index": 629,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 2.0,
+        "major_axis": "y",
+        "x_coef": 4.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0629.png",
+      "answer": "8;4;(0,pm*2*sqrt(3));{(pm*2,0),(0,pm*4)};sqrt(3)/2",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/16 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 630,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.8284271247461903,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 8.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0630.png",
+      "answer": "-1/2",
+      "fact_expr": "P: Point;Coordinate(P) = (1, 1);PointOnCurve(P, l) = True;l: Line;G: Ellipse;Expression(G) = (x^2/8 + y^2/4 = 1);Intersection(l, G) = {A, B};A: Point;B: Point;MidPoint(LineSegmentOf(A, B)) = P",
+      "coords": {
+        "P": [
+          1.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 631,
+      "success": true,
+      "conic_type": "parabola",
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+      "params": {
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+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0631.png",
+      "answer": "12",
+      "fact_expr": "C: Parabola;P: Point;G: Point;Expression(C) = (y^2 = 4*x);PointOnCurve(P, C);Coordinate(G) = (6, 12)",
+      "coords": {
+        "G": [
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+          12.0
+        ]
+      }
+    },
+    {
+      "index": 632,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0632.png",
+      "answer": "x^2/20 - y^2/16 = 1",
+      "fact_expr": "C: Hyperbola;a: Number;a > 0;Expression(C) = (x^2/a^2 - y^2/16 = 1);Distance(RightVertex(C), OneOf(Asymptote(C))) = 4*sqrt(5)/3",
+      "coords": {}
+    },
+    {
+      "index": 633,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 634,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "major_axis": "y",
+        "x_coef": 2.0,
+        "y_coef": 6.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0634.png",
+      "answer": "1/8",
+      "fact_expr": "G: Parabola;m: Number;H: Ellipse;Expression(G) = (y = m*x^2);Expression(H) = (x^2/2 + y^2/6 = 1);Focus(G) = UpperFocus(H)",
+      "coords": {}
+    },
+    {
+      "index": 635,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.75,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0635.png",
+      "answer": "9/8",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 3*x);F: Point;Focus(C) = F;G: Line;PointOnCurve(F, G);A: Point;B: Point;Intersection(G, C) = {A, B};O: Origin",
+      "coords": {}
+    },
+    {
+      "index": 636,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
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+          0.0
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+        "radius": 6.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/circle/problem_0636.png",
+      "answer": "x^2/16 + y^2/15 = 1",
+      "fact_expr": "M:Circle;C1:Circle;Expression(C1) = ((x+1)^2 + y^2 = 36);IsInTangent(M,C1);C2: Circle;Expression(C2) = ((x-1)^2 + y^2 = 4);IsOutTangent(M,C2);Center(M) = M1;M1: Point",
+      "coords": {}
+    },
+    {
+      "index": 637,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0637.png",
+      "answer": "10",
+      "fact_expr": "A: Point;Coordinate(A) = (8, 2);G: Parabola;Expression(G) = (y^2 = 8*x);F: Point;Focus(G) = F;P: Point;PointOnCurve(P, G) = True",
+      "coords": {
+        "A": [
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+          2.0
+        ]
+      }
+    },
+    {
+      "index": 638,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 639,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0639.png",
+      "answer": "x^2/6+y^2/2=1",
+      "fact_expr": "l: Line;C: Ellipse;b: Number;a: Number;A: Point;F1: Point;F2: Point;O:Origin;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(O,l);Inclination(l)=ApplyUnit(30,degree);OneOf(Intersection(l,C))=A;IsPerpendicular(LineSegmentOf(A, F1), LineSegmentOf(A, F2));Area(TriangleOf(F1,A,F2)) = 2",
+      "coords": {}
+    },
+    {
+      "index": 640,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 2.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0640.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "C2: Hyperbola;C1: Ellipse;a:Number;b:Number;a>b;b>0;P: Point;F2: Point;Expression(C1) = (x^2/a^2+y^2/b^2=1);Expression(C2) = (x^2-y^2=4);RightFocus(C1)=F2;RightFocus(C2)=F2;Intersection(C1, C2)=P;Quadrant(P) = 1;Abs(LineSegmentOf(P, F2)) = 2",
+      "coords": {}
+    },
+    {
+      "index": 641,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.33732376966822,
+        "b": 1.7454107534785184,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 1.8920117601792354e-05,
+        "converged": true,
+        "iterations": 51
+      },
+      "output_path": "results/dev/hyperbola/problem_0641.png",
+      "answer": "5/4",
+      "fact_expr": "G: Hyperbola;PointOnCurve(Focus(G), xAxis);Expression(Asymptote(G)) = (y = pm*(3/4)*x)",
+      "coords": {}
+    },
+    {
+      "index": 642,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0642.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/4 + x^2/a^2 = 1);a: Number;a>0;H: Point;Coordinate(H) = (1, 2);PointOnCurve(H, Asymptote(G))",
+      "coords": {
+        "H": [
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+          2.0
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+      }
+    },
+    {
+      "index": 643,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0643.png",
+      "answer": "2",
+      "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B};Abs(LineSegmentOf(A, B)) = 6",
+      "coords": {}
+    },
+    {
+      "index": 644,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0644.png",
+      "answer": "sqrt(3)+1",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Focus(C)={F1,F2};PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(P,F1),LineSegmentOf(P,F2));AngleOf(P,F1,F2)=ApplyUnit(30,degree)",
+      "coords": {}
+    },
+    {
+      "index": 645,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.2352789464429823,
+        "b": 2.2352789464429823,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 1.4312591569844774e-06,
+        "converged": true,
+        "iterations": 89
+      },
+      "output_path": "results/dev/hyperbola/problem_0645.png",
+      "answer": "x^2/9 - y^2 = 1",
+      "fact_expr": "G: Hyperbola;F1: Point;F2: Point;Coordinate(F1) = (-sqrt(10), 0);Coordinate(F2) = (sqrt(10), 0);Focus(G) = {F1, F2};Expression(Asymptote(G)) = (y = pm*(x/3))",
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+        "F2": [
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+          0.0
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+      }
+    },
+    {
+      "index": 646,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0646.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;O: Origin;P: Point;F: Point;Expression(G) = (x^2 - y^2/3 = 1);RightFocus(G) = F;PointOnCurve(P,Asymptote(G));Abs(LineSegmentOf(O, P)) = 2",
+      "coords": {}
+    },
+    {
+      "index": 647,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0647.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;H: Ellipse;a: Number;Expression(G) = (x^2/2 - y^2 = 1);a>0;Expression(H) = (y^2 + x^2/a^2 = 1);Focus(G) = Focus(H)",
+      "coords": {}
+    },
+    {
+      "index": 648,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0648.png",
+      "answer": "3",
+      "fact_expr": "G: Ellipse;A: Point;P: Point;F: Point;Expression(G) = (x^2/4 + y^2/3 = 1);Coordinate(A) = (1, 1);LeftFocus(G) = F;PointOnCurve(P, G)",
+      "coords": {
+        "A": [
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+          1.0
+        ]
+      }
+    },
+    {
+      "index": 649,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0649.png",
+      "answer": "10",
+      "fact_expr": "C2: Hyperbola;C1: Ellipse;A: Point;Expression(C1) = (x^2/9 + y^2/4 = 1);Expression(C2) = (x^2/4 - y^2 = 1);OneOf(Intersection(C1, C2)) = A;F1: Point;F2: Point;Focus(C1) = {F1, F2};Distance(A, F1) + Distance(A, F2) = m ;m: Number;n: Number;Abs(Distance(A, F1) - Distance(A, F2)) = n",
+      "coords": {}
+    },
+    {
+      "index": 650,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
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+          0.0
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+        "radius": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/circle/problem_0650.png",
+      "answer": "2",
+      "fact_expr": "l: Line;Expression(l) = (y = k*(x + 4));k:Number;G: Circle;Expression(G) = (y^2 + (x + 2)^2 = 4);A: Point;B: Point;Intersection(l, G) = {A, B};M: Point;MidPoint(LineSegmentOf(A, B)) = M;H: Line;Expression(H) = (3*x + 4*y - 6 = 0)",
+      "coords": {}
+    },
+    {
+      "index": 651,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0651.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G:Hyperbola;b: Number;a: Number;F: Point;O: Origin;A: Point;c:Number;L:Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G)=F;Coordinate(F) = (c, 0);PointOnCurve(F,L);IsPerpendicular(OneOf(Asymptote(G)),L);FootPoint(OneOf(Asymptote(G)),L)=A;Area(TriangleOf(O, A, F)) = c^2/4",
+      "coords": {}
+    },
+    {
+      "index": 652,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "left"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0652.png",
+      "answer": "(-2,0)",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = -8*x)",
+      "coords": {}
+    },
+    {
+      "index": 653,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0653.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;L: Line;M: Point;N: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(RightFocus(G), L);IsPerpendicular(L,xAxis);Intersection(L,G) = {N, M};IsDiameter(LineSegmentOf(M,N),H);PointOnCurve(LeftVertex(G),H)",
+      "coords": {}
+    },
+    {
+      "index": 654,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0654.png",
+      "answer": "1",
+      "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 + y^2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(F1, P), LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 655,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "left"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0655.png",
+      "answer": "3",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = -4*x);F: Point;Focus(C) = F;A: Point;Coordinate(A) = (-2, 1);P: Point;PointOnCurve(P, C)",
+      "coords": {
+        "A": [
+          -2.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 656,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0656.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;H: Line;b: Number;a: Number;F: Point;Expression(G) = (y^2 = 4*x);a>0;b>0;Expression(H) = (b + a*x + y - 1 = 0);Focus(G) = F;PointOnCurve(F, H)",
+      "coords": {}
+    },
+    {
+      "index": 657,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0657.png",
+      "answer": "3",
+      "fact_expr": "C: Parabola;M: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(M, C);XCoordinate(M) = 2",
+      "coords": {}
+    },
+    {
+      "index": 658,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0658.png",
+      "answer": "{9/16, 3/2}",
+      "fact_expr": "C: Parabola;A: Point;M: Point;F: Point;Expression(C) = (y^2 = 4*x);Coordinate(A) = (2, 3/2);Focus(C) = F;PointOnCurve(M, C);WhenMin(Abs(LineSegmentOf(M,A))+Abs(LineSegmentOf(M,F)))",
+      "coords": {
+        "A": [
+          2.0,
+          1.5
+        ]
+      }
+    },
+    {
+      "index": 659,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0659.png",
+      "answer": "English:",
+      "fact_expr": "G: Hyperbola;b: Number;b>0;Expression(G) = (x^2/4 - y^2/b^2 = 1);Expression(Asymptote(G)) = (y = pm*(x/2))",
+      "coords": {}
+    },
+    {
+      "index": 660,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0660.png",
+      "answer": "pm*(3/4)",
+      "fact_expr": "l: Line;C: Parabola;p: Number;k:Number;A: Point;B: Point;M: Point;F:Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;PointOnCurve(M, l);Intersection(Directrix(C), xAxis) = M;Slope(l)=k;Intersection(l,C)={A,B};Abs(LineSegmentOf(A, M)) = (5/4)*Abs(LineSegmentOf(A, F))",
+      "coords": {}
+    },
+    {
+      "index": 661,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "major_axis": "x",
+        "x_coef": 8.0,
+        "y_coef": 4.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0661.png",
+      "answer": "x-2*y+3=0",
+      "fact_expr": "l: Line;G: Ellipse;A: Point;B: Point;Expression(G) = (x^2/8 + y^2/4 = 1);Intersection(l, G) = {A, B};Coordinate(MidPoint(LineSegmentOf(A, B))) = (-1,1)",
+      "coords": {}
+    },
+    {
+      "index": 662,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0662.png",
+      "answer": "y = pm*(sqrt(5)/2)*x",
+      "fact_expr": "G: Hyperbola;b: Number;H: Parabola;Expression(G) = (x^2/4 - y^2/b^2 = 1);Expression(H) = (y^2 = 12*x);RightFocus(G) = Focus(H)",
+      "coords": {}
+    },
+    {
+      "index": 663,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0663.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "G1: Hyperbola;Expression(G1) = (-y^2/b^2 + x^2/a^2 = 1);G2: Hyperbola;Expression(G2) = (y^2/b^2 - x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;e1: Number;e2: Number;Eccentricity(G1) = e1;Eccentricity(G2) = e2",
+      "coords": {}
+    },
+    {
+      "index": 664,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 665,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 666,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_point_off_curve",
+      "params": {
+        "a": 1.9960020780563354,
+        "b": 1.4970015287399292,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "iterations": 2
+      },
+      "output_path": "results/dev/hyperbola/problem_0666.png",
+      "answer": "x^2/2 - y^2/4 = 1",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;F1: Point;B: Point;A: Point;D: Point;F2: Point;L:Line;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(P) = (2, 2);PointOnCurve(P,C);LeftFocus(C)=F1;RightFocus(C)=F2;PointOnCurve(F2,L);IsPerpendicular(L,xAxis);Intersection(L,C)={A,B};Intersection(LineSegmentOf(F1, B), yAxis) = D;IsPerpendicular(LineSegmentOf(A, D), LineSegmentOf(F1, B))",
+      "coords": {
+        "P": [
+          2.0,
+          2.0
+        ]
+      },
+      "validation_reasons": [
+        "hyperbola_point_off_curve"
+      ]
+    },
+    {
+      "index": 667,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0667.png",
+      "answer": "sqrt(3)/3",
+      "fact_expr": "l: Line;C: Ellipse;b: Number;a: Number;B: Point;A: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(MidPoint(LineSegmentOf(A,B))) = (1, 2);Slope(l)=-1/3;Intersection(l, C) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 668,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0668.png",
+      "answer": "x^2/9 - y^2/36 = 1",
+      "fact_expr": "G: Hyperbola;a: Number;b: Number;Q: Point;F: Point;M: Point;N: Point;l:Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G)=F;PointOnCurve(F,l);IsPerpendicular(l,OneOf(Asymptote(G)));FootPoint(l,OneOf(Asymptote(G)))=Q;Intersection(LineOf(F,Q), LeftPart(G)) = M;Intersection(LineOf(F,Q), RightPart(G)) = N;Abs(LineSegmentOf(M, Q)) = 3*Abs(LineSegmentOf(Q, N));Abs(LineSegmentOf(F, N)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 669,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          0.0,
+          0.0
+        ],
+        "radius": 5.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/circle/problem_0669.png",
+      "answer": "2*x-3*y+25=0",
+      "fact_expr": "C: Circle;Expression(C) = (x^2 + y^2 = 25);M: Point;l: Line;Coordinate(M) = (-2, 3);PointOnCurve(M, l);A: Point;B: Point;Intersection(l, C) = {A, B};L1: Line;L2: Line;TangentOnPoint(A,C)=L1;TangentOnPoint(B,C)=L2;N: Point;Intersection(L1, L2) = N",
+      "coords": {}
+    },
+    {
+      "index": 670,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 671,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 672,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0672.png",
+      "answer": "(1, 1/4)",
+      "fact_expr": "G: Parabola;Q: Point;P: Point;F: Point;Expression(G) = (y = x^2/4);Coordinate(Q) = (1, 1);Focus(G) = F;PointOnCurve(P, G);WhenMin(Perimeter(TriangleOf(P, Q, F)))",
+      "coords": {
+        "Q": [
+          1.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 673,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0673.png",
+      "answer": "x^2-y^2/4=1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, RightPart(G));Abs(LineSegmentOf(P, F1)) - Abs(LineSegmentOf(P, F2)) = b;FocalLength(G) = 2*sqrt(5)",
+      "coords": {}
+    },
+    {
+      "index": 674,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0674.png",
+      "answer": "1",
+      "fact_expr": "G: Parabola;p: Number;l: Line;A: Point;B: Point;F: Point;M: Point;N: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;PointOnCurve(A, G);PointOnCurve(B, G);AngleOf(A, F, B) = pi/3;Directrix(G) = l;MidPoint(LineSegmentOf(A, B)) = M;Projection(M, l) = N",
+      "coords": {}
+    },
+    {
+      "index": 675,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.366584062576294,
+        "b": 2.364173412322998,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.00017838670464698225,
+        "converged": true,
+        "iterations": 89
+      },
+      "output_path": "results/dev/hyperbola/problem_0675.png",
+      "answer": "pm*sqrt(35)",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Eccentricity(C) = 2;F: Point;RightFocus(C) = F;G: Line;PointOnCurve(F, G);k: Number;Slope(G) = k;A: Point;B: Point;Intersection(G, RightPart(C)) = {A, B};VectorOf(A, F) = 2*VectorOf(F, B)",
+      "coords": {}
+    },
+    {
+      "index": 676,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "y",
+        "x_coef": 3.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0676.png",
+      "answer": "3/5",
+      "fact_expr": "F1: Point;F2: Point;G: Ellipse;Expression(G) = (x^2/3 + y^2/4 = 1);Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) - Abs(LineSegmentOf(P, F2)) = 1",
+      "coords": {}
+    },
+    {
+      "index": 677,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0677.png",
+      "answer": "[4 + sqrt(61), +oo)",
+      "fact_expr": "G: Hyperbola;H: Circle;M: Point;F1: Point;F2: Point;A: Point;Expression(H) = (x^2 + (y - 6)^2 = 4);Expression(G) = (x^2/9 - y^2/16 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(A, H);PointOnCurve(M, RightPart(G))",
+      "coords": {}
+    },
+    {
+      "index": 678,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0678.png",
+      "answer": "1",
+      "fact_expr": "G: Hyperbola;a: Number;a>0;Expression(G) = (-y^2/3 + x^2/a^2 = 1);Coordinate(LeftFocus(G))=(-2,0)",
+      "coords": {}
+    },
+    {
+      "index": 679,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 680,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0680.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G);P: Point;Coordinate(P) = (3/2, 1);MidPoint(LineSegmentOf(A, B)) = P",
+      "coords": {
+        "P": [
+          1.5,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 681,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 682,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0682.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);P: Point;PointOnCurve(P, G) = True;H: Line;Expression(H) = (x + 2 = 0);Distance(P, H) = 5;F: Point;Focus(G) = F",
+      "coords": {}
+    },
+    {
+      "index": 683,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "down"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0683.png",
+      "answer": "(0, -1)",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = -4*y)",
+      "coords": {}
+    },
+    {
+      "index": 684,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 685,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0685.png",
+      "answer": "sqrt(21)/9",
+      "fact_expr": "A: Point;B: Point;G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;PointOnCurve(A, G) = True;PointOnCurve(B, G) = True;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;VectorOf(A, F1) = 2*VectorOf(F1, B);AngleOf(F1, A, F2) = pi/3",
+      "coords": {}
+    },
+    {
+      "index": 686,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.75,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0686.png",
+      "answer": "12",
+      "fact_expr": "C: Parabola;G: Line;A: Point;B: Point;F: Point;Expression(C) = (y^2 = 3*x);Focus(C) = F;PointOnCurve(F, G);Inclination(G) = ApplyUnit(30, degree);Intersection(G, C) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 687,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0687.png",
+      "answer": "{x+y-1=0,x-y-1=0}",
+      "fact_expr": "l: Line;C: Parabola;A: Point;B: Point;Expression(C) = (y^2 = 4*x);PointOnCurve(Focus(C),l);Intersection(l,C)={A,B};Abs(LineSegmentOf(A, B)) = 8",
+      "coords": {}
+    },
+    {
+      "index": 688,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0688.png",
+      "answer": "5/3",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2/9 - y^2/b^2 = 1);b: Number;b>0;F1: Point;F2: Point;Focus(C) = {F1, F2};A: Point;x0: Number;Coordinate(A) = (x0, 4);x0>0;PointOnCurve(A, C);Area(TriangleOf(F1, A, F2)) = 20;e: Number;Eccentricity(C) = e",
+      "coords": {}
+    },
+    {
+      "index": 689,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0689.png",
+      "answer": "1",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);P: Point;PointOnCurve(P, G);H: Circle;Expression(H) = (x^2 + (y - 3)^2 = 1);L: Line;TangentOfPoint(P, H) = L;A: Point;TangentPoint(L, H) = A",
+      "coords": {}
+    },
+    {
+      "index": 690,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0690.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "C: Ellipse;G: Line;A: Point;F: Point;B: Point;Expression(C) = (x^2/2 + y^2 = 1);Expression(G) = (x = 2);RightFocus(C)=F;PointOnCurve(A, G);Intersection(LineSegmentOf(A,F), C) = B;VectorOf(F, A) = 3*VectorOf(F, B)",
+      "coords": {}
+    },
+    {
+      "index": 691,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0691.png",
+      "answer": "(2/3)*sqrt(6)",
+      "fact_expr": "G: Hyperbola;a: Number;b: Number;e:Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Inclination(OneOf(Asymptote(G)))=pi/3;Eccentricity(G)=e",
+      "coords": {}
+    },
+    {
+      "index": 692,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_point_off_curve",
+      "params": {
+        "a": 3.0,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0692.png",
+      "answer": "Preserve all mathematical expressions, LaTeX formulas, symbols, and equations EXACTLY.  \nDo not add explanations, do not rewrite formulas, do not omit any content.  \n\nText:",
+      "fact_expr": "G: Ellipse;C:Curve;H: Point;Coordinate(H) = (-3, 2);PointOnCurve(H, G);Expression(C)=(x^2/9 + y^2/4 = 1);Focus(G)=Focus(C)",
+      "coords": {
+        "H": [
+          -3.0,
+          2.0
+        ]
+      },
+      "validation_reasons": [
+        "ellipse_point_off_curve"
+      ]
+    },
+    {
+      "index": 693,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0693.png",
+      "answer": "(2, 4)",
+      "fact_expr": "G: Ellipse;b: Real;H: Line;k: Number;b>0;Expression(G) = (x^2/16 + y^2/b^2 = 1);Expression(H) = (y = k*x + 2);PointOnCurve(Focus(G), xAxis);NumIntersection(H, G) = 2",
+      "coords": {}
+    },
+    {
+      "index": 694,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0694.png",
+      "answer": "-4",
+      "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;Expression(G) = (x^2 = 4*y);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);PointOnCurve(Focus(G), l);Intersection(l, G) = {A,B};x1:Number;y1:Number;x2:Number;y2:Number",
+      "coords": {}
+    },
+    {
+      "index": 695,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0695.png",
+      "answer": "(pm*sqrt(7),0)",
+      "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (x^2/4 - y^2/m = 1);Expression(Asymptote(G)) = (y = pm*(x*(sqrt(3)/2)))",
+      "coords": {}
+    },
+    {
+      "index": 696,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.125,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0696.png",
+      "answer": "y=-1/8",
+      "fact_expr": "G: Parabola;Expression(G) = (y = 2*x^2)",
+      "coords": {}
+    },
+    {
+      "index": 697,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0697.png",
+      "answer": "7/5",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;e: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, RightPart(G));Abs(LineSegmentOf(P, F1)) = 6*Abs(LineSegmentOf(P, F2));Eccentricity(G) = e",
+      "coords": {}
+    },
+    {
+      "index": 698,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0698.png",
+      "answer": "sqrt(3)/2",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F: Point;D: Point;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F;Intersection(RightDirectrix(G), xAxis) = D;PointOnCurve(P,G);AngleOf(P,F,D)=ApplyUnit(60,degree);Sin(AngleOf(P, D, F)) = 3/5",
+      "coords": {}
+    },
+    {
+      "index": 699,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 700,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 701,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 702,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 2.23606797749979,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0702.png",
+      "answer": "21",
+      "fact_expr": "E: Ellipse;Expression(E) = (x^2/25 + y^2/16 = 1);G: Hyperbola;Expression(G) = (x^2/4 - y^2/5 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};Focus(E) = {F1, F2};P: Point;OneOf(Intersection(E, G)) = P",
+      "coords": {}
+    },
+    {
+      "index": 703,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 704,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.4142135623730951,
+        "major_axis": "y",
+        "x_coef": 2.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0704.png",
+      "answer": "8",
+      "fact_expr": "l: Line;G: Ellipse;F1: Point;A: Point;B: Point;Expression(G) = (x^2/2 + y^2/4 = 1);OneOf(Focus(G)) = F1;Negation(PointOnCurve(F1, l));Intersection(l, G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 705,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 706,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 707,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0707.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(Asymptote(G))=(y=pm*x);Coordinate(OneOf(Focus(G))) = (4*sqrt(2), 0)",
+      "coords": {}
+    },
+    {
+      "index": 708,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0708.png",
+      "answer": "m - a^2",
+      "fact_expr": "H: Ellipse;Expression(H) = (y^2/n + x^2/m = 1);m: Number;n: Number;m > n;n > 0;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(H) = F1;RightFocus(H) = F2;LeftFocus(G) = F1;RightFocus(G)= F2;P: Point;OneOf(Intersection(H, G)) = P",
+      "coords": {}
+    },
+    {
+      "index": 709,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0709.png",
+      "answer": "(sqrt(2)/2,1)",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;LeftFocus(G) = F;Coordinate(F) = (-c, 0);c: Number;H: Circle;Center(H) = F;Radius(H) = c;NumIntersection(H,LeftDirectrix(G)) = 2",
+      "coords": {}
+    },
+    {
+      "index": 710,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 711,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0711.png",
+      "answer": "[-11/5,1]",
+      "fact_expr": "A: Point;B: Point;G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 + y^2 = 1);Focus(G) = {F1, F2};LeftVertex(G)=A;UpperVertex(G)=B;PointOnCurve(P, LineSegmentOf(A, B))",
+      "coords": {}
+    },
+    {
+      "index": 712,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.552875593585339,
+        "b": 2.3303634978370953,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 1.3057248743840209e-06,
+        "converged": true,
+        "iterations": 98
+      },
+      "output_path": "results/dev/hyperbola/problem_0712.png",
+      "answer": "sqrt(13)/2, sqrt(13)/3",
+      "fact_expr": "G: Hyperbola;Expression(Asymptote(G)) = (y = pm*(3/2)*x)",
+      "coords": {}
+    },
+    {
+      "index": 713,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0713.png",
+      "answer": "y=pm*(sqrt(3)/3*x)",
+      "fact_expr": "G: Hyperbola;C:Parabola;b: Number;Expression(C)=(y^2 = 16*x);Expression(G) = (x^2/12 - y^2/b^2 = 1);Focus(C) = RightFocus(G)",
+      "coords": {}
+    },
+    {
+      "index": 714,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0714.png",
+      "answer": "sqrt(5)/2",
+      "fact_expr": "C: Hyperbola;a: Number;G: Parabola;l: Line;e: Number;Expression(C) = (-y^2 + x^2/a^2 = 1);Expression(G) = (y^2 = 8*x);Directrix(G) = l;IsTangent(l, C);Eccentricity(C) = e",
+      "coords": {}
+    },
+    {
+      "index": 715,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0715.png",
+      "answer": "y^2/2 - x^2/4 = 1",
+      "fact_expr": "G: Hyperbola;M: Point;C:Hyperbola;Expression(G) = (x^2 - 2*y^2 = 2);Coordinate(M) = (2,-2);Asymptote(C) = Asymptote(G);PointOnCurve(M,C)",
+      "coords": {
+        "M": [
+          2.0,
+          -2.0
+        ]
+      }
+    },
+    {
+      "index": 716,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 717,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 718,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0718.png",
+      "answer": "y^2=8*x",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;G: Circle;PointOnCurve(Center(G), C);O: Origin;PointOnCurve(O, G);IsTangent(G, Directrix(C));H: Line;Expression(H) = (x - sqrt(2)*y = 0);Length(InterceptChord(H, G)) = 2*sqrt(6)",
+      "coords": {}
+    },
+    {
+      "index": 719,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0719.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "C:Hyperbola;l:Line;a,b:Number;e:Number;a>0;b>0;Expression(C)=(x^2/a^2-y^2/b^2=1);Expression(l)=(x-2*y+2020=0);IsPerpendicular(OneOf(Asymptote(C)),l)=True;Eccentricity(C)=e",
+      "coords": {}
+    },
+    {
+      "index": 720,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0720.png",
+      "answer": "sqrt(3)-1",
+      "fact_expr": "F1: Point;F2: Point;G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;LeftFocus(G) = F1;RightFocus(G) = F2;OneOf(Intersection(H, G)) = P;P: Point;IsDiameter(LineSegmentOf(F1, F2), H);H: Circle;AngleOf(P, F1, F2) = 2*AngleOf(P, F2, F1)",
+      "coords": {}
+    },
+    {
+      "index": 721,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0721.png",
+      "answer": "3*sqrt(2)",
+      "fact_expr": "G: Parabola;H: Circle;F: Point;P: Point;Expression(G) = (y = x^2/8);Expression(H) = ((x + 3)^2 + (y + 3)^2 = 16);Focus(G) = F;L: Line;TangentOfPoint(F, H) = L;TangentPoint(H, L) = P",
+      "coords": {}
+    },
+    {
+      "index": 722,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0722.png",
+      "answer": "4\n1",
+      "fact_expr": "P: Point;PointOnCurve(P, G);G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2}",
+      "coords": {}
+    },
+    {
+      "index": 723,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0723.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);x1: Number;y1: Number;P: Point;Coordinate(P) = (x1, y1);PointOnCurve(P, G);A: Point;Coordinate(A) = (3, 0);M: Point;Abs(VectorOf(A, M)) = 1;DotProduct(VectorOf(P, M), VectorOf(A, M)) = 0",
+      "coords": {
+        "A": [
+          3.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 724,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 8.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 64.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0724.png",
+      "answer": "x^2/36 - y^2/12 = 1",
+      "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/64 + y^2/16 = 1);Focus(G)=Focus(H);Expression(Asymptote(G))=(x+pm*sqrt(3)*y=0)",
+      "coords": {}
+    },
+    {
+      "index": 725,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0725.png",
+      "answer": "2*sqrt(5)",
+      "fact_expr": "C: Ellipse;a: Number;P: Point;F1:Point;F2:Point;x0:Number;a>2;Expression(C) = (y^2/4 + x^2/a^2 = 1);Coordinate(P) = (x0, 1);PointOnCurve(P, C);Focus(C)={F1,F2};Area(TriangleOf(P,F1,F2))=1",
+      "coords": {}
+    },
+    {
+      "index": 726,
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+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0726.png",
+      "answer": "2",
+      "fact_expr": "C: Ellipse;a: Number;b: Real;F1: Point;P: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Focus(C)={F1,F2};PointOnCurve(P, C);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0;Area(TriangleOf(F1, P, F2)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 727,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 728,
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+        "x_coef": 25.0,
+        "y_coef": 16.0
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+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0728.png",
+      "answer": "3/5",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 729,
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+        "x_coef": 9.0,
+        "y_coef": 5.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0729.png",
+      "answer": "sqrt(5);3",
+      "fact_expr": "G: Ellipse;O: Origin;M: Point;Expression(G) = (x^2/9 + y^2/5 = 1);PointOnCurve(M, G)",
+      "coords": {}
+    },
+    {
+      "index": 730,
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0730.png",
+      "answer": "(1/2, +oo)",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = x);A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G);Negation(IsPerpendicular(LineSegmentOf(A, B), xAxis))",
+      "coords": {}
+    },
+    {
+      "index": 731,
+      "success": true,
+      "conic_type": "ellipse",
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+        "b": 2.6457513110645907,
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+        "x_coef": 9.0,
+        "y_coef": 7.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0731.png",
+      "answer": "7/3",
+      "fact_expr": "G: Ellipse;H: Line;M: Point;O: Origin;B: Point;A: Point;Expression(G) = (x^2/9 + y^2/7 = 1);Slope(H)=-1/3;Negation(PointOnCurve(O,H));Intersection(H, G) = {A, B};MidPoint(LineSegmentOf(A,B))=M",
+      "coords": {}
+    },
+    {
+      "index": 732,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0732.png",
+      "answer": "2",
+      "fact_expr": "l: Line;G: Hyperbola;b: Number;a: Number;H: Point;I: Point;O: Origin;0 < a;a < b;c: Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(H) = (a, 0);Coordinate(I) = (0, b);HalfFocalLength(G) = c;PointOnCurve(H, l);PointOnCurve(I, l);Distance(O, l) = (3/4)*c",
+      "coords": {}
+    },
+    {
+      "index": 733,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 734,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0734.png",
+      "answer": "sqrt(13)/2",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;F2: Point;P: Point;M: Point;F1: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;L:Line;PointOnCurve(F2,L);IsPerpendicular(Asymptote(C),L);FootPoint(Asymptote(C),L)=M;Intersection(L,RightPart(C))=P;VectorOf(F2, P) = 2*VectorOf(P, M);AngleOf(F1, P, F2) = ApplyUnit(120, degree)",
+      "coords": {}
+    },
+    {
+      "index": 735,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0735.png",
+      "answer": "sqrt(2)-1",
+      "fact_expr": "F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;O: Origin;l: Line;PointOnCurve(O, l);Inclination(l) = ApplyUnit(60, degree);M: Point;OneOf(Intersection(l, C)) = M;Abs(VectorOf(M, F1) + VectorOf(M, F2)) = Abs(VectorOf(M, F1) - VectorOf(M, F2))",
+      "coords": {}
+    },
+    {
+      "index": 736,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 2.364173412322998,
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+        "converged": true,
+        "iterations": 89
+      },
+      "output_path": "results/dev/hyperbola/problem_0736.png",
+      "answer": "8",
+      "fact_expr": "C1: Hyperbola;C2: Parabola;a:Number;b:Number;p:Number;p>0;a>0;b>0;Expression(C1)=(x^2/a^2-y^2/b^2=1);Eccentricity(C1)=2;Expression(C2)=(x^2=2*(p*y));Distance(Focus(C2),Asymptote(C1))=2",
+      "coords": {}
+    },
+    {
+      "index": 737,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 738,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_ecc_mismatch",
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+        "converged": true,
+        "iterations": 2
+      },
+      "output_path": "results/dev/ellipse/problem_0738.png",
+      "answer": "{3,16/3}",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/m = 1);m: Number;Eccentricity(G) = 1/2",
+      "coords": {},
+      "validation_reasons": [
+        "ellipse_ecc_mismatch"
+      ]
+    },
+    {
+      "index": 739,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 740,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0740.png",
+      "answer": "10",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;A: Point;B: Point;y1: Number;y2: Number;Coordinate(A) = (1, y1);Coordinate(B) = (9, y2);PointOnCurve(A, G);PointOnCurve(B, G);y2>y1;y1>0;F: Point;Focus(G) = F;Abs(LineSegmentOf(B, F)) = 5*Abs(LineSegmentOf(A, F))",
+      "coords": {}
+    },
+    {
+      "index": 741,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0741.png",
+      "answer": "(-oo,2]",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);Q: Point;PointOnCurve(Q, G);P: Point;Coordinate(P) = (a, 0);a: Number;Abs(LineSegmentOf(P, Q)) >= Abs(a)",
+      "coords": {}
+    },
+    {
+      "index": 742,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0742.png",
+      "answer": "2",
+      "fact_expr": "G: Ellipse;O: Point;P: Point;F: Point;Expression(G) = (x^2/2 + y^2 = 1);Center(G) = O;LeftFocus(G) = F;PointOnCurve(P, G)",
+      "coords": {}
+    },
+    {
+      "index": 743,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0743.png",
+      "answer": "8;x = -4",
+      "fact_expr": "C: Parabola;p: Number;G: Line;Expression(C) = (y^2 = 2*(p*x));Expression(G) = (x + 2*y - 4 = 0);PointOnCurve(Focus(C), G)",
+      "coords": {}
+    },
+    {
+      "index": 744,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 745,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0745.png",
+      "answer": "pm*3*sqrt(2)/3",
+      "fact_expr": "G: Parabola;M: Point;N: Point;F: Point;Expression(G) = (y^2 = 8*x);Focus(G) = F;PointOnCurve(M, G);Abs(LineSegmentOf(M, F)) = 3;Coordinate(N) = (-2, 0)",
+      "coords": {
+        "N": [
+          -2.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 746,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 747,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 2.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0747.png",
+      "answer": "y^2 = 4*sqrt(5)*x",
+      "fact_expr": "G: Hyperbola;H: Parabola;Expression(G) = (x^2 - y^2/4 = 1);Focus(H) = RightFocus(G)",
+      "coords": {}
+    },
+    {
+      "index": 748,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 749,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0749.png",
+      "answer": "(0, sqrt(6)/3]",
+      "fact_expr": "F1: Point;F2: Point;b: Number;a: Number;a > b;b > 0;H: Line;P: Point;c: Number;G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);Expression(H) = (x = (2*a^2)/c);PointOnCurve(P, H);Abs(LineSegmentOf(P, F2)) = 2*c;LeftFocus(G) = F1;RightFocus(G) = F2",
+      "coords": {}
+    },
+    {
+      "index": 750,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0750.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "F:Point;G:Parabola;Expression(G) = (x^2=4*y);B: Point;Coordinate(B) = (0,-1);A:Point;PointOnCurve(A,G) = True;WhenMin(Abs(LineSegmentOf(A,F)) / Abs(LineSegmentOf(A,B))) = True;Focus(G) = F",
+      "coords": {
+        "B": [
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+          -1.0
+        ]
+      }
+    },
+    {
+      "index": 751,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0751.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;m: Real;Expression(G) = (y^2 = m*x);Coordinate(Focus(G))=(1/2,0)",
+      "coords": {}
+    },
+    {
+      "index": 752,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0752.png",
+      "answer": "20",
+      "fact_expr": "G: Ellipse;A: Point;B: Point;F2: Point;F1: Point;Expression(G) = (y^2/9 + x^2/a^2 = 1);Focus(G) = {F1, F2};Abs(LineSegmentOf(F1, F2)) = 8;PointOnCurve(F1, LineSegmentOf(A, B));IsChordOf(LineSegmentOf(A, B), G);a:Number",
+      "coords": {}
+    },
+    {
+      "index": 753,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 6.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 36.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0753.png",
+      "answer": "x+2*y-8=0",
+      "fact_expr": "l: Line;G: Ellipse;Expression(G) = (x^2/36 + y^2/9 = 1);Coordinate(MidPoint(InterceptChord(l,G)))=(4,2)",
+      "coords": {}
+    },
+    {
+      "index": 754,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0754.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;a: Number;H: Parabola;Expression(G) = (-y^2/4 + x^2/a^2 = 1);Expression(H) = (y^2 = 12*x);RightFocus(G) = Focus(H)",
+      "coords": {}
+    },
+    {
+      "index": 755,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0755.png",
+      "answer": "[-2/3,0)",
+      "fact_expr": "F: Point;M: Hyperbola;Expression(M) = (x^2/9 - y^2/16 = 1);LeftFocus(M) = F;O: Origin;PointOnCurve(O, l) = True;l: Line;Intersection(l,LeftPart(M)) = A;Intersection(l, RightPart(M)) = B;A: Point;B: Point",
+      "coords": {}
+    },
+    {
+      "index": 756,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0756.png",
+      "answer": "sqrt(2)+1",
+      "fact_expr": "C: Parabola;p: Number;A: Point;B: Point;D: Point;F: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;Intersection(Directrix(C), xAxis) = A;TangentOfPoint(A,C)=LineOf(A,B);TangentPoint(LineOf(A,B),C)=B;Quadrant(B)=1;PointOnCurve(D,C);Quadrant(D)=4;IsParallel(LineSegmentOf(D,F),LineSegmentOf(A,B))",
+      "coords": {}
+    },
+    {
+      "index": 757,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 2.5275303393962245,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 4.481942266905428e-06,
+        "converged": true,
+        "iterations": 95
+      },
+      "output_path": "results/dev/hyperbola/problem_0757.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/b^2 = 1);b: Number;b>0;Expression(OneOf(Asymptote(G))) = (y = 2*x)",
+      "coords": {}
+    },
+    {
+      "index": 758,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0758.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "F1: Point;F2: Point;G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G) = True;AngleOf(F1, P, F2) = ApplyUnit(60, degree)",
+      "coords": {}
+    },
+    {
+      "index": 759,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 760,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.2175992727279663,
+        "b": 2.434081792831421,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 3.452532837400213e-06,
+        "converged": true,
+        "iterations": 124
+      },
+      "output_path": "results/dev/hyperbola/problem_0760.png",
+      "answer": "x^2-y^2/4=1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;Expression(OneOf(Asymptote(G))) = (y = 2*x);Distance(Focus(G), Asymptote(G)) = 2",
+      "coords": {}
+    },
+    {
+      "index": 761,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0761.png",
+      "answer": "{2,(2*sqrt(3)/3)}",
+      "fact_expr": "F: Point;C: Hyperbola;a: Number;b: Number;a>0;b>0;l: Line;A: Point;B: Point;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;PointOnCurve(F, l);Slope(l) = a/b;l1: Line;l2: Line;Asymptote(C) = {l1, l2};Intersection(l, l1) = A;Intersection(l,l2) = B;Abs(VectorOf(A, F)) = 2*Abs(VectorOf(B, F))",
+      "coords": {}
+    },
+    {
+      "index": 762,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0762.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;m: Number;m>0;Expression(G) = (-y^2/3 + x^2/m = 1);Expression(OneOf(Asymptote(G))) = (y = x*(sqrt(3)/2))",
+      "coords": {}
+    },
+    {
+      "index": 763,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "vertical"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0763.png",
+      "answer": "1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/3 + y^2/t^2 = 1);t: Real;t>0;H: Parabola;Expression(H) = (y = x^2/8);OneOf(Focus(G)) = Focus(H)",
+      "coords": {}
+    },
+    {
+      "index": 764,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 765,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0765.png",
+      "answer": "Preserve all mathematical expressions, LaTeX formulas, symbols, and equations EXACTLY.  \nDo not add explanations, do not rewrite formulas, do not omit any content.  \n\nText:",
+      "fact_expr": "G: Hyperbola;H: Circle;Expression(G) = (x^2/9 - y^2/16 = 1);PointOnCurve(Center(H), G);PointOnCurve(RightVertex(G),H);PointOnCurve(RightFocus(G),H)",
+      "coords": {}
+    },
+    {
+      "index": 766,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0766.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "G: Line;Expression(G) = (y = (1/2)*x + 1);C: Parabola;Expression(C) = (x^2 = 4*y);A: Point;B: Point;Intersection(G, C) = {A, B};O: Origin",
+      "coords": {}
+    },
+    {
+      "index": 767,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 768,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0768.png",
+      "answer": "x - y + 2 = 0",
+      "fact_expr": "G: Parabola;l: Line;P1: Point;P2: Point;Expression(G) = (x^2 = y);IsChordOf(LineSegmentOf(P1, P2), G);PerpendicularBisector(LineSegmentOf(P1, P2)) = l;Expression(l) = (y = x + 3)",
+      "coords": {}
+    },
+    {
+      "index": 769,
+      "success": false,
+      "conic_type": "parabola",
+      "error": "validation: parabola_point_off_curve",
+      "params": {
+        "p": 14.875267028808594,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 7.913284778594971,
+        "converged": false,
+        "iterations": 500
+      },
+      "output_path": "results/dev/parabola/problem_0769.png",
+      "answer": "(-4,0)",
+      "fact_expr": "G: Parabola;a: Number;H: Point;Expression(G) = (y^2 = 2*(a*x));Coordinate(H) = (-1, 4);PointOnCurve(H, G)",
+      "coords": {
+        "H": [
+          -1.0,
+          4.0
+        ]
+      },
+      "validation_reasons": [
+        "parabola_point_off_curve"
+      ]
+    },
+    {
+      "index": 770,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0770.png",
+      "answer": "2*sqrt(3)",
+      "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2 - y^2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 771,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0771.png",
+      "answer": "2",
+      "fact_expr": "A1: Point;A2: Point;G: Ellipse;Expression(G) = (y^2 + x^2/a^2 = 1);a>1;LeftVertex(G) = A1;RightVertex(G) = A2;P: Point;PointOnCurve(P, G) = True;k1: Number;Slope(LineOf(P,A1)) = k1;k2: Number;Slope(LineOf(P,A2)) = k2;k1 * k2 = -1/4;a: Real",
+      "coords": {}
+    },
+    {
+      "index": 772,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0772.png",
+      "answer": "3",
+      "fact_expr": "G: Ellipse;H: Line;m: Number;F1: Point;A: Point;B: Point;F2: Point;Expression(G) = (x^2/4 + y^2/3 = 1);Expression(H) = (x = m);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2, H);Intersection(H, G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 773,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0773.png",
+      "answer": "2",
+      "fact_expr": "C: Hyperbola;a: Number;b: Number;G: Line;A: Point;F1: Point;F2: Point;B: Point;a>b;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F2, G);Intersection(G,RightPart(C)) = {A, B};Abs(LineSegmentOf(A, F1)) = 2*Abs(LineSegmentOf(A, F2));AngleOf(A, F1, F2) = AngleOf(F1, B, F2)",
+      "coords": {}
+    },
+    {
+      "index": 774,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0774.png",
+      "answer": "1/2",
+      "fact_expr": "P: Point;a: Number;b: Number;Coordinate(P) = (a, b);G: Hyperbola;Expression(G) = (x^2 - y^2 = 1);PointOnCurve(P, RightPart(G));Distance(P,Asymptote(G)) = (sqrt(2))",
+      "coords": {}
+    },
+    {
+      "index": 775,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 2.23606797749979,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 5.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0775.png",
+      "answer": "x^2/12-y^2/8=1",
+      "fact_expr": "H: Ellipse;Expression(H) = (x^2/25 + y^2/5 = 1);Focus(H) = Focus(G);I: Point;Coordinate(I) = (3*sqrt(2), 2);PointOnCurve(I, G);G: Hyperbola",
+      "coords": {
+        "I": [
+          4.242640687119286,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 776,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0776.png",
+      "answer": "y^2=2*x",
+      "fact_expr": "l: Line;G: Parabola;p: Number;A: Point;B: Point;F: Point;p>0;Expression(G) = (y^2 = 2*p*x);Focus(G) = F;PointOnCurve(F, l);Inclination(l) = pi/4;Intersection(l, G) = {A, B};Abs(LineSegmentOf(A, B)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 777,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0777.png",
+      "answer": "{(4,4),(4,-4)}",
+      "fact_expr": "G: Parabola;P: Point;F: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(P, G);Focus(G)=F;Abs(LineSegmentOf(P, F)) = 5",
+      "coords": {}
+    },
+    {
+      "index": 778,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0778.png",
+      "answer": "3/4",
+      "fact_expr": "l: Line;G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);Intersection(l, G) = {A, B};B: Point;A: Point;M: Point;Coordinate(M)=(1,-1);MidPoint(LineSegmentOf(A,B)) = M",
+      "coords": {
+        "M": [
+          1.0,
+          -1.0
+        ]
+      }
+    },
+    {
+      "index": 779,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0779.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;p: Real;H: Point;Expression(G) = (y^2 = 2*(p*x));Coordinate(H) = (-2, 4);PointOnCurve(H, Directrix(G))",
+      "coords": {
+        "H": [
+          -2.0,
+          4.0
+        ]
+      }
+    },
+    {
+      "index": 780,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0780.png",
+      "answer": "(1/2,1)",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;H: Circle;c: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 + (-c + x)^2 = c^2);c=sqrt(a^2-b^2);NumIntersection(H,G)=2",
+      "coords": {}
+    },
+    {
+      "index": 781,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0781.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "O: Origin;F: Point;Focus(G) = F;G: Parabola;Expression(G) = (y^2 = 4*x);l: Line;Expression(l) = (y = m*(x - 1));m: Number;Intersection(l, G) = {A, B};A: Point;B: Point;Quadrant(A) = 1;Abs(LineSegmentOf(F, A)) = 3*Abs(LineSegmentOf(F, B))",
+      "coords": {}
+    },
+    {
+      "index": 782,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0782.png",
+      "answer": "1",
+      "fact_expr": "H: Parabola;Expression(H) = (x^2 = 8*y);G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 783,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0783.png",
+      "answer": "5",
+      "fact_expr": "C: Parabola;A: Point;P: Point;F: Point;Expression(C) = (y^2 = 4*x);Coordinate(A) = (4, 2);Focus(C) = F;PointOnCurve(P, C)",
+      "coords": {
+        "A": [
+          4.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 784,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0784.png",
+      "answer": "2*sqrt(3)/3",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;B: Point;F: Point;A: Point;O: Origin;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F;L1:Line;L2:Line;Asymptote(G)={L1,L2};PointOnCurve(A,L1);PointOnCurve(B,L2);VectorOf(B,F)=2*VectorOf(F,A);DotProduct(VectorOf(O, A), VectorOf(A, B)) = 0",
+      "coords": {}
+    },
+    {
+      "index": 785,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0785.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;M: Point;N: Point;C:Circle;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(LeftFocus(G), H);IsPerpendicular(H,xAxis);Intersection(H, G) = {M, N};IsDiameter(LineSegmentOf(M,N),C);PointOnCurve(RightVertex(G),C)",
+      "coords": {}
+    },
+    {
+      "index": 786,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 787,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 788,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0788.png",
+      "answer": "sqrt(7)/2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(H) = (2, sqrt(3));PointOnCurve(H, OneOf(Asymptote(G)))",
+      "coords": {
+        "H": [
+          2.0,
+          1.7320508075688772
+        ]
+      }
+    },
+    {
+      "index": 789,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0789.png",
+      "answer": "sqrt(5)/3",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G) = True;DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0;Tan(AngleOf(P, F1, F2)) = 2",
+      "coords": {}
+    },
+    {
+      "index": 790,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0790.png",
+      "answer": "4/3",
+      "fact_expr": "C: Hyperbola;a: Number;b: Number;a > 0;b > 0;Expression(C) = (x^2/a^2 - y^2/b^2 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;l: Line;PointOnCurve(F1, l);l1, l2: Line;Asymptote(C) = {l1, l2};A: Point;B: Point;Intersection(l, l1) = A;Intersection(l, l2) = B;3*VectorOf(F1, A) = 2*VectorOf(A, B);DotProduct(VectorOf(F1, B), VectorOf(F2, B)) = 0",
+      "coords": {}
+    },
+    {
+      "index": 791,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0791.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;A: Point;B: Point;PointOnCurve(A,G) = True;PointOnCurve(B, G) = True;AngleOf(A, F, B) = pi/2;IsChordOf(LineSegmentOf(A, B), G) = True;MidPoint(LineSegmentOf(A,B)) = M;M: Point;Projection(M,Directrix(G)) = N;N: Point",
+      "coords": {}
+    },
+    {
+      "index": 792,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0792.png",
+      "answer": "((2-sqrt(2))/4, 1)",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;OneOf(Focus(G)) = F;H: Line;Expression(H) = (y = k*x);k: Number;A: Point;B: Point;Intersection(H, G) = {A, B};AngleOf(A, F, B) = ApplyUnit(135, degree);e: Number;Eccentricity(G) = e",
+      "coords": {}
+    },
+    {
+      "index": 793,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 794,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0794.png",
+      "answer": "16",
+      "fact_expr": "C: Parabola;G: Line;A: Point;B: Point;F:Point;Expression(C) = (y^2 = 4*x);Focus(C)=F;PointOnCurve(F,G);Slope(G) = sqrt(3)/3;Intersection(G,C)={A,B}",
+      "coords": {}
+    },
+    {
+      "index": 795,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0795.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "E: Hyperbola;b: Number;a: Number;M: Point;F1: Point;F2: Point;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(E)=F1;RightFocus(E)=F2;PointOnCurve(M, E);IsPerpendicular(LineSegmentOf(M,F1),xAxis);Sin(AngleOf(M, F2, F1)) = 1/3",
+      "coords": {}
+    },
+    {
+      "index": 796,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0796.png",
+      "answer": "1",
+      "fact_expr": "G: Parabola;P: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(P, G) ;Focus(G) = F;F:Point",
+      "coords": {}
+    },
+    {
+      "index": 797,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0797.png",
+      "answer": "sqrt(7)",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;M: Point;N: Point;PointOnCurve(M, LeftPart(C)) = True;PointOnCurve(N, RightPart(C)) = True;IsParallel(LineSegmentOf(M, N), LineSegmentOf(F1, F2)) = True;Abs(LineSegmentOf(M, N)) = Abs(LineSegmentOf(F1, F2))/2;Intersection(LineSegmentOf(F1, N), C) = Q;Q: Point;Abs(LineSegmentOf(F1, Q)) = (2/5)*Abs(LineSegmentOf(F1, N))",
+      "coords": {}
+    },
+    {
+      "index": 798,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.449489742783178,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0798.png",
+      "answer": "16",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/6 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, l) = True;l: Line;Intersection(l, LeftPart(G)) = {A, B};A: Point;B: Point",
+      "coords": {}
+    },
+    {
+      "index": 799,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0799.png",
+      "answer": "5/7",
+      "fact_expr": "C1: Ellipse;Expression(C1) = (x^2/36 + y^2/b^2 = 1);b: Number;F1: Point;F2: Point;Focus(C1) = {F1, F2};C2: Parabola;Expression(C2) = (y^2 = 2*p*x);p: Number;p > 0;Focus(C2) = F2;P: Point;Intersection(C1, C2) = P;Abs(LineSegmentOf(P, F1)) = 7",
+      "coords": {}
+    },
+    {
+      "index": 800,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.463192105293274,
+        "b": 2.070901870727539,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 1.4661259228887502e-06,
+        "converged": true,
+        "iterations": 106
+      },
+      "output_path": "results/dev/hyperbola/problem_0800.png",
+      "answer": "y=pm*sqrt(2)*x",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(C) = sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 801,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 802,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0802.png",
+      "answer": "ApplyUnit(90, degree)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1))*Abs(LineSegmentOf(P, F2)) = 32",
+      "coords": {}
+    },
+    {
+      "index": 803,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 804,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 3.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0804.png",
+      "answer": "8",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 12*y);F: Point;Focus(G) = F;P: Point;Coordinate(P) = (2, 1);A: Point;B: Point;l: Line;PointOnCurve(P, l);Intersection(l, G) = {A, B};MidPoint(LineSegmentOf(A, B)) = P",
+      "coords": {
+        "P": [
+          2.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 805,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0805.png",
+      "answer": "Preserve all mathematical expressions, LaTeX formulas, symbols, and equations EXACTLY.  \nDo not add explanations, do not rewrite formulas, do not omit any content.  \n\nText:",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;G: Circle;N: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(N) = (0, 2);Eccentricity(C) = sqrt(2)/2;IsIntersect(G,C) = True;Center(G) = N;Max(Radius(G)) = sqrt(26)",
+      "coords": {
+        "N": [
+          0.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 806,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0806.png",
+      "answer": "(1, 1/4)",
+      "fact_expr": "G: Parabola;Q: Point;P: Point;Expression(G) = (x^2 = 4*y);Coordinate(Q) = (1, 2);PointOnCurve(P, G);WhenMin(Distance(P, Q) + Distance(P, Focus(G)))",
+      "coords": {
+        "Q": [
+          1.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 807,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0807.png",
+      "answer": "y^2=4*x",
+      "fact_expr": "C: Parabola;G: Line;A: Point;B: Point;P: Point;O:Origin;Expression(G) = (y = x);Coordinate(P) = (2, 2);Vertex(C)=O;PointOnCurve(Focus(C), xAxis);Intersection(G, C) = {A, B};MidPoint(LineSegmentOf(A, B)) = P",
+      "coords": {
+        "P": [
+          2.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 808,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 0.929953396320343,
+        "b": 2.629857301712036,
+        "orientation": "vertical"
+      },
+      "optimization": {
+        "final_loss": 4.594453912432073e-06,
+        "converged": true,
+        "iterations": 175
+      },
+      "output_path": "results/dev/hyperbola/problem_0808.png",
+      "answer": "y = pm*(sqrt(2)/4)*x",
+      "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (y^2/16 - x^2/m = 1);Eccentricity(G) = 3",
+      "coords": {}
+    },
+    {
+      "index": 809,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0809.png",
+      "answer": "4",
+      "fact_expr": "C: Parabola;P: Point;F: Point;x0:Number;y0:Number;Expression(C) = (y^2 = 4*x);Coordinate(P) = (x0, y0);Focus(C) = F;PointOnCurve(P, C);Abs(LineSegmentOf(P, F)) = 5",
+      "coords": {}
+    },
+    {
+      "index": 810,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0810.png",
+      "answer": "24/7",
+      "fact_expr": "l: Line;G: Ellipse;A: Point;B: Point;Expression(G) = (x^2/4 + y^2/3 = 1);Expression(l) = (x - y + 1 = 0);Intersection(l, G) = {A, B};IsChordOf(LineSegmentOf(A,B),G)",
+      "coords": {}
+    },
+    {
+      "index": 811,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0811.png",
+      "answer": "3/2",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(C) = F;P: Point;PointOnCurve(P, C);Abs(LineSegmentOf(P, F)) = 5*a;AngleOf(P, F, O) = ApplyUnit(120, degree);O: Origin",
+      "coords": {}
+    },
+    {
+      "index": 812,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0812.png",
+      "answer": "x^2/2-y^2=1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Length(ImageinaryAxis(G)) = 2;FocalLength(G) = 2*sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 813,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0813.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Parabola;A: Point;P: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (0, 1);PointOnCurve(P, G);Distance(P,Directrix(G))=d;d:Number",
+      "coords": {
+        "A": [
+          0.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 814,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_point_off_curve",
+      "params": {
+        "a": 1.9960020780563354,
+        "b": 1.4970015287399292,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "iterations": 2
+      },
+      "output_path": "results/dev/hyperbola/problem_0814.png",
+      "answer": "2*sqrt(13)",
+      "fact_expr": "G: Hyperbola;a: Number;H: Point;Expression(G) = (-y^2/4 + x^2/a^2 = 1);Coordinate(H) = (-3*sqrt(2), 2);PointOnCurve(H, G)",
+      "coords": {
+        "H": [
+          -4.242640687119286,
+          2.0
+        ]
+      },
+      "validation_reasons": [
+        "hyperbola_point_off_curve"
+      ]
+    },
+    {
+      "index": 815,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0815.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;m: Real;Expression(G) = (-y^2/4 + x^2/m = 1);Expression(OneOf(Asymptote(G))) = (y = x)",
+      "coords": {}
+    },
+    {
+      "index": 816,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0816.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 1);H: Line;PointOnCurve(Focus(G), H);IsPerpendicular(H, xAxis);A: Point;B: Point;Intersection(H, G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 817,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "y",
+        "x_coef": 1.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0817.png",
+      "answer": "2*sqrt(3)",
+      "fact_expr": "C: Curve;P: Point;Coordinate(P) = (x1, y1);Expression(C) = (x^2 + y^2/3 = 1);x1:Number;y1:Number;PointOnCurve(P, C)",
+      "coords": {}
+    },
+    {
+      "index": 818,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0818.png",
+      "answer": "Preserve all mathematical expressions, LaTeX formulas, symbols, and equations EXACTLY.  \nDo not add explanations, do not rewrite formulas, do not omit any content.  \n\nText:",
+      "fact_expr": "G: Ellipse;P: Point;M: Point;N: Point;Expression(G) = (x^2/4 + y^2/3 = 1);Endpoint(MajorAxis(G))={M,N};PointOnCurve(P,G);Negation(M=P);Negation(N=P)",
+      "coords": {}
+    },
+    {
+      "index": 819,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0819.png",
+      "answer": "1",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;A: Point;B: Point;M: Point;N: Point;F: Point;p>0;Focus(G)=F;PointOnCurve(A,G);PointOnCurve(B,G);AngleOf(A, F, B) = ApplyUnit(60, degree);IsChordOf(LineSegmentOf(A,B),G);MidPoint(LineSegmentOf(A,B))=M;IsPerpendicular(LineSegmentOf(M,N),Directrix(G));FootPoint(LineSegmentOf(M,N),Directrix(G))=N;PointOnCurve(M, LineSegmentOf(M, N))",
+      "coords": {}
+    },
+    {
+      "index": 820,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 821,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 822,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0822.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "E: Ellipse;b: Number;a: Number;A: Point;B: Point;O: Origin;M: Point;F1:Point;a > b;b > 0;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);RightVertex(E)=A;UpperVertex(E)=B;LeftFocus(E)=F1;PointOnCurve(M, E);L:Line;PointOnCurve(M,L);IsPerpendicular(L,xAxis);PointOnCurve(F1,L);IsParallel(LineSegmentOf(A, B),LineSegmentOf(O,M))",
+      "coords": {}
+    },
+    {
+      "index": 823,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "down"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0823.png",
+      "answer": "2*y - 1 = 0",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = -2*y)",
+      "coords": {}
+    },
+    {
+      "index": 824,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0824.png",
+      "answer": "2022",
+      "fact_expr": "G: Parabola;P: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(P) = (2021, t);PointOnCurve(P, G);Focus(G) = F;t:Number",
+      "coords": {}
+    },
+    {
+      "index": 825,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 826,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 827,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0827.png",
+      "answer": "(sqrt(2)-1, 1)",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;c: Number;Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);a/Sin(AngleOf(P, F1, F2)) = c/Sin(AngleOf(P, F2, F1))",
+      "coords": {}
+    },
+    {
+      "index": 828,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0828.png",
+      "answer": "(pm*sqrt(2), 0)\ny = pm*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 829,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0829.png",
+      "answer": "1/4",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2 + x^2/a^2 = 1);F1:Point;F2:Point;LeftFocus(G)=F1;RightFocus(G)=F2;a: Number;a>1;A:Point;B:Point;M:Point;N:Point;PointOnCurve(M, G);PointOnCurve(N, G);Max(Area(TriangleOf(M, A, B))) = 4*(1 + sqrt(2)) ;LeftVertex(G)=A;UpperVertex(G)=B",
+      "coords": {}
+    },
+    {
+      "index": 830,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 0.5,
+        "b": 0.8660254037844386,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0830.png",
+      "answer": "x^2/5 + y^2/4 = 1",
+      "fact_expr": "l: Line;G: Hyperbola;H: Ellipse;P: Point;Expression(G) = (12*x^2 - 4*y^2 = 3);Expression(l) = (y = x + 3);PointOnCurve(P, l);PointOnCurve(P, H);Focus(G) = Focus(H);WhenMin(MajorAxis(H))",
+      "coords": {}
+    },
+    {
+      "index": 831,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0831.png",
+      "answer": "sqrt(3)/2 + 1",
+      "fact_expr": "G: Parabola;H: Line;A: Point;F: Point;B: Point;Expression(G) = (y^2 = x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 832,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0832.png",
+      "answer": "2",
+      "fact_expr": "l: Line;G: Parabola;p: Number;O: Origin;A: Point;B: Point;p>0;Expression(G) = (y^2 = 2*(p*x));F: Point;Focus(G) = F;PointOnCurve(F, l);Slope(l) = 2;Intersection(l, G) = {A, B};Area(TriangleOf(O, A, B)) = sqrt(5)",
+      "coords": {}
+    },
+    {
+      "index": 833,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0833.png",
+      "answer": "32",
+      "fact_expr": "C: Ellipse;P: Point;F2: Point;F1: Point;Expression(C) = (x^2/25 + y^2/16 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);DotProduct(VectorOf(P, F1),VectorOf(P, F2))=0",
+      "coords": {}
+    },
+    {
+      "index": 834,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 835,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 836,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 837,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 2.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0837.png",
+      "answer": "8*sqrt(2)/3",
+      "fact_expr": "G: Hyperbola;H: Line;Expression(G) = (x^2 - y^2/4 = 1);Expression(H) = (y = x + 1)",
+      "coords": {}
+    },
+    {
+      "index": 838,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0838.png",
+      "answer": "{2,25/2}",
+      "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2/5 + y^2/m = 1);Eccentricity(G) = e ;e:Number;e= sqrt(15)/5",
+      "coords": {}
+    },
+    {
+      "index": 839,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0839.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "G: Ellipse;H: Circle;P: Point;M: Point;x1:Number;y1:Number;l1:Line;Expression(G) = (x^2/25 + y^2/16 = 1);Expression(H) = (y^2 + (x - 3)^2 = 1);Coordinate(P) = (x1, y1);PointOnCurve(P, G);TangentOfPoint(P,H)=l1;TangentPoint(l1,H)=M",
+      "coords": {}
+    },
+    {
+      "index": 840,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.1209959984138265,
+        "b": 2.1209959984138265,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 8.099017116797696e-06,
+        "converged": true,
+        "iterations": 98
+      },
+      "output_path": "results/dev/hyperbola/problem_0840.png",
+      "answer": "x^2/4 - y^2/5 = 1",
+      "fact_expr": "C: Hyperbola;F1: Point;F2: Point;Coordinate(F1) = (-3, 0);Coordinate(F2) = (3, 0);Focus(C) = {F1,F2};G: Line;Expression(G) = (y = x*(sqrt(5)/2));OneOf(Asymptote(C)) = G",
+      "coords": {
+        "F1": [
+          -3.0,
+          0.0
+        ],
+        "F2": [
+          3.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 841,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0841.png",
+      "answer": "1",
+      "fact_expr": "G: Parabola;p: Number;H: Line;N: Point;F: Point;M: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (y = 2);Focus(G) = F;Intersection(H, yAxis) = M;Intersection(H,G)=N;4*Abs(LineSegmentOf(N, F)) = 5*Abs(LineSegmentOf(M, N))",
+      "coords": {}
+    },
+    {
+      "index": 842,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0842.png",
+      "answer": "y^2 = -44*x",
+      "fact_expr": "G: Parabola;a: Number;M: Point;a<0;Expression(G) = (y^2 = a*x);Coordinate(M) = (5, 3);Distance(M, Directrix(G)) = 6",
+      "coords": {
+        "M": [
+          5.0,
+          3.0
+        ]
+      }
+    },
+    {
+      "index": 843,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0843.png",
+      "answer": "8",
+      "fact_expr": "H: Parabola;Expression(H) = (x^2 = a*y);a: Number;G: Hyperbola;Expression(G) = (-x^2 + y^2 = 2);Focus(H) = UpperFocus(G)",
+      "coords": {}
+    },
+    {
+      "index": 844,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0844.png",
+      "answer": "(1/4, 1)",
+      "fact_expr": "C: Parabola;B: Point;Q: Point;F: Point;Expression(C) = (y^2 = 4*x);Coordinate(B) = (4, 1);PointOnCurve(Q, C);Focus(C) = F;WhenMin(Distance(Q, B) + Distance(Q, F))",
+      "coords": {
+        "B": [
+          4.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 845,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 0.9999431050898532,
+        "b": 1.731428600448389,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 6.486633488920056e-05,
+        "converged": true,
+        "iterations": 80
+      },
+      "output_path": "results/dev/hyperbola/problem_0845.png",
+      "answer": "x^2 - y^2/3 = 1",
+      "fact_expr": "E: Hyperbola;Eccentricity(E) = 2;F1: Point;F2: Point;Focus(E) = {F1, F2};Coordinate(F1) = (-2, 0);Coordinate(F2) = (2, 0)",
+      "coords": {
+        "F1": [
+          -2.0,
+          0.0
+        ],
+        "F2": [
+          2.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 846,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0846.png",
+      "answer": "16",
+      "fact_expr": "l: Line;C: Parabola;A: Point;B: Point;Expression(C) = (y^2 = 4*x);Intersection(l, C) = {A, B};PointOnCurve(Focus(C),l);Inclination(l)=pi/6",
+      "coords": {}
+    },
+    {
+      "index": 847,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0847.png",
+      "answer": "12",
+      "fact_expr": "G: Hyperbola;m: Number;H: Parabola;p: Real;Expression(G) = (x^2/(16 - m) - y^2/(m + 20) = 1);p>0;Expression(H) = (y^2 = 2*(p*x));Focus(H) = RightFocus(G)",
+      "coords": {}
+    },
+    {
+      "index": 848,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 849,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 850,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 851,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.2175992727279663,
+        "b": 2.434081792831421,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 3.452532837400213e-06,
+        "converged": true,
+        "iterations": 124
+      },
+      "output_path": "results/dev/hyperbola/problem_0851.png",
+      "answer": "x^2/4 - y^2/16 = 1",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;l:Line;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);OneOf(Asymptote(C))=l;Expression(l)=(y=2*x);Length(RealAxis(C))<6",
+      "coords": {}
+    },
+    {
+      "index": 852,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 853,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0853.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;H: Circle;Expression(H) = (y^2 + (x - 3)^2 = 16);IsTangent(Directrix(G), H)",
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+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0854.png",
+      "answer": "y^2/9 - x^2/27 = 1",
+      "fact_expr": "G: Hyperbola;H: Parabola;Expression(H) = (x^2 = 24*y);OneOf(Focus(G)) = Focus(H);Inclination(OneOf(Asymptote(G)))=ApplyUnit(30,degree)",
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+    {
+      "index": 855,
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "results/dev/parabola/problem_0855.png",
+      "answer": "sqrt(5)/5",
+      "fact_expr": "C: Parabola;p: Number;B: Point;A: Point;M: Point;F:Point;p>0;Expression(C) = (y^2 = 2*(p*x));Coordinate(M) = (2, -1);PointOnCurve(A, C);PointOnCurve(B, C);Focus(C)=F;Distance(A,F)+Distance(B,F)=6;MidPoint(LineSegmentOf(A,B)) = M;Negation(A=B)",
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+          -1.0
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+      "index": 856,
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+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 857,
+      "success": true,
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+        "b": 1.5,
+        "orientation": "vertical"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0857.png",
+      "answer": "2*sqrt(3)/3",
+      "fact_expr": "C: Hyperbola;a: Number;b: Number;a>0;b>0;Expression(C) = (-x^2/b^2 + y^2/a^2 = 1);Inclination(OneOf(Asymptote(C)))=ApplyUnit(60,degree)",
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+    {
+      "index": 858,
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+      "optimization": {
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+        "iterations": 94
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+      "output_path": "results/dev/hyperbola/problem_0858.png",
+      "answer": "4/5",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/(m + 4) + x^2/(2*m) = 1);m: Real;Expression(OneOf(Asymptote(G))) = (y = sqrt(3)*x)",
+      "coords": {}
+    },
+    {
+      "index": 859,
+      "success": true,
+      "conic_type": "hyperbola",
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+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0859.png",
+      "answer": "x^2/4-y^2/16=1\nsqrt(5)",
+      "fact_expr": "C: Hyperbola;a: Number;P: Point;F1: Point;F2: Point;a>0;Expression(C) = (-y^2/16 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);Abs(LineSegmentOf(P, F1)) - Abs(LineSegmentOf(P, F2)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 860,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0860.png",
+      "answer": "9/4",
+      "fact_expr": "G: Parabola;Expression(G) = (y = x^2/2);H: Ellipse;Expression(H) = (x^2/2 + y^2/m = 1);m: Number;Focus(G) = OneOf(Focus(H))",
+      "coords": {}
+    },
+    {
+      "index": 861,
+      "success": true,
+      "conic_type": "parabola",
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0861.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);P: Point;PointOnCurve(P, G);A: Point;Coordinate(A) = (0, -1);H: Line;Expression(H) = (x = -1)",
+      "coords": {
+        "A": [
+          0.0,
+          -1.0
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+    },
+    {
+      "index": 862,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0862.png",
+      "answer": "2*x-y-1=0",
+      "fact_expr": "G: Parabola;A: Point;B: Point;P: Point;Expression(G) = (y^2 = 4*x);IsChordOf(LineSegmentOf(A,B),G);Coordinate(P) = (1, 1);MidPoint(LineSegmentOf(A,B))=P",
+      "coords": {
+        "P": [
+          1.0,
+          1.0
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+      }
+    },
+    {
+      "index": 863,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0863.png",
+      "answer": "x=-1",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x)",
+      "coords": {}
+    },
+    {
+      "index": 864,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0864.png",
+      "answer": "(2, pm*2*sqrt(2))",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);P: Point;PointOnCurve(P, G);Distance(P, Focus(G)) = 3",
+      "coords": {}
+    },
+    {
+      "index": 865,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 866,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0866.png",
+      "answer": "sqrt(3)/2",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;H: Line;Expression(H) = (-sqrt(a*b) + a*x + b*y = 0);G: Circle;Expression(G) = (x^2 + y^2 = 2/5);IsTangent(H, G)",
+      "coords": {}
+    },
+    {
+      "index": 867,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 868,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0868.png",
+      "answer": "1/2",
+      "fact_expr": "G: Parabola;A: Point;F: Point;B: Point;lambda:Number;Expression(G) = (x^2=4*y);Focus(G) = F;PointOnCurve(A,G);PointOnCurve(B,G);VectorOf(A, F) = lambda*VectorOf(F, B);Abs(VectorOf(A, F)) = 3/2",
+      "coords": {}
+    },
+    {
+      "index": 869,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 870,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.5,
+        "direction": "down"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0870.png",
+      "answer": "y= 3/2",
+      "fact_expr": "G: Parabola;Expression(G) = (y = -x^2/6)",
+      "coords": {}
+    },
+    {
+      "index": 871,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0871.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "G: Hyperbola;H: Circle;Expression(G) = (x^2 - y^2 = 1);Expression(H) = (y^2 + (x - 2)^2 = 4)",
+      "coords": {}
+    },
+    {
+      "index": 872,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.25,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0872.png",
+      "answer": "15/4",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 5*x);A: Point;B: Point;PointOnCurve(A, G) = True;PointOnCurve(B, G) = True;Distance(A, Focus(G)) + Distance(B, Focus(G)) = 10",
+      "coords": {}
+    },
+    {
+      "index": 873,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 874,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0874.png",
+      "answer": "4",
+      "fact_expr": "l: Line;G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;F: Point;Focus(G) = F;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, G) = {A, B};Negation(A=B);y0: Number;Coordinate(A) = (2, y0);Abs(LineSegmentOf(A, F)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 875,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0875.png",
+      "answer": "x^2/4 - y^2/2 = 1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/2 - y^2 = 1);Z: Hyperbola;Asymptote(Z) = Asymptote(G);H: Point;Coordinate(H) = (2, 0);PointOnCurve(H, Z)",
+      "coords": {
+        "H": [
+          2.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 876,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0876.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;e: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(P,G);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2));Area(TriangleOf(P, F1, F2)) = 2*(a*b);Eccentricity(G) = e",
+      "coords": {}
+    },
+    {
+      "index": 877,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0877.png",
+      "answer": "(pm*sqrt(7),0)",
+      "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (x^2/4 - y^2/m = 1);Expression(Asymptote(G)) = (y = pm*(x*(sqrt(3)/2)))",
+      "coords": {}
+    },
+    {
+      "index": 878,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0878.png",
+      "answer": "[2,6]",
+      "fact_expr": "G: Ellipse;O: Point;P: Point;F: Point;Center(G)=O;Expression(G) = (x^2/4 + y^2/3 = 1);LeftFocus(G)=F;PointOnCurve(P, G)",
+      "coords": {}
+    },
+    {
+      "index": 879,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 880,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0880.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;p: Number;H: Ellipse;Expression(G) = (y^2 = 2*p*x);Expression(H) = (x^2/5 + y^2/p^2 = 1);PointOnCurve(LeftFocus(H),Directrix(G))",
+      "coords": {}
+    },
+    {
+      "index": 881,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0881.png",
+      "answer": "3*sqrt(2)/4",
+      "fact_expr": "G: Hyperbola;m: Number;m>0;Expression(G) = (x^2/9 - y^2/m^2 = 1);Distance(OneOf(Vertex(G)), OneOf(Asymptote(G))) = 1",
+      "coords": {}
+    },
+    {
+      "index": 882,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0882.png",
+      "answer": "3*sqrt(3)",
+      "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/16 + y^2/9 = 1);PointOnCurve(P, G);Focus(G) = {F1, F2};AngleOf(F1, P, F2) = ApplyUnit(60, degree)",
+      "coords": {}
+    },
+    {
+      "index": 883,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 884,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 885,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0885.png",
+      "answer": "(sqrt(5) - 1)/2",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;G: Circle;P: Point;F1: Point;F2: Point;B: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;LowerVertex(C) = B;PointOnCurve(F1, G);PointOnCurve(F2, G);PointOnCurve(B, G);Intersection(G, C) = P;IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(F1, F2))",
+      "coords": {}
+    },
+    {
+      "index": 886,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 887,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 888,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "vertical"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0888.png",
+      "answer": "y=pm*sqrt(2)*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/2 + y^2/4 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 889,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0889.png",
+      "answer": "(1, 2)",
+      "fact_expr": "G: Hyperbola;a: Number;b: Number;P: Point;F1: Point;F2: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);Negation(Vertex(G) = P);AngleOf(P, F2, F1) = 3*AngleOf(P, F1, F2)",
+      "coords": {}
+    },
+    {
+      "index": 890,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0890.png",
+      "answer": "x^2/3-y^2=1",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 8*x);Expression(OneOf(Asymptote(G))) = (y = x*(sqrt(3)/3));PointOnCurve(OneOf(Focus(G)), Directrix(H))",
+      "coords": {}
+    },
+    {
+      "index": 891,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 892,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 893,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0893.png",
+      "answer": "3",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);x1: Number;y1: Number;M: Point;Coordinate(M) = (x1, y1);PointOnCurve(M, G);Distance(M, Focus(G)) = 3;O: Origin",
+      "coords": {}
+    },
+    {
+      "index": 894,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0894.png",
+      "answer": "x^2/(1/16)-y^2/(1/4)=1",
+      "fact_expr": "G: Hyperbola;H: Parabola;Expression(H) = (y = x^2);Expression(OneOf(Asymptote(G)))=(x+2*y=0);NumIntersection(G,Directrix(H)) = 1",
+      "coords": {}
+    },
+    {
+      "index": 895,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 896,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 8.0,
+        "b": 6.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0896.png",
+      "answer": "31",
+      "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/64 - y^2/36 = 1);PointOnCurve(P, G);Focus(G) = {F1, F2};Abs(LineSegmentOf(P, F1)) = 15",
+      "coords": {}
+    },
+    {
+      "index": 897,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0897.png",
+      "answer": "(x - 4)^2 + (y - 4)^2 = 25",
+      "fact_expr": "G: Parabola;p: Number;C: Circle;A: Point;F: Point;m: Number;p > 0;m > 0;Expression(G) = (y^2 = 2*(p*x));Coordinate(A) = (4, m);PointOnCurve(A, G);Focus(G) = F;Center(C) = A;Radius(C) = Abs(LineSegmentOf(A, F));Length(InterceptChord(C, yAxis)) = 6",
+      "coords": {}
+    },
+    {
+      "index": 898,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0898.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;H: Circle;a: Number;Expression(G) = (x^2 - y^2 = 1);a > 0;Expression(H) = (y^2 + (x - 1)^2 = a^2);NumIntersection(G, H) = 3",
+      "coords": {}
+    },
+    {
+      "index": 899,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0899.png",
+      "answer": "Preserve all mathematical expressions, LaTeX formulas, symbols, and equations EXACTLY.  \nDo not add explanations, do not rewrite formulas, do not omit any content.  \n\nText:",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;C: Circle;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(C) = (-8*x + x^2 + y^2 + 12 = 0);IsTangent(Asymptote(G), C);RightFocus(G) = Center(C)",
+      "coords": {}
+    },
+    {
+      "index": 900,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0900.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "G: Hyperbola;b: Number;H: Parabola;Expression(G) = (x^2/4 - y^2/b^2 = 1);Expression(H) = (y^2 = 12*x);RightFocus(G) = Focus(H)",
+      "coords": {}
+    },
+    {
+      "index": 901,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0901.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "F: Point;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;RightFocus(G) = F;O: Origin;PointOnCurve(O, l) = True;l: Line;Intersection(l, G) = {M, N};M: Point;N: Point;DotProduct(VectorOf(M, F), VectorOf(N, F)) = 0;Area(TriangleOf(M, N, F)) = a*b",
+      "coords": {}
+    },
+    {
+      "index": 902,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0902.png",
+      "answer": "4*sqrt(7)/7",
+      "fact_expr": "P: Point;PointOnCurve(P,C1) = True;C1: Curve;Expression(C1) = (y^2 = 8*x);Q: Point;PointOnCurve(Q,C2) = True;C2: Curve;Expression(C2) = (y^2 + (x - 2)^2 = 1);O: Origin",
+      "coords": {}
+    },
+    {
+      "index": 903,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0903.png",
+      "answer": "3",
+      "fact_expr": "K: Parabola;A: Point;B: Point;F: Point;Expression(K) = (x^2 = 4*y);PointOnCurve(A,K);PointOnCurve(B,K);Negation(YCoordinate(A)=YCoordinate(B));Abs(LineSegmentOf(A, F)) + Abs(LineSegmentOf(B, F)) = 4;Focus(K)=F",
+      "coords": {}
+    },
+    {
+      "index": 904,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0904.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 - y^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P,G);Abs(LineSegmentOf(P,F1))+Abs(LineSegmentOf(P,F2))=6",
+      "coords": {}
+    },
+    {
+      "index": 905,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0905.png",
+      "answer": "[sqrt(2), sqrt(3)+1]",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(C) = F;G: Line;Expression(G) = (y = k*x);k: Number;In(k, [sqrt(3)/3, sqrt(3)]);A: Point;B: Point;Intersection(G, C) = {A, B};IsPerpendicular(LineSegmentOf(A, F), LineSegmentOf(B, F))",
+      "coords": {}
+    },
+    {
+      "index": 906,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0906.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);P: Point;PointOnCurve(P, RightPart(G));C1: Circle;Expression(C1) = ((x + 5)^2 + y^2 = 4);C2: Circle;Expression(C2) = ((x - 5)^2 + y^2 = r^2);r: Number;r>0;Z1: Line;Z2: Line;TangentOfPoint(P, C1) = Z1;TangentOfPoint(P, C2) = Z2;M: Point;TangentPoint(Z1, C1) = M;N: Point;TangentPoint(Z2, C2) = N;Min(Abs(LineSegmentOf(P, M))^2 - Abs(LineSegmentOf(P, N))^2) = 58",
+      "coords": {}
+    },
+    {
+      "index": 907,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0907.png",
+      "answer": "2",
+      "fact_expr": "C: Parabola;G: Line;B: Point;A: Point;P: Point;F: Point;Expression(C) = (y^2 = 8*x);Focus(C)=F;PointOnCurve(F, G);Intersection(G, C) = {A, B};Intersection(PerpendicularBisector(LineSegmentOf(A,B)), xAxis) = P",
+      "coords": {}
+    },
+    {
+      "index": 908,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0908.png",
+      "answer": "(1, 2] + [3, 6)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);P: Point;PointOnCurve(P, RightPart(G));Distance(P, LeftFocus(G)) = 6*Distance(P, RightDirectrix(G))",
+      "coords": {}
+    },
+    {
+      "index": 909,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0909.png",
+      "answer": "x^2+(y-1)^2=3",
+      "fact_expr": "G: Parabola;H: Circle;I: Line;Expression(G) = (x^2 = 4*y);Expression(I) = (y = x + 3);Center(H)=Focus(G);Length(InterceptChord(I,H))=2",
+      "coords": {}
+    },
+    {
+      "index": 910,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0910.png",
+      "answer": "2",
+      "fact_expr": "G: Ellipse;t: Number;t>1;Expression(G) = (y^2 + x^2/t = 1);F:Point;A:Point;B:Point;OneOf(Focus(G))=F;Endpoint(MinorAxis(G))={A,B};Area(TriangleOf(F,A,B))=1",
+      "coords": {}
+    },
+    {
+      "index": 911,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0911.png",
+      "answer": "(-2/3, 1/3)",
+      "fact_expr": "G: Ellipse;H: Line;Expression(G) = (x^2 + 2*y^2 = 4);Expression(H) = (y = x + 1)",
+      "coords": {}
+    },
+    {
+      "index": 912,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0912.png",
+      "answer": "3/2",
+      "fact_expr": "l: Line;C: Parabola;p: Number;A: Point;F: Point;M: Point;B: Point;p>0;Expression(C) = (x^2 = 2*(p*y));Focus(C) = F;PointOnCurve(F, l);Intersection(l,C)={A,B};Intersection(l,Directrix(C))= M;VectorOf(A, F) = VectorOf(F, M)",
+      "coords": {}
+    },
+    {
+      "index": 913,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0913.png",
+      "answer": "2",
+      "fact_expr": "G: Line;C: Curve;A: Point;B: Point;F: Point;Expression(C) = (y^2 = 2*x);Inclination(G) = ApplyUnit(60, degree);PointOnCurve(F, G);Intersection(G, C) = {A, B};Quadrant(A) = 1;Negation(A = B);Focus(C) = F",
+      "coords": {}
+    },
+    {
+      "index": 914,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0914.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;O: Origin;F: Point;A:Point;e: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F;IsDiameter(LineSegmentOf(O,F),H);Intersection(OneOf(Asymptote(G)),H)=A;Negation(A=O);IsPerpendicular(LineSegmentOf(A,MidPoint(LineSegmentOf(O,F))),LineSegmentOf(O,F));Eccentricity(G) = e",
+      "coords": {}
+    },
+    {
+      "index": 915,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0915.png",
+      "answer": "1",
+      "fact_expr": "G: Parabola;p: Number;A: Point;B: Point;M: Point;N: Point;F: Point;p>0;Expression(G) = (y^2 = 2*p*x);Focus(G) = F;PointOnCurve(A,G);PointOnCurve(B,G);AngleOf(A, F, B) = ApplyUnit(60, degree);IsChordOf(LineSegmentOf(A,B),G);MidPoint(LineSegmentOf(A,B))=M;PointOnCurve(M,LineSegmentOf(M,N));IsPerpendicular(LineSegmentOf(M,N),Directrix(G));FootPoint(LineSegmentOf(M,N),Directrix(G))=N",
+      "coords": {}
+    },
+    {
+      "index": 916,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 917,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0917.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;p: Number;H: LineSegment;Expression(G) = (y^2 = 2*p*x);Coordinate(Z) = (3, 1);Z: Point;MidPoint(H) = Z;IsChordOf(H,G);Slope(OverlappingLine(H))=2",
+      "coords": {
+        "Z": [
+          3.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 918,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 2.8284271247461903,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0918.png",
+      "answer": "y^2/8 - x^2/4 = 1",
+      "fact_expr": "C1: Hyperbola;C2: Hyperbola;H: Point;Expression(C1) = (x^2/4 - y^2/8 = 1);Coordinate(H) = (2, 4);Asymptote(C1)=Asymptote(C2);PointOnCurve(H,C2)",
+      "coords": {
+        "H": [
+          2.0,
+          4.0
+        ]
+      }
+    },
+    {
+      "index": 919,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.6457513110645907,
+        "b": 2.0,
+        "major_axis": "y",
+        "x_coef": 4.0,
+        "y_coef": 7.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0919.png",
+      "answer": "24*sqrt(13)/13",
+      "fact_expr": "G: Ellipse;H: Line;Q: Point;Expression(G) = (x^2/4 + y^2/7 = 1);Expression(H) = (3*x - 2*y - 16 = 0);PointOnCurve(Q, G)",
+      "coords": {}
+    },
+    {
+      "index": 920,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0920.png",
+      "answer": "2*sqrt(3)",
+      "fact_expr": "G: Parabola;H: Circle;Expression(G) = (y^2 = 4*x);Expression(H) = (y^2 + (x - 2)^2 = 4);C:Line;PointOnCurve(Focus(G),C)",
+      "coords": {}
+    },
+    {
+      "index": 921,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 922,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 923,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0923.png",
+      "answer": "7",
+      "fact_expr": "G: Hyperbola;P: Point;Q: Point;F1: Point;F2:Point;Expression(G) = (x^2 - y^2/3 = 1);Coordinate(Q) = (-2, 3);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(P,RightPart(G))",
+      "coords": {
+        "Q": [
+          -2.0,
+          3.0
+        ]
+      }
+    },
+    {
+      "index": 924,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0924.png",
+      "answer": "(y - x^2)*(1 + 4*x^2) = 1",
+      "fact_expr": "C: Parabola;B: Point;A: Point;P1: Point;P2: Point;M: Point;Expression(C) = (y = x^2);Length(LineSegmentOf(A, B)) = 2;Endpoint(LineSegmentOf(A, B)) = {P1, P2};PointOnCurve(P1, C);PointOnCurve(P2, C);MidPoint(LineSegmentOf(A, B)) = M",
+      "coords": {}
+    },
+    {
+      "index": 925,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 926,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0926.png",
+      "answer": "2",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/2 + x^2/a^2 = 1);a: Number;a>sqrt(2);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;H: Line;PointOnCurve(F1, H) = True;Slope(H) = -2;Intersection(H, G) = {A, B};A: Point;B: Point;MidPoint(LineSegmentOf(A, B)) = P;P: Point;O: Origin;Slope(LineOf(O, P)) = 1/4",
+      "coords": {}
+    },
+    {
+      "index": 927,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 928,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0928.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "C: Parabola;B: Point;A: Point;M: Point;P: Point;Expression(C) = (y^2 = 4*x);Coordinate(M) = (0, 1);PointOnCurve(P, Directrix(C));L1:Line;L2:Line;TangentOfPoint(P, C) = {L1,L2};TangentPoint(L1,C)=A;TangentPoint(L2,C)=B",
+      "coords": {
+        "M": [
+          0.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 929,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0929.png",
+      "answer": "{((x-1)^2+y^2=2),((x+1)^2+y^2=2)}",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);F: Point;Focus(G) = F;Intersection(Directrix(G), yAxis) = Q;Q: Point;PointOnCurve(Q, L) = True;L: Line;TangentPoint(L, G) = P;P: Point",
+      "coords": {}
+    },
+    {
+      "index": 930,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 931,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0931.png",
+      "answer": "1",
+      "fact_expr": "C: Hyperbola;F2: Point;Expression(C) = (x^2/4 - y^2 = 1);RightFocus(C) = F2",
+      "coords": {}
+    },
+    {
+      "index": 932,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 933,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 9.0,
+        "b": 5.0,
+        "major_axis": "x",
+        "x_coef": 81.0,
+        "y_coef": 25.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0933.png",
+      "answer": "12",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/81 + y^2/25 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 934,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 5.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0934.png",
+      "answer": "10",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 20*x)",
+      "coords": {}
+    },
+    {
+      "index": 935,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 936,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0936.png",
+      "answer": "x^2-y^2/3=1",
+      "fact_expr": "H: Parabola;Expression(H) = (y^2 = 8*x);G: Hyperbola;RightFocus(G) = Focus(H);Eccentricity(G) = 2",
+      "coords": {}
+    },
+    {
+      "index": 937,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 938,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 2.23606797749979,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0938.png",
+      "answer": "14",
+      "fact_expr": "C: Hyperbola;A: Point;P: Point;F: Point;Expression(C) = (x^2/4 - y^2/5 = 1);Coordinate(A) = (0, 4);LeftFocus(C)=F;PointOnCurve(P, RightPart(C))",
+      "coords": {
+        "A": [
+          0.0,
+          4.0
+        ]
+      }
+    },
+    {
+      "index": 939,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 940,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0940.png",
+      "answer": "9*sqrt(3)/16",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/3 = 1);P: Point;PointOnCurve(P, RightPart(G));Z1: Line;Z2: Line;K1: Line;K2: Line;Asymptote(G) = {K1, K2};PointOnCurve(P, Z1);PointOnCurve(P, Z2);IsPerpendicular(K1, Z1);IsPerpendicular(K2, Z2);M: Point;N: Point;FootPoint(K1, Z1) = M;FootPoint(K2, Z2) = N",
+      "coords": {}
+    },
+    {
+      "index": 941,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 942,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0942.png",
+      "answer": "8",
+      "fact_expr": "G: Hyperbola;H: Parabola;a: Number;Expression(H) = (y^2 = a*x);Focus(H) = RightFocus(G)",
+      "coords": {}
+    },
+    {
+      "index": 943,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0943.png",
+      "answer": "(1 + sqrt(5))/2",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;c: Number;B: Point;F: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (c, 0);RightFocus(C) = F;OneOf(Endpoint(ImageinaryAxis(C))) = B;IsPerpendicular(LineOf(B, F), OneOf(Asymptote(C)))",
+      "coords": {}
+    },
+    {
+      "index": 944,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0944.png",
+      "answer": "sqrt(5)/5",
+      "fact_expr": "H: Parabola;Expression(H) = (y^2 = 4*x);G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 945,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0945.png",
+      "answer": "sqrt(5)/5",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;O: Origin;Center(G) = O;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;A: Point;B: Point;UpperVertex(G) = A;RightVertex(G) = B;P: Point;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F1), xAxis);IsParallel(LineSegmentOf(P, F2), LineSegmentOf(A, B))",
+      "coords": {}
+    },
+    {
+      "index": 946,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 947,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0947.png",
+      "answer": "x^2+(y-5)^2=16",
+      "fact_expr": "G: Hyperbola;H: Circle;A: Point;Expression(G) = (x^2/16 - y^2/9 = 1);Coordinate(A) = (0, 5);Center(H)=A;IsTangent(Asymptote(G),H)",
+      "coords": {
+        "A": [
+          0.0,
+          5.0
+        ]
+      }
+    },
+    {
+      "index": 948,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0948.png",
+      "answer": "7",
+      "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/25 + y^2/16 = 1);PointOnCurve(P, G);F1: Point;F2: Point;OneOf(Focus(G)) = F1;Distance(P, F1) = 3;OneOf(Focus(G)) = F2;Negation(F1 = F2)",
+      "coords": {}
+    },
+    {
+      "index": 949,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 3.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0949.png",
+      "answer": "3/2",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 12*x);M: Point;Coordinate(M) = (-3, 4);PointOnCurve(Focus(C), L);Slope(L) = k;k: Number;L: Line;Intersection(L, C) = {A, B};A: Point;B: Point;DotProduct(VectorOf(M, A), VectorOf(M, B)) = 0",
+      "coords": {
+        "M": [
+          -3.0,
+          4.0
+        ]
+      }
+    },
+    {
+      "index": 950,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0950.png",
+      "answer": "m - a^2",
+      "fact_expr": "H: Ellipse;Expression(H) = (y^2/n + x^2/m = 1);m: Number;n: Number;m > n;n > 0;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Focus(H) = {F1, F2};Focus(G) = {F1, F2};F1: Point;F2: Point;P: Point;OneOf(Intersection(H, G)) = P",
+      "coords": {}
+    },
+    {
+      "index": 951,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0951.png",
+      "answer": "5",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, RightPart(G));O: Origin;DotProduct((VectorOf(O, F2) + VectorOf(O, P)),VectorOf(P, F2)) = 0;3*Abs(VectorOf(P, F1)) = 4*Abs(VectorOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 952,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0952.png",
+      "answer": "y^2=8*x",
+      "fact_expr": "C: Parabola;p: Number;G: Line;Q: Point;F: Point;P: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Expression(G) = (y = 4);Focus(C) = F;Intersection(G, yAxis) = P;Intersection(G,C)=Q;Abs(LineSegmentOf(Q, F)) = 2*Abs(LineSegmentOf(P, Q))",
+      "coords": {}
+    },
+    {
+      "index": 953,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0953.png",
+      "answer": "3",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = m*x);m: Number;m>0;F: Point;Focus(G) = F;Z: Line;PointOnCurve(F, Z);Slope(Z) = 2*sqrt(2);A: Point;B: Point;Intersection(Z, G) = {A, B};H: Circle;IsDiameter(LineSegmentOf(A, B), H);l: Line;Directrix(G) = l;M: Point;Intersection(H, l) = M;Abs(LineSegmentOf(M, F)) = sqrt(2)",
+      "coords": {}
+    },
+    {
+      "index": 954,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0954.png",
+      "answer": "3/8, 2/3",
+      "fact_expr": "A: Point;Coordinate(A) = (2, 0);B: Point;Coordinate(B) = (0, 1);G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;In(A, Vertex(G));H: Line;Expression(H) = (y = k*x);k: Number;k>0;Intersection(H, LineOf(A, B)) = D;D: Point;Intersection(H, G) = {E, F};E: Point;F: Point;VectorOf(E, D) = 6*VectorOf(D, F);Slope(H) = k;In(B, Vertex(G))",
+      "coords": {
+        "A": [
+          2.0,
+          0.0
+        ],
+        "B": [
+          0.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 955,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 956,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.449489742783178,
+        "b": 2.449489742783178,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0956.png",
+      "answer": "(-sqrt(15)/3,-1)",
+      "fact_expr": "G: Hyperbola;H: Line;k: Number;Expression(G) = (x^2 - y^2 = 6);Expression(H) = (y = k*x + 2);NumIntersection(H, RightPart(G)) = 2",
+      "coords": {}
+    },
+    {
+      "index": 957,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 958,
+      "success": true,
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+        "direction": "right"
+      },
+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0958.png",
+      "answer": "3/2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 6*x);P:Point;PointOnCurve(P,G);Distance(P,Focus(G))=2*Distance(P,yAxis)",
+      "coords": {}
+    },
+    {
+      "index": 959,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 2.449489742783178,
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0959.png",
+      "answer": "y= pm*(sqrt(3)/2)*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/8 - y^2/6 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 960,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 2.070901870727539,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 1.4661259228887502e-06,
+        "converged": true,
+        "iterations": 106
+      },
+      "output_path": "results/dev/hyperbola/problem_0960.png",
+      "answer": "x + pm*sqrt(2)*y = 0",
+      "fact_expr": "C2: Hyperbola;C1: Ellipse;a: Number;b: Number;a > b;b > 0;Expression(C1) = (y^2/b^2 + x^2/a^2 = 1);Expression(C2) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(C1)*Eccentricity(C2) = sqrt(3)/2",
+      "coords": {}
+    },
+    {
+      "index": 961,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 8.0,
+        "b": 6.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0961.png",
+      "answer": "36",
+      "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/64 - y^2/36 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 962,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0962.png",
+      "answer": "2",
+      "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;M: Point;N: Point;F: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftVertex(C) = M;RightFocus(C) = F;PointOnCurve(LeftVertex(C),l);Slope(l)=1;Intersection(l, RightPart(C)) = N;Area(TriangleOf(M, N, F)) = (3/2)*b^2",
+      "coords": {}
+    },
+    {
+      "index": 963,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 964,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 965,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0965.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "F1: Point;F2: Point;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2, H);IsParallel(H, OneOf(Asymptote(G)));H: Line;Intersection(H, G) = P;P: Point;Abs(LineSegmentOf(P, F1)) = 3*Abs(LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 966,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 967,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "vertical"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0967.png",
+      "answer": "y=pm*(sqrt(7)/7)*x\n8",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/14 + y^2/2 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 968,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 969,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 970,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 6.0,
+        "y_coef": 2.0
+      },
+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0970.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "C1: Curve;C2: Curve;P: Point;F1: Point;F2: Point;Expression(C1) = (x^2/6+y^2/2=1);Expression(C2) = (x^2/3-y^2=1);Focus(C1) = {F2,F1};OneOf(Intersection(C1,C2))=P",
+      "coords": {}
+    },
+    {
+      "index": 971,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0971.png",
+      "answer": "2",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;M: Point;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;Coordinate(M)=(a,b);AngleOf(M,F1,F2)=ApplyUnit(30,degree)",
+      "coords": {}
+    },
+    {
+      "index": 972,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 973,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.2352789464429823,
+        "b": 2.2352789464429823,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 1.4312591569844774e-06,
+        "converged": true,
+        "iterations": 89
+      },
+      "output_path": "results/dev/hyperbola/problem_0973.png",
+      "answer": "x^2/9 - y^2 = 1",
+      "fact_expr": "G: Hyperbola;F1: Point;F2: Point;M: Point;Coordinate(F1) = (-sqrt(10), 0);Coordinate(F2) = (sqrt(10), 0);Focus(G) = {F1,F2};PointOnCurve(M, G);DotProduct(VectorOf(M, F1), VectorOf(M, F2)) = 0;Abs(VectorOf(M, F1))*Abs(VectorOf(M, F2)) = 2",
+      "coords": {
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+        "F2": [
+          3.1622776601683795,
+          0.0
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+      }
+    },
+    {
+      "index": 974,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0974.png",
+      "answer": "{(1,1),(-1,1)}",
+      "fact_expr": "C: Parabola;M: Point;F: Point;Expression(C) = (y = x^2);Focus(C) = F;PointOnCurve(M, C);Abs(LineSegmentOf(M, F)) = 5/4",
+      "coords": {}
+    },
+    {
+      "index": 975,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0975.png",
+      "answer": "3+sqrt(6)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2 + 3*y^2 = 9);F1: Point;LeftFocus(G) = F1;P: Point;PointOnCurve(P, G);Negation(In(P,Vertex(G)));O: Origin;D: Point;MidPoint(LineSegmentOf(P, F1)) = D",
+      "coords": {}
+    },
+    {
+      "index": 976,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 977,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 0.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0977.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - 4*y^2 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 978,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.2135674953460693,
+        "b": 2.4270472526550293,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 4.453341261978494e-06,
+        "converged": true,
+        "iterations": 125
+      },
+      "output_path": "results/dev/hyperbola/problem_0978.png",
+      "answer": "x+pm*2*y=0",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Eccentricity(C) = sqrt(5)/2",
+      "coords": {}
+    },
+    {
+      "index": 979,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 980,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0980.png",
+      "answer": "44",
+      "fact_expr": "F: Point;C: Hyperbola;Expression(C) = (x^2/9 - y^2/16 = 1);LeftFocus(C) = F;Q: Point;P: Point;PointOnCurve(P, C);PointOnCurve(Q, C);Length(LineSegmentOf(P, Q)) = Length(ImageinaryAxis(C)) * 2;A: Point;Coordinate(A) = (5, 0);PointOnCurve(A, LineSegmentOf(P, Q))",
+      "coords": {
+        "A": [
+          5.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 981,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0981.png",
+      "answer": "y^2/7 - x^2/14 = 1",
+      "fact_expr": "G: Hyperbola;H: Point;Q: Hyperbola;Expression(G) = (x^2/2 - y^2 = 1);Coordinate(H) = (2, 3);PointOnCurve(H, Q);Asymptote(G) = Asymptote(Q)",
+      "coords": {
+        "H": [
+          2.0,
+          3.0
+        ]
+      }
+    },
+    {
+      "index": 982,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 983,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0983.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;p: Number;H: Line;A: Point;F: Point;C: Point;B: Point;p>0;Expression(G) = (y^2 = 2*p*x);Focus(G) = F;PointOnCurve(F,H);Intersection(H,G) = {A, B};Abs(LineSegmentOf(A,F))=4;PointOnCurve(C,Directrix(G));VectorOf(C,B)=2*VectorOf(B,F)",
+      "coords": {}
+    },
+    {
+      "index": 984,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.4641016151377544,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 12.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0984.png",
+      "answer": "3",
+      "fact_expr": "C: Ellipse;P: Point;F1: Point;F2: Point;O: Origin;Expression(C) = (x^2/12 + y^2/3 = 1);Focus(C) = {F1, F2};PointOnCurve(P, C);Abs(LineSegmentOf(O, P)) = 3",
+      "coords": {}
+    },
+    {
+      "index": 985,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0985.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;Focus(C) = {F1, F2};P: Point;PointOnCurve(P, C) = True;AngleOf(P, F1, F2) = ApplyUnit(15, degree);Area(TriangleOf(P, F1, F2)) = b^2",
+      "coords": {}
+    },
+    {
+      "index": 986,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0986.png",
+      "answer": "y = -1/2",
+      "fact_expr": "G: Parabola;Expression(G) = (y = x^2/2)",
+      "coords": {}
+    },
+    {
+      "index": 987,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0987.png",
+      "answer": "(-sqrt(2), sqrt(2))&Negation(m=pm*1)",
+      "fact_expr": "G: Hyperbola;H: Line;m: Real;Expression(G) = (x^2 - y^2 = 1);Expression(H) = (y = m*x + 1);NumIntersection(H, G) = 2",
+      "coords": {}
+    },
+    {
+      "index": 988,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0988.png",
+      "answer": "x^2/32+y^2/8=1",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Eccentricity(C) = sqrt(3)/2;l: Line;Expression(l) = (y = x + 2);A: Point;B: Point;Intersection(l, C) = {A, B};O: Origin;Slope(LineOf(O, A)) + Slope(LineOf(O, B)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 989,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 990,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_0990.png",
+      "answer": "9",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/25 + y^2/9 = 1);PointOnCurve(P,G);Focus(G)={F1,F2};IsPerpendicular(LineSegmentOf(P,F1),LineSegmentOf(P,F2))",
+      "coords": {}
+    },
+    {
+      "index": 991,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.03125,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0991.png",
+      "answer": "y=-1/32",
+      "fact_expr": "G: Parabola;Expression(G) = (y = 8*x^2)",
+      "coords": {}
+    },
+    {
+      "index": 992,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 993,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 2.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0993.png",
+      "answer": "(1/3, 1)",
+      "fact_expr": "G: Hyperbola;H: Line;b: Number;A: Point;C: Point;B: Point;b>0;Expression(H) = (y = b + x);Expression(G) = (x^2 - y^2/4 = 1);Intersection(H,LeftPart(G))=A;Intersection(H,RightPart(G))=B;Intersection(H,yAxis)=C",
+      "coords": {}
+    },
+    {
+      "index": 994,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 995,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_0995.png",
+      "answer": "1/3",
+      "fact_expr": "C: Parabola;p: Number;A: Point;B: Point;F: Point;D: Point;p>0;Expression(C) = (y^2 = 2*(p*x));PointOnCurve(A, C);PointOnCurve(B, C);Focus(C) = F;AngleOf(A, F, B) = ApplyUnit(120, degree);MidPoint(LineSegmentOf(A, B))=D;Distance(D, Directrix(C)) = d;d:Number",
+      "coords": {}
+    },
+    {
+      "index": 996,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0996.png",
+      "answer": "3",
+      "fact_expr": "M: Point;Coordinate(M) = (-1, 0);PointOnCurve(M, l) = True;l: Line;C: Parabola;Expression(C) = (y^2 = 4*x);Intersection(l, C) = {A, B};A: Point;B: Point;Between(A, M, B) = True;F: Point;Focus(C) = F;Area(TriangleOf(M, B, F)) = 4*Area(TriangleOf(M, A, F))",
+      "coords": {
+        "M": [
+          -1.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 997,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_0997.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "Gamma: Curve;Expression(Gamma) = (x^2 = 8*y);F: Point;Focus(Gamma) = F;P: Point;PointOnCurve(P,Gamma) = True;A: Point;Coordinate(A) = (0, -2)",
+      "coords": {
+        "A": [
+          0.0,
+          -2.0
+        ]
+      }
+    },
+    {
+      "index": 998,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0998.png",
+      "answer": "2*sqrt(3)/3",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;O: Origin;A: Point;B: Point;F: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F;L:Line;PointOnCurve(F, L);IsPerpendicular(L, xAxis);L1:Line;L2:Line;Intersection(L,L1)=A;PointOnCurve(B, L2);lambda:Number;lambda>0;VectorOf(O, A) = lambda*VectorOf(B, F);DotProduct(VectorOf(A, B), VectorOf(O, B)) = 0;OneOf(Asymptote(C))=L1;OneOf(Asymptote(C))=L2;Negation(L1=L2)",
+      "coords": {}
+    },
+    {
+      "index": 999,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_0999.png",
+      "answer": "x^2/4-(3*y^2)/4=1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;Expression(Asymptote(G)) = (y = pm*((sqrt(3)/3)*x));Distance(Vertex(G), Asymptote(G)) = 1",
+      "coords": {}
+    },
+    {
+      "index": 1000,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_1000.png",
+      "answer": "(x-5/2)^2+(y-5/2)^2=13/2",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/4 = 1);A: Point;B: Point;C: Point;RightVertex(G) = A;UpperVertex(G) = B;Coordinate(C) = (2, 5);H: Circle;PointOnCurve(A, H);PointOnCurve(B, H);PointOnCurve(C, H)",
+      "coords": {
+        "C": [
+          2.0,
+          5.0
+        ]
+      }
+    },
+    {
+      "index": 1001,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_1001.png",
+      "answer": "(x+1)^2+(y-1/2)^2=1",
+      "fact_expr": "G: Parabola;H: Circle;Expression(G) = (y = x^2/2)&(x<0);PointOnCurve(Center(H), G);IsTangent(Directrix(G),H);IsTangent(yAxis,H)",
+      "coords": {}
+    },
+    {
+      "index": 1002,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_1002.png",
+      "answer": "y^2 = 8*x",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(C) = F;A: Point;PointOnCurve(A, C);Coordinate(MidPoint(LineSegmentOf(A, F))) = (2, 2)",
+      "coords": {}
+    },
+    {
+      "index": 1003,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_1003.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2 - y^2 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, OneOf(Asymptote(C)));G: Circle;IsDiameter(LineSegmentOf(F1, F2), G);PointOnCurve(P, G)",
+      "coords": {}
+    },
+    {
+      "index": 1004,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1005,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_1005.png",
+      "answer": "Preserve all mathematical expressions, LaTeX formulas, symbols, and equations EXACTLY.  \nDo not add explanations, do not rewrite formulas, do not omit any content.  \n\nText:",
+      "fact_expr": "G: Ellipse;H: Line;m:Number;M:Point;m>0;Expression(H) = (y = x*(sqrt(2)/2));Expression(G) = (x^2/16 + y^2/m^2 = 1);OneOf(Intersection(G,H))=M;Projection(M,xAxis)=RightFocus(G)",
+      "coords": {}
+    },
+    {
+      "index": 1006,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_1006.png",
+      "answer": "8",
+      "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(Focus(G), l);Intersection(l, G) = {A, B};XCoordinate(MidPoint(LineSegmentOf(A, B))) = 3",
+      "coords": {}
+    },
+    {
+      "index": 1007,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 2.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_1007.png",
+      "answer": "24",
+      "fact_expr": "l: Line;G: Hyperbola;A: Point;B: Point;Expression(G) = (x^2/16 - y^2/4 = 1);PointOnCurve(RightFocus(G),l);Intersection(l,RightPart(G))={A,B};Abs(LineSegmentOf(A, B)) = 4;F:Point;LeftFocus(G)=F",
+      "coords": {}
+    },
+    {
+      "index": 1008,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_1008.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;P: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G)=F;PointOnCurve(P, G);Abs(LineSegmentOf(P, F)) = 3",
+      "coords": {}
+    },
+    {
+      "index": 1009,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_1009.png",
+      "answer": "9",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G)",
+      "coords": {}
+    },
+    {
+      "index": 1010,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_1010.png",
+      "answer": "8",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;G: Ellipse;Expression(G) = (x^2/16 + y^2/12 = 1);Focus(C) = Focus(G);InterReciprocal(Eccentricity(C), Eccentricity(G));F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P,RightPart(C))",
+      "coords": {}
+    },
+    {
+      "index": 1011,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 0.5,
+        "major_axis": "x",
+        "x_coef": 1.0,
+        "y_coef": 0.25
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_1011.png",
+      "answer": "sqrt(3)/2",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2 + 4*y^2 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 1012,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_1012.png",
+      "answer": "4",
+      "fact_expr": "C1:Parabola;C2:Hyperbola;Expression(C1) = (y^2=2*p*x);p:Number;p>0;Expression(C2) = (x^2/3-y^2/b^2=1);F:Point;Focus(C1)=F;OneOf(Focus(C2))=F;Distance(F,OneOf(Asymptote(C2)))=1",
+      "coords": {}
+    },
+    {
+      "index": 1013,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.4142135623730951,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_1013.png",
+      "answer": "3*sqrt(2)/4",
+      "fact_expr": "C: Hyperbola;P: Point;F: Point;O: Origin;Expression(C) = (x^2/4 - y^2/2 = 1);RightFocus(C) = F;PointOnCurve(P, OneOf(Asymptote(C)));Abs(LineSegmentOf(P, O)) = Abs(LineSegmentOf(P, F))",
+      "coords": {}
+    },
+    {
+      "index": 1014,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/dev/parabola/problem_1014.png",
+      "answer": "2",
+      "fact_expr": "A: Point;B: Point;G: Parabola;p: Number;H: Line;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;PointOnCurve(F, H);Inclination(H) = ApplyUnit(45, degree);Intersection(H, G) = {A, B};Length(LineSegmentOf(A, B)) = 8",
+      "coords": {}
+    },
+    {
+      "index": 1015,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_focus_mismatch",
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_1015.png",
+      "answer": "x^2/18+y^2/9=1",
+      "fact_expr": "E: Ellipse;b: Number;a: Number;G: Line;A: Point;B: Point;F: Point;a > b;b > 0;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(MidPoint(LineSegmentOf(A,B))) = (1, -1);Coordinate(F) = (3, 0);RightFocus(E) = F;PointOnCurve(F, G);Intersection(G, E) = {A, B}",
+      "coords": {
+        "F": [
+          3.0,
+          0.0
+        ]
+      },
+      "validation_reasons": [
+        "ellipse_focus_mismatch"
+      ]
+    },
+    {
+      "index": 1016,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_1016.png",
+      "answer": "y=pm*(4/3)*x",
+      "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;P: Point;M: Point;F1: Point;F2:Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C)=F1;RightFocus(C)=F2;PointOnCurve(F1,l);Slope(l)=a/b;Intersection(l,RightPart(C))=P;Intersection(l,OneOf(Asymptote(C)))=M;VectorOf(P, M) = 3*VectorOf(M, F1)",
+      "coords": {}
+    },
+    {
+      "index": 1017,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.4142135623730951,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_1017.png",
+      "answer": "{pm*1,pm*sqrt(6)/2}",
+      "fact_expr": "G: Hyperbola;H: Line;k: Real;Expression(G) = (x^2 - y^2 = 2);Expression(H) = (y = k*x + 1);NumIntersection(H, G)=1",
+      "coords": {}
+    },
+    {
+      "index": 1018,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1019,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_focus_mismatch",
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_1019.png",
+      "answer": "x^2/16 - y^2/4 = 1",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;F: Point;e:Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F)=(2*sqrt(5),0);OneOf(Focus(G))=F;Eccentricity(G)=e;e=sqrt(5)/2",
+      "coords": {
+        "F": [
+          4.47213595499958,
+          0.0
+        ]
+      },
+      "validation_reasons": [
+        "hyperbola_focus_mismatch"
+      ]
+    },
+    {
+      "index": 1020,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1021,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1022,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_1022.png",
+      "answer": "10",
+      "fact_expr": "C: Parabola;P: Point;PointOnCurve(P, C);H: Line;PointOnCurve(P, H);IsPerpendicular(H, Directrix(C)) = True;M: Point;FootPoint(H, Directrix(C)) = M;F: Point;Focus(C) = F;Abs(LineSegmentOf(P, F)) = 5;Expression(C)=(y^2=4*x)",
+      "coords": {}
+    },
+    {
+      "index": 1023,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 4.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_1023.png",
+      "answer": "13",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 16*x);P: Point;PointOnCurve(P, G);Distance(P, xAxis) = 12",
+      "coords": {}
+    },
+    {
+      "index": 1024,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_1024.png",
+      "answer": "[1/2,1)",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;m:Number;n:Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(P, G);Max(Abs(VectorOf(P,F1))*Abs(VectorOf(P,F2)))=m;Min(DotProduct(VectorOf(P, F1), VectorOf(P, F2))) = n;m >= 2*n",
+      "coords": {}
+    },
+    {
+      "index": 1025,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 4.47213595499958,
+        "b": 2.8284271247461903,
+        "major_axis": "x",
+        "x_coef": 20.0,
+        "y_coef": 8.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_1025.png",
+      "answer": "8",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/20 + y^2/8 = 1);Focus(G)={F1,F2};PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 1026,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_1026.png",
+      "answer": "pm*sqrt(2)/2",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;k: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = k*x);Eccentricity(G) = sqrt(2)/2;XCoordinate(OneOf(Intersection(H,G))) = b",
+      "coords": {}
+    },
+    {
+      "index": 1027,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.125,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_1027.png",
+      "answer": "3/8",
+      "fact_expr": "G: Parabola;H: Line;A: Point;F: Point;B: Point;Expression(G) = (y = 2*x^2);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};VectorOf(A, F) = 2*VectorOf(F, B)",
+      "coords": {}
+    },
+    {
+      "index": 1028,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_1028.png",
+      "answer": "{4,5}",
+      "fact_expr": "G: Curve;n: Number;m: Number;F1: Point;F2: Point;P: Point;m>0;n>0;Expression(G) = (y^2/n^2 + x^2/m^2 = 1);Focus(G) = {F1, F2};PointOnCurve(P,G);Perimeter(TriangleOf(P,F1,F2))=16;Min(Distance(P,F1))=2",
+      "coords": {}
+    },
+    {
+      "index": 1029,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1030,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/parabola/problem_1030.png",
+      "answer": "1/2",
+      "fact_expr": "P: Point;Coordinate(P) = (1, -1);C: Parabola;Expression(C) = (y = x^2/4);G: Line;PointOnCurve(Focus(C), G);k: Number;Slope(G) = k;A: Point;B: Point;Intersection(G, C) = {A, B};DotProduct(VectorOf(P, A), VectorOf(P, B)) = 0",
+      "coords": {
+        "P": [
+          1.0,
+          -1.0
+        ]
+      }
+    },
+    {
+      "index": 1031,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "y",
+        "x_coef": 3.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_1031.png",
+      "answer": "(-2,2)",
+      "fact_expr": "G: Ellipse;H: Line;m: Real;Expression(G) = (x^2/3 + y^2/4 = 1);Expression(H) = (y = m);NumIntersection(H, G) = 2",
+      "coords": {}
+    },
+    {
+      "index": 1032,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1033,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 2.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/hyperbola/problem_1033.png",
+      "answer": "8*x-y-15=0",
+      "fact_expr": "l: Line;P: Point;Coordinate(P) = (2, 1);PointOnCurve(P, l);G: Hyperbola;Expression(G) = (x^2 - y^2/4 = 1);Intersection(l, G) = {A, B};A: Point;B: Point;MidPoint(LineSegmentOf(A, B)) = P",
+      "coords": {
+        "P": [
+          2.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 1034,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/dev/ellipse/problem_1034.png",
+      "answer": "36/25",
+      "fact_expr": "G: Ellipse;H:Line;A: Point;B: Point;F1: Point;F2: Point;Expression(G) = (x^2/25 + y^2/9 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(F1,H);IsPerpendicular(H,MajorAxis(G));Intersection(H,G) = {A, B}",
+      "coords": {}
+    }
+  ]
+}
\ No newline at end of file
diff --git a/samples/test/summary.json b/samples/test/summary.json
new file mode 100644
index 0000000000000000000000000000000000000000..124027b12bbca214338e6a95e1e570a8867bff5c
--- /dev/null
+++ b/samples/test/summary.json
@@ -0,0 +1,37008 @@
+{
+  "stats": {
+    "total": 2069,
+    "success": 1596,
+    "failed": 473
+  },
+  "type_stats": {
+    "ellipse": 467,
+    "hyperbola": 565,
+    "parabola": 532,
+    "circle": 32
+  },
+  "reason_counts": {
+    "ellipse_ecc_mismatch": 8,
+    "ellipse_point_off_curve": 8,
+    "hyperbola_point_off_curve": 14,
+    "parabola_point_off_curve": 4,
+    "ellipse_focus_mismatch": 7,
+    "hyperbola_focus_mismatch": 8,
+    "parabola_focus_mismatch": 2
+  },
+  "results": [
+    {
+      "index": 0,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.732385540716972,
+        "b": 2.4497476064802375,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 2.601017381791051e-06,
+        "converged": true,
+        "iterations": 91
+      },
+      "output_path": "results/test/hyperbola/problem_0000.png",
+      "answer": "x^2/3-y^2/6=1",
+      "fact_expr": "G: Hyperbola;F1: Point;F2: Point;Coordinate(F1) = (-3, 0);Coordinate(F2) = (3, 0);Focus(G)={F1,F2};Expression(OneOf(Asymptote(G))) = (y = sqrt(2)*x)",
+      "coords": {
+        "F1": [
+          -3.0,
+          0.0
+        ],
+        "F2": [
+          3.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 1,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0001.png",
+      "answer": "sqrt(10)",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2 + x^2/k^2 = 1);k: Number;k>0;Coordinate(OneOf(Focus(G))) = (3,0)",
+      "coords": {}
+    },
+    {
+      "index": 2,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0002.png",
+      "answer": "sqrt(34)-1",
+      "fact_expr": "G: Parabola;A: Point;P: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (4, 5);PointOnCurve(P,G);Distance(P, yAxis) = d;d:Number",
+      "coords": {
+        "A": [
+          4.0,
+          5.0
+        ]
+      }
+    },
+    {
+      "index": 3,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0003.png",
+      "answer": "3*x+4*y-5=0",
+      "fact_expr": "G: Hyperbola;H: LineSegment;M: Point;Expression(G) = (x^2/4 - y^2 = 1);Coordinate(M) = (3, -1);IsChordOf(H,G);MidPoint(H)=M;PointOnCurve(M,H)",
+      "coords": {
+        "M": [
+          3.0,
+          -1.0
+        ]
+      }
+    },
+    {
+      "index": 4,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          0.0,
+          0.0
+        ],
+        "radius": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/circle/problem_0004.png",
+      "answer": "(x^2+y^2=1)&Negation(x=pm*1)",
+      "fact_expr": "P: Point;Slope(LineSegmentOf(P, A))*Slope(LineSegmentOf(P, B)) = -1;A: Point;Coordinate(A) = (-1, 0);B: Point;Coordinate(B) = (1, 0)",
+      "coords": {}
+    },
+    {
+      "index": 5,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0005.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 6,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0006.png",
+      "answer": "(0,sqrt(2)/2]",
+      "fact_expr": "P: Point;PointOnCurve(P, G) = True;Coordinate(P) = (x1, y1);x1: Number;y1: Number;G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;Focus(G) = {F1, F2};AngleOf(F1, P, F2) <= ApplyUnit(90, degree)",
+      "coords": {}
+    },
+    {
+      "index": 7,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 0.9872854805122186,
+        "b": 2.6119589473247777,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 4.271082513482038e-06,
+        "converged": true,
+        "iterations": 115
+      },
+      "output_path": "results/test/hyperbola/problem_0007.png",
+      "answer": "x^2/2-y^2/14=1",
+      "fact_expr": "C: Hyperbola;G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);Focus(C) = Focus(G);Expression(OneOf(Asymptote(C))) = (y = sqrt(7)*x)",
+      "coords": {}
+    },
+    {
+      "index": 8,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_ecc_mismatch",
+      "params": {
+        "a": 1.9257268905639648,
+        "b": 1.7993911322594118,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 2.5576271215184017e-15,
+        "converged": true,
+        "iterations": 2
+      },
+      "output_path": "results/test/ellipse/problem_0008.png",
+      "answer": "x^2/16+y^2/12=1",
+      "fact_expr": "H: Ellipse;Expression(H) = (y^2/n + x^2/m = 1);m: Number;n: Number;m>0;n>0;Eccentricity(H) = 1/2;G: Parabola;Expression(G) = (y^2 = 8*x);OneOf(Focus(H)) = Focus(G)",
+      "coords": {},
+      "validation_reasons": [
+        "ellipse_ecc_mismatch"
+      ]
+    },
+    {
+      "index": 9,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0009.png",
+      "answer": "y^2=-8*x",
+      "fact_expr": "G: Parabola;P: Point;O: Origin;m:Number;Coordinate(P) = (-3, m);Vertex(G) = O;PointOnCurve(Focus(G), xAxis);PointOnCurve(P, G);Distance(P, Focus(G))=5",
+      "coords": {}
+    },
+    {
+      "index": 10,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0010.png",
+      "answer": "6",
+      "fact_expr": "G: Parabola;A: Point;M: Point;F: Point;Expression(G) = (x^2 = 4*y);Coordinate(A) = (1, 5);PointOnCurve(M, G);Focus(G) = F",
+      "coords": {
+        "A": [
+          1.0,
+          5.0
+        ]
+      }
+    },
+    {
+      "index": 11,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0011.png",
+      "answer": "8",
+      "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (y = x - 1);Intersection(H, G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 12,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0012.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;G: Line;Expression(G) = (y = b*sqrt(6)/3);Intersection(G, C) = {A, B};A: Point;B: Point;O: Origin;IsPerpendicular(LineSegmentOf(O, A), LineSegmentOf(O, B)) = True",
+      "coords": {}
+    },
+    {
+      "index": 13,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.125,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0013.png",
+      "answer": "(-1, 1)+{17/8}",
+      "fact_expr": "G: Parabola;Expression(G) = (y = 2*x^2);H: Circle;Expression(H) = (x^2 + (-a + y)^2 = 1);a: Number;NumIntersection(G, H) = 2",
+      "coords": {}
+    },
+    {
+      "index": 14,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.23606797749979,
+        "major_axis": "y",
+        "x_coef": 5.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0014.png",
+      "answer": "9",
+      "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/5 + y^2/9 = 1);Coordinate(P) = (x1, y1);PointOnCurve(P, G);x1:Number;y1:Number",
+      "coords": {}
+    },
+    {
+      "index": 15,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0015.png",
+      "answer": "(sqrt(5)-1)/2",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;LeftFocus(G) = F1;O: Origin;P: Point;PointOnCurve(P, G);Q: Point;PointOnCurve(Q, RightDirectrix(G));VectorOf(P, Q) = 2*VectorOf(F1, O);lambda: Number;lambda>0;VectorOf(F1, Q) = lambda*(VectorOf(F1, O)/Abs(VectorOf(F1, O)) + VectorOf(F1, P)/Abs(VectorOf(F1, P)))",
+      "coords": {}
+    },
+    {
+      "index": 16,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
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+        "note": "using explicit params"
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+      "answer": "2*sqrt(6)",
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+      "answer": "-2*p/t",
+      "fact_expr": "G: Parabola;p: Number;H: Line;O: Origin;A: Point;M: Point;B: Point;t:Number;p>0;Expression(G) = (y^2 = 2*(p*x));PointOnCurve(M,SymmetryAxis(G));Coordinate(M) = (t, 0);PointOnCurve(M, H);Intersection(H, G) = {A, B}",
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+      "answer": "[21/2, 27/2]",
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+      "answer": "x^2+y^2/4=1",
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+        "note": "using explicit params"
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+      "output_path": "results/test/parabola/problem_0023.png",
+      "answer": "3/5",
+      "fact_expr": "G: Parabola;P: Point;M: Point;F: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(P,G);L:Line;PointOnCurve(P,L);IsPerpendicular(L,Directrix(G));FootPoint(L,Directrix(G))=M;Abs(LineSegmentOf(P, M)) = 5;Focus(G) = F",
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+    {
+      "index": 24,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 25,
+      "success": true,
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+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0025.png",
+      "answer": "20",
+      "fact_expr": "F2: Point;F1: Point;Focus(G) = {F1, F2};G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);H: Line;PointOnCurve(F1, H) = True;A: Point;B: Point;Intersection(H, G) = {A, B}",
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+      "output_path": "results/test/parabola/problem_0026.png",
+      "answer": "(0,1)",
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+          1.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0027.png",
+      "answer": "2*sqrt(3)/3",
+      "fact_expr": "G: Hyperbola;m: Number;H: Parabola;m>0;Expression(G) = (-x^2 + y^2/m = 1);Expression(H) = (y = x^2/8);OneOf(Focus(G)) = Focus(H)",
+      "coords": {}
+    },
+    {
+      "index": 28,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 1.5,
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0028.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "C: Hyperbola;a: Number;b: Number;a > 0;b > 0;Expression(C) = (x^2/a^2 - y^2/b^2 = 1);F: Point;LeftFocus(C) = F;l: Line;Slope(l) = sqrt(2);PointOnCurve(F, l);A: Point;B: Point;Intersection(l, LeftPart(C)) = {A, B};E: Circle;IsDiameter(LineSegmentOf(A, B), E);O: Origin;PointOnCurve(O, E)",
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+    },
+    {
+      "index": 29,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 30,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0030.png",
+      "answer": "6/5",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;A: Point;B: Point;F: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;PointOnCurve(F, G);Slope(G) = sqrt(3);Intersection(G, C) = {A, B};VectorOf(A, B) = 5*(VectorOf(F, B))",
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+    },
+    {
+      "index": 31,
+      "success": true,
+      "conic_type": "parabola",
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+        "direction": "right"
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+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0031.png",
+      "answer": "{1,3}",
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+    {
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+      "conic_type": "hyperbola",
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+        "b": 3.0,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0032.png",
+      "answer": "y=pm*(3/4)*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 33,
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+        "b": 4.0,
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+        "note": "using explicit params"
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+      "output_path": "results/test/hyperbola/problem_0033.png",
+      "answer": "5/3",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 34,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 35,
+      "success": true,
+      "conic_type": "ellipse",
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+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0035.png",
+      "answer": "-1",
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+    },
+    {
+      "index": 36,
+      "success": true,
+      "conic_type": "ellipse",
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+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0036.png",
+      "answer": "(-oo,-sqrt(2)/2]+[sqrt(2)/2,+oo)",
+      "fact_expr": "G: Ellipse;H: Line;B: Point;A: Point;Expression(G) = (x^2/2 + y^2 = 1);Expression(H) = (x = -1/2);PointOnCurve(A, G);PointOnCurve(B, G);PointOnCurve(MidPoint(LineSegmentOf(A,B)),H)",
+      "coords": {}
+    },
+    {
+      "index": 37,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "converged": true,
+        "iterations": 125
+      },
+      "output_path": "results/test/hyperbola/problem_0037.png",
+      "answer": "y=pm*2*x",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = sqrt(5)",
+      "coords": {}
+    },
+    {
+      "index": 38,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0038.png",
+      "answer": "(1,3]",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Focus(G) = {F1,F2};PointOnCurve(P,RightPart(G));Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 39,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_point_off_curve",
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+        "b": 1.632524068785969,
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+        "converged": true,
+        "iterations": 84
+      },
+      "output_path": "results/test/hyperbola/problem_0039.png",
+      "answer": "4*sqrt(3)",
+      "fact_expr": "C: Hyperbola;G: Point;Coordinate(G) = (4, 2);Expression(Asymptote(C)) = (y = pm*(sqrt(2)/2)*X);PointOnCurve(G, C);PointOnCurve(Focus(C), axis)",
+      "coords": {
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+          2.0
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+      },
+      "validation_reasons": [
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+    },
+    {
+      "index": 40,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 2.2412689930799665,
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+      "optimization": {
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+        "converged": true,
+        "iterations": 95
+      },
+      "output_path": "results/test/hyperbola/problem_0040.png",
+      "answer": "{5/3,5/4}",
+      "fact_expr": "C: Hyperbola;O:Origin;Center(C)=O;Expression(OneOf(Asymptote(C)))=(y=4/3*x);SymmetryAxis(C)=axis",
+      "coords": {}
+    },
+    {
+      "index": 41,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0041.png",
+      "answer": "1/3",
+      "fact_expr": "E: Parabola;p: Number;C: Circle;G: Line;A: Point;F: Point;P: Point;M: Point;N: Point;p>0;Expression(E) = (x^2 = 2*(p*y));Expression(C) = ((x + 2)^2 + (y + 2)^2 = 1);Coordinate(P) = (3/2, -1);PointOnCurve(P, Directrix(E));TangentOfPoint(P , E) = G;TangentPoint(G, E) = A;Quadrant(A) = 1;Focus(E) = F;PointOnCurve(M, LineOf(A, F));PointOnCurve(N, C)",
+      "coords": {
+        "P": [
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+          -1.0
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+    },
+    {
+      "index": 42,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
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+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0042.png",
+      "answer": "3",
+      "fact_expr": "l: Line;C: Parabola;p: Number;p>0;Expression(C) = (y^2 = 2*p*x);Expression(l)=(x-2*y+6=0);IsTangent(l,C)",
+      "coords": {}
+    },
+    {
+      "index": 43,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0043.png",
+      "answer": "(7+2*sqrt(3))/3",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;F1: Point;P: Point;F2: Point;e:Number;L:Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = -x*b/a);LeftFocus(G)=F1;RightFocus(G)=F2;Eccentricity(G)=e;PointOnCurve(F1,L);IsPerpendicular(L,H);Intersection(L,RightPart(G))=P;AngleOf(F1, P, F2) = pi/3",
+      "coords": {}
+    },
+    {
+      "index": 44,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0044.png",
+      "answer": "[sqrt(3)/3,1)",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, RightDirectrix(G)) = True;P: Point;PointOnCurve(F2, PerpendicularBisector(LineSegmentOf(P, F1))) = True",
+      "coords": {}
+    },
+    {
+      "index": 45,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.366584062576294,
+        "b": 2.364173412322998,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.00017838670464698225,
+        "converged": true,
+        "iterations": 89
+      },
+      "output_path": "results/test/hyperbola/problem_0045.png",
+      "answer": "3",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;k1: Number;k2:Number;A: Point;P: Point;B: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y = k*x);Intersection(G, C) = {A, B};PointOnCurve(P, C);Negation(P = A);Negation(P = B);Slope(LineOf(P,A)) = k1;Slope(LineOf(P,B))=k2;Eccentricity(C) = 2;k:Number",
+      "coords": {}
+    },
+    {
+      "index": 46,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 47,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0047.png",
+      "answer": "(-3/2,pm*3*sqrt(3))",
+      "fact_expr": "l: Line;C: Parabola;A: Point;B: Point;F: Point;D: Point;Expression(C) = (y^2 = 6*x);Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};Intersection(l,Directrix(C))=D;Abs(LineSegmentOf(A, B)) = 8",
+      "coords": {}
+    },
+    {
+      "index": 48,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 49,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0049.png",
+      "answer": "a^2+1+1/(a^2-1)",
+      "fact_expr": "G: Ellipse;a: Number;A: Point;B: Point;a>1;Expression(G) = (y^2 + x^2/a^2 = 1);Eccentricity(G)>sqrt(2)/2;UpperVertex(G)=A;PointOnCurve(B, G)",
+      "coords": {}
+    },
+    {
+      "index": 50,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 51,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0051.png",
+      "answer": "16/3",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;G: Line;PointOnCurve(F, G);A: Point;B: Point;Intersection(G, C) = {A, B};Abs(LineSegmentOf(F, A)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 52,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.75,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0052.png",
+      "answer": "[sqrt(2)/2,1]",
+      "fact_expr": "C: Parabola;M: Point;A: Point;F: Point;Expression(C) = (x^2 = 3*y);Focus(C)=F;Intersection(Directrix(C),yAxis)=A;PointOnCurve(M, C)",
+      "coords": {}
+    },
+    {
+      "index": 53,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0053.png",
+      "answer": "(sqrt(5)+1)/2",
+      "fact_expr": "C1: Hyperbola;Expression(C1)=(x^2/a^2-y^2/b^2=1);a:Number;b:Number;a>0;b>0;F1:Point;LeftFocus(C1)=F1;H:Circle;M:Point;l:Line;Expression(H)=(x^2+y^2=a^2);TangentOfPoint(F1,H)=l;TangentPoint(l,H)=M;C2:Parabola;Expression(C2)=(y^2=2*p*x);p:Number;p>0;N:Point;Intersection(OverlappingLine(LineSegmentOf(F1,M)),C2)=N;OneOf(Focus(C1))=Focus(C2);Abs(LineSegmentOf(M, F1)) = Abs(LineSegmentOf(M, N))",
+      "coords": {}
+    },
+    {
+      "index": 54,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.4641016151377544,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 12.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0054.png",
+      "answer": "ApplyUnit(120, degree)",
+      "fact_expr": "G: Ellipse;B1: Point;F1: Point;B2: Point;Expression(G) = (x^2/12 + y^2/9 = 1);Endpoint(MinorAxis(G)) = {B1, B2};OneOf(Focus(G)) = F1",
+      "coords": {}
+    },
+    {
+      "index": 55,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.23606797749979,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 5.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0055.png",
+      "answer": "3",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/5 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;IsChordOf(LineSegmentOf(A, B), G);PointOnCurve(F1, LineSegmentOf(A, B));Perimeter(InscribedCircle(TriangleOf(A, B, F2))) = 2*pi;x1: Number;x2: Number;y1: Number;y2: Number;A: Point;B: Point;Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2)",
+      "coords": {}
+    },
+    {
+      "index": 56,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0056.png",
+      "answer": "3",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);M: Point;PointOnCurve(M, G);F: Point;Focus(G) = F;Distance(M, F) = 4",
+      "coords": {}
+    },
+    {
+      "index": 57,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0057.png",
+      "answer": "y = pm*2*sqrt(2)*(x - 1)",
+      "fact_expr": "G: Parabola;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, LineSegmentOf(A, B));LineSegmentOf(A, F) = 2*LineSegmentOf(F, B);IsChordOf(LineSegmentOf(A, B), G)",
+      "coords": {}
+    },
+    {
+      "index": 58,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0058.png",
+      "answer": "5",
+      "fact_expr": "P: Point;Coordinate(P) = (-1, 0);H: Line;PointOnCurve(P, H);G: Parabola;Expression(G) = (y^2 = 8*x);A: Point;B: Point;Intersection(H, G) = {A, B};2*Abs(LineSegmentOf(P, A)) = Abs(LineSegmentOf(A, B))",
+      "coords": {
+        "P": [
+          -1.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 59,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0059.png",
+      "answer": "sqrt(7)/2",
+      "fact_expr": "G: Parabola;A: Point;B: Point;Expression(G) = (y^2 = x);PointOnCurve(B,G);Coordinate(A)=(2,0)",
+      "coords": {
+        "A": [
+          2.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 60,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0060.png",
+      "answer": "sqrt(3) - 1",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;l1: Line;l2: Line;F1: Point;P: Point;F2: Point;e: Number;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, l1);Inclination(l1) = ApplyUnit(30, degree);PointOnCurve(F2, l2);Intersection(l1, l2) = P;PointOnCurve(P, C);AngleOf(F1, P, F2) = ApplyUnit(90, degree);Eccentricity(C) = e",
+      "coords": {}
+    },
+    {
+      "index": 61,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 62,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0062.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;A: Point;C:Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);Coordinate(A) = (-3, 2*sqrt(3));Asymptote(G)=Asymptote(C);PointOnCurve(A,C)",
+      "coords": {
+        "A": [
+          -3.0,
+          3.4641016151377544
+        ]
+      }
+    },
+    {
+      "index": 63,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.5129417181015015,
+        "b": 2.0152759552001953,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 3.66731001122389e-06,
+        "converged": true,
+        "iterations": 100
+      },
+      "output_path": "results/test/hyperbola/problem_0063.png",
+      "answer": "5/3",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(Asymptote(G)) = (y = pm*(4/3)*x)",
+      "coords": {}
+    },
+    {
+      "index": 64,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0064.png",
+      "answer": "8",
+      "fact_expr": "G: Parabola;p: Number;M: Circle;A: Point;B:Point;F: Point;x0:Number;y0:Number;p>0;Expression(G) = (y^2 = 2*(p*x));Expression(M) = (y^2 + (x + 2)^2 = 4);Coordinate(A) = (x0, y0);PointOnCurve(A,G);TangentOnPoint(A,G)=l;TangentPoint(l,M)=B;Focus(G)=F;Distance(F,A)=Abs(LineSegmentOf(A,F));l:Line",
+      "coords": {}
+    },
+    {
+      "index": 65,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "down"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0065.png",
+      "answer": "(0, -2)",
+      "fact_expr": "C: Parabola;Expression(C) = (y = -x^2/8)",
+      "coords": {}
+    },
+    {
+      "index": 66,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0066.png",
+      "answer": "2*x-y-15=0",
+      "fact_expr": "G: Hyperbola;H: Line;B: Point;A: Point;P: Point;Expression(G) = (x^2 - 4*y^2 = 4);Coordinate(P) = (8, 1);PointOnCurve(P,H);Intersection(H,G) = {A, B};MidPoint(LineSegmentOf(A,B))=P",
+      "coords": {
+        "P": [
+          8.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 67,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 68,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0068.png",
+      "answer": "3/4",
+      "fact_expr": "C: Parabola;G: Line;l: Line;A: Point;O: Origin;F: Point;B: Point;D: Point;Expression(C) = (y^2 = 4*x);Directrix(C) = l;Focus(C) = F;PointOnCurve(F, G);Slope(G) = sqrt(3);Intersection(G, C) = {A, B};Abs(LineSegmentOf(A, F)) > Abs(LineSegmentOf(B, F));Intersection(LineOf(A,O), l) = D",
+      "coords": {}
+    },
+    {
+      "index": 69,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0069.png",
+      "answer": "(4*sqrt(3))/3",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/3 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;L: Line;PointOnCurve(F1, L);A: Point;B: Point;Intersection(L, LeftPart(G)) = {A, B};AngleOf(A, F2, B) = ApplyUnit(60, degree)",
+      "coords": {}
+    },
+    {
+      "index": 70,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0070.png",
+      "answer": "5/4",
+      "fact_expr": "Length(LineSegmentOf(A, B)) = 3;A: Point;B: Point;Endpoint(LineSegmentOf(A, B)) = {A, B};PointOnCurve(A, G);PointOnCurve(B, G);G: Parabola;Expression(G) = (y^2 = x);M: Point;MidPoint(LineSegmentOf(A, B)) = M",
+      "coords": {}
+    },
+    {
+      "index": 71,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 72,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0072.png",
+      "answer": "3*sqrt(2)/2",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;b: Number;a: Number;a>0;b>0;A: Point;H: Line;Expression(H) = (x = (4/3)*a);OneOf(Intersection(H,C)) = A;G: Circle;IsDiameter(LineSegmentOf(F1, F2), G);PointOnCurve(A, G)",
+      "coords": {}
+    },
+    {
+      "index": 73,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.2631799883090584,
+        "b": 2.5275303393962245,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 4.481942266905428e-06,
+        "converged": true,
+        "iterations": 95
+      },
+      "output_path": "results/test/hyperbola/problem_0073.png",
+      "answer": "2*sqrt(5)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (m*y^2 + x^2 = 1);m: Number;Expression(OneOf(Asymptote(G))) = (y = 2*x)",
+      "coords": {}
+    },
+    {
+      "index": 74,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0074.png",
+      "answer": "{y=((sqrt(3)/3)*x)-sqrt(3)/3, y=((-sqrt(3)/3)*x)+sqrt(3)/3}",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, C) = {A, B};P: Point;Q: Point;Projection(MidPoint(LineSegmentOf(A, F)), yAxis) = P;Projection(MidPoint(LineSegmentOf(B, F)), yAxis) = Q;Abs(LineSegmentOf(P, Q)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 75,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.366584062576294,
+        "b": 2.364173412322998,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.00017838670464698225,
+        "converged": true,
+        "iterations": 89
+      },
+      "output_path": "results/test/hyperbola/problem_0075.png",
+      "answer": "x^2-y^2/3=1",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 8*x);PointOnCurve(OneOf(Focus(G)), Directrix(H));Eccentricity(G) = 2",
+      "coords": {}
+    },
+    {
+      "index": 76,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0076.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;H: Circle;Expression(H) = (x^2 + y^2 = a^2);M: Point;AngleOf(F1, M, F2) = pi/4;L:Line;TangentOfPoint(F1,H)=L;Intersection(L,RightPart(G))=M",
+      "coords": {}
+    },
+    {
+      "index": 77,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0077.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "A: Point;Coordinate(A) = (0, 2);G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;l: Line;Directrix(G) = l;B: Point;Intersection(LineSegmentOf(F, A), G) = B;Z: Line;PointOnCurve(B, Z);IsPerpendicular(Z, l);M: Point;FootPoint(Z, l) = M;IsPerpendicular(LineSegmentOf(A, M), LineSegmentOf(M, F))",
+      "coords": {
+        "A": [
+          0.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 78,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0078.png",
+      "answer": "(sqrt(13) + 1)/3",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;O: Origin;F: Point;A: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F;IsDiameter(LineSegmentOf(O, F), H);OneOf(Intersection(H, G)) = A;Inclination(LineSegmentOf(O, A)) = pi/6",
+      "coords": {}
+    },
+    {
+      "index": 79,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.7320508075688772,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0079.png",
+      "answer": "4*sqrt(3)/3",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2 = 1);H: Line;PointOnCurve(RightFocus(G), H);IsPerpendicular(H, xAxis);A: Point;B: Point;L1: Line;L2: Line;Asymptote(G) = {L1, L2};Intersection(H, L1) = A;Intersection(H, L2) = B",
+      "coords": {}
+    },
+    {
+      "index": 80,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0080.png",
+      "answer": "b^2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;F1: Point;F2: Point;P: Point;M: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, G);Negation(P=Vertex(G));LeftFocus(G) = F1;RightFocus(G) = F2;HalfFocalLength(G) = c;c:Number;TangentPoint(InscribedCircle(TriangleOf(P,F1,F2)),LineSegmentOf(F1,F2))=M",
+      "coords": {}
+    },
+    {
+      "index": 81,
+      "success": false,
+      "conic_type": "parabola",
+      "error": "validation: parabola_point_off_curve",
+      "params": {
+        "p": 0.9927864074707031,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.00015326803259085864,
+        "converged": true,
+        "iterations": 44
+      },
+      "output_path": "results/test/parabola/problem_0081.png",
+      "answer": "(125*pi)/16",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;P: Point;Coordinate(P) = (4, 4);PointOnCurve(P, C);F: Point;Focus(C) = F;M: Point;Coordinate(M) = (-1, 4)",
+      "coords": {
+        "P": [
+          4.0,
+          4.0
+        ],
+        "M": [
+          -1.0,
+          4.0
+        ]
+      },
+      "validation_reasons": [
+        "parabola_point_off_curve"
+      ]
+    },
+    {
+      "index": 82,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0082.png",
+      "answer": "4*x - y - 15 = 0",
+      "fact_expr": "G: Parabola;A: Point;B: Point;Q: Point;Expression(G) = (y^2 = 8*x);IsChordOf(LineSegmentOf(A,B),G);Coordinate(Q) = (4, 1);MidPoint(LineSegmentOf(A,B))=Q",
+      "coords": {
+        "Q": [
+          4.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 83,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0083.png",
+      "answer": "x=-3/2",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*p*x);p: Number;P: Point;F: Point;Q: Point;O: Origin;p>0;Focus(C)=F;PointOnCurve(P,C);IsPerpendicular(LineSegmentOf(P,F),xAxis);PointOnCurve(Q, xAxis);IsPerpendicular(LineSegmentOf(P, Q), LineSegmentOf(O, P));Abs(LineSegmentOf(F, Q)) = 6",
+      "coords": {}
+    },
+    {
+      "index": 84,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.366584062576294,
+        "b": 2.364173412322998,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.00017838670464698225,
+        "converged": true,
+        "iterations": 89
+      },
+      "output_path": "results/test/hyperbola/problem_0084.png",
+      "answer": "x^2/4-y^2/12=1",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y^2 + (-a + x)^2 = 3);Eccentricity(C) = 2;IsTangent(Asymptote(C),G)",
+      "coords": {}
+    },
+    {
+      "index": 85,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.7320508075688772,
+        "b": 2.0,
+        "orientation": "vertical"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0085.png",
+      "answer": "{(sqrt(3)/2, +oo), (-oo, -sqrt(3)/2)}",
+      "fact_expr": "l: Line;O: Origin;PointOnCurve(O, l) = True;G: Hyperbola;Expression(G) = (x^2/4 - y^2/3 = -1);NumIntersection(l, G) = 2",
+      "coords": {}
+    },
+    {
+      "index": 86,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0086.png",
+      "answer": "2",
+      "fact_expr": "M: Point;Coordinate(M) = (-1, 1);C: Parabola;Expression(C) = (y^2 = 4*x);G: Line;PointOnCurve(Focus(C),G) = True;Slope(G) = k;k: Number;Intersection(G, C) = {A, B};A: Point;B: Point;AngleOf(A, M, B) = ApplyUnit(90, degree)",
+      "coords": {
+        "M": [
+          -1.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 87,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0087.png",
+      "answer": "",
+      "fact_expr": "H: Parabola;Expression(H) = (y^2 = 2*(p*x));p: Number;p>0;M: Point;m: Number;m>0;Coordinate(M) = (1, m);PointOnCurve(M, H);Distance(M, Focus(H)) = 5;G: Hyperbola;Expression(G) = (-y^2 + x^2/a = 1);a: Real;A: Point;LeftVertex(G) = A;IsParallel(OneOf(Asymptote(G)), LineOf(A, M))",
+      "coords": {}
+    },
+    {
+      "index": 88,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 89,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0089.png",
+      "answer": "2*sqrt(3)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/b^2 = 1);b: Number;b>0;Expression(Asymptote(G)) = (sqrt(3)*x + pm*y = 0)",
+      "coords": {}
+    },
+    {
+      "index": 90,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 91,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0091.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (3, 1);Focus(G) = F;PointOnCurve(P, G)",
+      "coords": {
+        "A": [
+          3.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 92,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 93,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0093.png",
+      "answer": "(3*sqrt(3)/4, +oo)",
+      "fact_expr": "C: Hyperbola;M: Point;F1: Point;F2: Point;P: Point;Expression(C) = (x^2 - y^2/3 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(M, LeftPart(C));Perimeter(TriangleOf(M, F1, F2)) = 9;PointOnCurve(P, RightPart(C))",
+      "coords": {}
+    },
+    {
+      "index": 94,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 95,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 96,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.8284271247461903,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 8.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0096.png",
+      "answer": "8",
+      "fact_expr": "G: Ellipse;P: Point;O: Origin;A: Point;F1: Point;F2: Point;x0: Number;y0: Number;Expression(G) = (x^2/9 + y^2/8 = 1);Coordinate(P) = (x0, y0);LeftFocus(G) = F1;RightFocus(G) = F2;RightVertex(G) = A;DotProduct(VectorOf(O, P), VectorOf(O, A)) = 9;WhenMax(AngleOf(F1, P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 97,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0097.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;c:Number;A: Circle;F1: Point;F2: Point;P: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) =F1;RightFocus(C)=F2;Abs(LineSegmentOf(F1,F2))=2*c;O:Origin;Center(A)=O;Radius(A)=c;OneOf(Intersection(A, C)) = P;Area(TriangleOf(F1,P,F2)) = a^2",
+      "coords": {}
+    },
+    {
+      "index": 98,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0098.png",
+      "answer": "2*sqrt(10)",
+      "fact_expr": "G: Hyperbola;P: Point;F2: Point;F1: Point;Expression(G) = (x^2 - y^2/9 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0",
+      "coords": {}
+    },
+    {
+      "index": 99,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_point_off_curve",
+      "params": {
+        "a": 0.8176298651812968,
+        "b": 2.452642014841466,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 1.4992967561765154e-06,
+        "converged": true,
+        "iterations": 283
+      },
+      "output_path": "results/test/hyperbola/problem_0099.png",
+      "answer": "y^2/9 - x^2 = 1",
+      "fact_expr": "E: Hyperbola;F: Point;Coordinate(F) = (-sqrt(3), 6);PointOnCurve(F, E);Expression(Asymptote(E)) = (y = pm*3*x)",
+      "coords": {
+        "F": [
+          -1.7320508075688772,
+          6.0
+        ]
+      },
+      "validation_reasons": [
+        "hyperbola_point_off_curve"
+      ]
+    },
+    {
+      "index": 100,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 101,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0101.png",
+      "answer": "5",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*x);P: Point;t: Number;Coordinate(P) = (t, 3);PointOnCurve(P, C);F: Point;Focus(C) = F",
+      "coords": {}
+    },
+    {
+      "index": 102,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 103,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0103.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;b: Number;b>0;Expression(G) = (x^2/9 - y^2/b^2 = 1);Distance(RightFocus(G),Asymptote(G))=(1/2)*Distance(RightFocus(G),LeftVertex(G))",
+      "coords": {}
+    },
+    {
+      "index": 104,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 105,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 106,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 107,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0107.png",
+      "answer": "x^2/9 + y^2/4 = 1",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Focus(C) = {F1, F2};PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(F1, F2));Abs(LineSegmentOf(P, F1)) = 4/3;Abs(LineSegmentOf(P, F2)) = 14/3",
+      "coords": {}
+    },
+    {
+      "index": 108,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.8284271247461903,
+        "b": 2.8284271247461903,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0108.png",
+      "answer": "14+8*sqrt(2)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 8);RightFocus(G) = F2;F2: Point;P: Point;Q: Point;PointOnCurve(F2, LineSegmentOf(P, Q)) = True;IsChordOf(LineSegmentOf(P, Q), G) = True;Abs(LineSegmentOf(P, Q)) = 7;F1: Point;LeftFocus(G) = F1",
+      "coords": {}
+    },
+    {
+      "index": 109,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.7994307279586792,
+        "b": 1.695713604329748,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 1.1180587689958606e-05,
+        "converged": true,
+        "iterations": 86
+      },
+      "output_path": "results/test/ellipse/problem_0109.png",
+      "answer": "4",
+      "fact_expr": "G: Ellipse;I: Point;P: Point;F1: Point;F2: Point;Q: Point;lambda:Number;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) =F1;RightFocus(G)=F2;PointOnCurve(P,G);Quadrant(P)=1;Incenter(TriangleOf(P, F1, F2)) = I;Intersection(LineOf(P,I), xAxis) = Q;Eccentricity(G) = 1/3;VectorOf(P, Q) = lambda*VectorOf(I, Q);a:Number;b:Number;a>b;b>0",
+      "coords": {}
+    },
+    {
+      "index": 110,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0110.png",
+      "answer": "pm*sqrt(5)/2",
+      "fact_expr": "l: Line;G: Parabola;A: Point;F: Point;B: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, l);Intersection(l,G)={A,B};VectorOf(A, F) = 5*VectorOf(F, B)",
+      "coords": {}
+    },
+    {
+      "index": 111,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0111.png",
+      "answer": "(x-5)^2+y^2=16",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);H: Circle;RightFocus(G) = Center(H);IsTangent(Asymptote(G), H)",
+      "coords": {}
+    },
+    {
+      "index": 112,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0112.png",
+      "answer": "sqrt(6)/2",
+      "fact_expr": "C: Ellipse;a: Real;a>1;Expression(C) = (y^2 + x^2/a^2 = 1);Eccentricity(C) = sqrt(3)/3",
+      "coords": {}
+    },
+    {
+      "index": 113,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0113.png",
+      "answer": "2*sqrt(3)",
+      "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2 - y^2 = 1);Focus(G)={F1,F2};PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 114,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 115,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0115.png",
+      "answer": "2*sqrt(3)/3",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;LeftFocus(G) = F;L: Line;PointOnCurve(F, L) ;IsPerpendicular(L, OneOf(Asymptote(G))) ;Asymptote(G) = {l1, l2};Intersection(L, l1) = A;Intersection(L, l2) = B;A: Point;B: Point;l1: Line;l2: Line;Abs(LineSegmentOf(A, F))/Abs(LineSegmentOf(B, F)) = 1/2",
+      "coords": {}
+    },
+    {
+      "index": 116,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0116.png",
+      "answer": "(0, pm*b)",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;F: Point;c: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (c, 0);OneOf(Focus(G)) = F;P: Point;PointOnCurve(P, G);Max(Distance(F, P)) = m ;m: Number;n: Number;Min(Distance(F, P)) = n;P0: Point;PointOnCurve(P0, G);Distance(F, P0) = (m + n)/2",
+      "coords": {}
+    },
+    {
+      "index": 117,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          0.0,
+          0.0
+        ],
+        "radius": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/circle/problem_0117.png",
+      "answer": "(x^2/4 + y^2/3 = 1)&Negation(y = 0)",
+      "fact_expr": "G: Parabola;H: Circle;A: Point;B: Point;Coordinate(A) = (-1, 0);Coordinate(B) = (1, 0);Expression(H) = (x^2 + y^2 = 4);PointOnCurve(A, G);PointOnCurve(B, G);IsTangent(H, Directrix(G))",
+      "coords": {}
+    },
+    {
+      "index": 118,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0118.png",
+      "answer": "{sqrt(6)/2, -sqrt(6)/2, sqrt(2)/2, -sqrt(2)/2}",
+      "fact_expr": "H: Line;Expression(H) = (y = k*x + 1);k: Real;G: Hyperbola;Expression(G) = (x^2 - y^2 = 1);NumIntersection(H, G) = 1",
+      "coords": {}
+    },
+    {
+      "index": 119,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0119.png",
+      "answer": "3, 25/3",
+      "fact_expr": "G: Ellipse;m: Number;e: Number;Expression(G) = (x^2/5 + y^2/m = 1);Eccentricity(G) = e ;e = sqrt(10)/5",
+      "coords": {}
+    },
+    {
+      "index": 120,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0120.png",
+      "answer": "(-oo,-1]+[3,+oo)",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = x);B: Point;Coordinate(B) = (1, 1);PointOnCurve(B, G) = True;PointOnCurve(P, G) = True;PointOnCurve(Q, G) = True;P: Point;Q: Point;IsPerpendicular(LineSegmentOf(B, P), LineSegmentOf(P, Q)) = True",
+      "coords": {
+        "B": [
+          1.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 121,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "down"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0121.png",
+      "answer": "(0,-2)",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = -8*y)",
+      "coords": {}
+    },
+    {
+      "index": 122,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0122.png",
+      "answer": "5*sqrt(2)/2-1",
+      "fact_expr": "l: Line;G: Parabola;P: Point;d1:Number;d2:Number;Expression(G) = (y^2 = 4*x);Expression(l) = (x - y + 4 = 0);PointOnCurve(P, G);Distance(P, yAxis) = d1;Distance(P, l) = d2",
+      "coords": {}
+    },
+    {
+      "index": 123,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0123.png",
+      "answer": "",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G) = F2",
+      "coords": {}
+    },
+    {
+      "index": 124,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0124.png",
+      "answer": "1",
+      "fact_expr": "C: Parabola;E: Point;M: Point;N: Point;lambda: Real;lambda>=0;Expression(C) = (y^2 = 4*x);Coordinate(E) = (-lambda, 0);PointOnCurve(M, C);PointOnCurve(N, C);Min(DotProduct(VectorOf(E, M), VectorOf(E, N))) = 0",
+      "coords": {}
+    },
+    {
+      "index": 125,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 3.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0125.png",
+      "answer": "3",
+      "fact_expr": "C: Parabola;F: Point;N: Point;M: Point;l: Line;lambda: Number;lambda > 0;Expression(C) = (y^2 = 12*x);Focus(C) = F;Directrix(C) = l;PointOnCurve(M, C);PointOnCurve(N, l);LineSegmentOf(F, N) = lambda*LineSegmentOf(F, M);Abs(LineSegmentOf(M, F)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 126,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 127,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 128,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "y",
+        "x_coef": 9.0,
+        "y_coef": 25.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0128.png",
+      "answer": "(pm*3,0)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/25 = 1);P:Point;F1:Point;F2:Point;PointOnCurve(P,G);Focus(G)={F1,F2};WhenMax(Distance(P,F1)*Distance(P,F2))",
+      "coords": {}
+    },
+    {
+      "index": 129,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0129.png",
+      "answer": "y=pm*(3/5)*x",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2/25 - y^2/9 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 130,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.23606797749979,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 5.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0130.png",
+      "answer": "6",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/9 + y^2/5 = 1);M: Point;PointOnCurve(M, C) = True;F1: Point;F2: Point;Focus(C) = {F1, F2}",
+      "coords": {}
+    },
+    {
+      "index": 131,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0131.png",
+      "answer": "y^2=16*x",
+      "fact_expr": "G: Parabola;H: Ellipse;Expression(H) = (x^2/25 + y^2/9 = 1);Vertex(G) = Center(H);Focus(G) = RightFocus(H)",
+      "coords": {}
+    },
+    {
+      "index": 132,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0132.png",
+      "answer": "10",
+      "fact_expr": "G: Hyperbola;P: Point;Expression(G) = (x^2/9 - y^2/16 = 1);PointOnCurve(P, G);Distance(P, RightFocus(G)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 133,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 134,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 135,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 136,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0136.png",
+      "answer": "p=1/2",
+      "fact_expr": "C: Parabola;p: Number;A: Point;p>0;Expression(C) = (y^2 = 2*p*x);m:Number;Coordinate(A) = (4, m);PointOnCurve(A, C);Distance(A, Focus(C)) = 17/4",
+      "coords": {}
+    },
+    {
+      "index": 137,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0137.png",
+      "answer": "4*sqrt(10)/5",
+      "fact_expr": "G: Ellipse;H: Line;t: Number;A: Point;B: Point;Expression(G) = (x^2/4 + y^2 = 1);Expression(H) = (y = t + x);Intersection(H, G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 138,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 4.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0138.png",
+      "answer": "pm*4*sqrt(2)",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 16*x);F: Point;Focus(C) = F;M: Point;PointOnCurve(M, C);O: Origin;N: Point;Intersection(OverlappingLine(LineSegmentOf(F, M)), yAxis) = N;Abs(LineSegmentOf(F, N)) = 2*Abs(LineSegmentOf(O, M))",
+      "coords": {}
+    },
+    {
+      "index": 139,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_point_off_curve",
+      "params": {
+        "a": 2.8706156235768385,
+        "b": 0.9134866531697662,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 8.509210183089842e-07,
+        "converged": true,
+        "iterations": 93
+      },
+      "output_path": "results/test/ellipse/problem_0139.png",
+      "answer": "x^2/9+y^2=1",
+      "fact_expr": "G: Ellipse;P: Point;Coordinate(P) = (3, 0);PointOnCurve(Focus(G),xAxis);PointOnCurve(P,G);Length(MajorAxis(G))=3*Length(MinorAxis(G))",
+      "coords": {
+        "P": [
+          3.0,
+          0.0
+        ]
+      },
+      "validation_reasons": [
+        "ellipse_point_off_curve"
+      ]
+    },
+    {
+      "index": 140,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 141,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 2.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0141.png",
+      "answer": "(-oo, -sqrt(2)]",
+      "fact_expr": "P: Point;Coordinate(P) = (x0, y0);x0: Number;y0: Number;G: Hyperbola;Expression(G) = (x^2/2 - y^2/4 = 1);PointOnCurve(P, LeftPart(G))",
+      "coords": {}
+    },
+    {
+      "index": 142,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 2.8284271247461903,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 8.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0142.png",
+      "answer": "16",
+      "fact_expr": "l: Line;PointOnCurve(F2, l);G: Ellipse;Expression(G) = (x^2/16 + y^2/8 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;A: Point;B: Point;Intersection(l, G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 143,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 144,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0144.png",
+      "answer": "5/8",
+      "fact_expr": "l: Line;C: Parabola;p: Number;G: Circle;B: Point;A: Point;Q: Point;M: Point;N: Point;F: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;PointOnCurve(F, l);Slope(l) = sqrt(3);Intersection(l, C) = {A, B};IsDiameter(LineSegmentOf(A, B), G);Intersection(G, yAxis) = {M, N};MidPoint(LineSegmentOf(A, B)) = Q;Distance(F, Directrix(C)) = 3",
+      "coords": {}
+    },
+    {
+      "index": 145,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0145.png",
+      "answer": "y = pm*x*8/15",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;O: Origin;A: Point;F: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F;Slope(G) = 1/4;PointOnCurve(F, G);Intersection(G, OneOf(Asymptote(C))) = A;Quadrant(A) = 1;Abs(LineSegmentOf(O, A)) = Abs(LineSegmentOf(O, F))",
+      "coords": {}
+    },
+    {
+      "index": 146,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0146.png",
+      "answer": "5/3",
+      "fact_expr": "G: Hyperbola;Expression(G) = (16*x^2 - 9*y^2 = 144)",
+      "coords": {}
+    },
+    {
+      "index": 147,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 148,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 149,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0149.png",
+      "answer": "3/2",
+      "fact_expr": "l: Line;G: Parabola;B: Point;C: Point;F: Point;A: Point;Expression(G) = (y^2 = 4*x);Intersection(l, G) = {A, B};Intersection(l,Directrix(G))=C;Focus(G) = F;VectorOf(B, C) = 3*VectorOf(F, B)",
+      "coords": {}
+    },
+    {
+      "index": 150,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "y",
+        "x_coef": 3.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0150.png",
+      "answer": "8",
+      "fact_expr": "G: Ellipse;H: Line;A: Point;l:Line;B: Point;C: Point;D: Point;Expression(G) = (x^2/3 + y^2/4 = 1);Expression(H) = (x + y + 1 = 0);Expression(l) = (x+y-1=0);UpperFocus(G) = F;Intersection(H,G)={A,B};Intersection(l,G)={C,D};F:Point",
+      "coords": {}
+    },
+    {
+      "index": 151,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "y",
+        "x_coef": 1.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0151.png",
+      "answer": "",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2 + y^2/9 = 1);P: Point;Coordinate(P) = (1/2, 1/2);L: Line;PointOnCurve(P,L);Intersection(P,G) = {A,B};A: Point;B: Point;IsChordOf(LineSegmentOf(A,B),G);MidPoint(LineSegmentOf(A,B)) = P",
+      "coords": {
+        "P": [
+          0.5,
+          0.5
+        ]
+      }
+    },
+    {
+      "index": 152,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          -2.0,
+          0.0
+        ],
+        "radius": 1.4142135623730951
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/circle/problem_0152.png",
+      "answer": "{5, 1/3}",
+      "fact_expr": "l: Line;C: Circle;A: Point;B: Point;P: Point;Expression(C) = (y^2 + (x - 2)^2 = 2);Expression(l) = (x - y + 2 = 0);Intersection(l, xAxis) = A;PointOnCurve(P, l);PointOnCurve(B, C);IsPerpendicular(LineSegmentOf(A, B), LineSegmentOf(B, P))",
+      "coords": {}
+    },
+    {
+      "index": 153,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 154,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 155,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0155.png",
+      "answer": "2",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;G: Circle;Expression(G) = (x^2 + y^2 = a^2);L1: Line;L2: Line;TangentOfPoint(OneOf(Focus(C)), G) = {L1, L2};A: Point;B: Point;TangentPoint(L1, G) = A;TangentPoint(L2, G) = B;AngleOf(A, O, B) = ApplyUnit(120, degree);O: Origin",
+      "coords": {}
+    },
+    {
+      "index": 156,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0156.png",
+      "answer": "1",
+      "fact_expr": "P: Point;Q: Point;PointOnCurve(P,G) = True;PointOnCurve(Q,G) = True;G: Parabola;Expression(G) = (x^2 = 2*y);XCoordinate(P) = 4;XCoordinate(Q) = -2;L1: Line;L2: Line;TangentOfPoint(P,G) = L1;TangentOfPoint(Q,G) = L2;Intersection(L1,L2) = A;A:Point",
+      "coords": {}
+    },
+    {
+      "index": 157,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0157.png",
+      "answer": "(sqrt(17)-3)/2",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;M: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(M, G);Distance(M,LeftFocus(G))=2*Distance(M,RightDirectrix(G))",
+      "coords": {}
+    },
+    {
+      "index": 158,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0158.png",
+      "answer": "x^2/16-y^2/9=1",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Point;C:Parabola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(H) = (4, 3);PointOnCurve(H,OneOf(Asymptote(G)));Expression(C)=(y^2=20*x);PointOnCurve(OneOf(Focus(G)),Directrix(C))",
+      "coords": {
+        "H": [
+          4.0,
+          3.0
+        ]
+      }
+    },
+    {
+      "index": 159,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0159.png",
+      "answer": "(0, 1]+[9, +oo)",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/3 + y^2/m = 1);m: Number;A: Point;B: Point;Endpoint(MajorAxis(C)) = {A, B};M: Point;PointOnCurve(M, C);AngleOf(A, M, B) = ApplyUnit(120, degree)",
+      "coords": {}
+    },
+    {
+      "index": 160,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.23606797749979,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 5.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0160.png",
+      "answer": "2/3",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/5 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 161,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0161.png",
+      "answer": "(1, 2]",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 3*Abs(LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 162,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0162.png",
+      "answer": "9",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(90, degree)",
+      "coords": {}
+    },
+    {
+      "index": 163,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.23606797749979,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0163.png",
+      "answer": "x^2/10 - y^2/6 = 1",
+      "fact_expr": "G: Hyperbola;Z: Hyperbola;Expression(G) = (x^2/5 - y^2/3 = 1);Asymptote(Z) = Asymptote(G);Expression(OneOf(Directrix(Z))) = (x = 5/2)",
+      "coords": {}
+    },
+    {
+      "index": 164,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0164.png",
+      "answer": "8",
+      "fact_expr": "G: Parabola;A: Point;F: Point;P: Point;l: Line;Expression(G) = (y^2 = 8*x);Focus(G) = F;Directrix(G)=l;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P,A),l);FootPoint(LineSegmentOf(P,A),l)=A;Inclination(LineSegmentOf(A, F)) = (2*pi)/3",
+      "coords": {}
+    },
+    {
+      "index": 165,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          0.0,
+          0.0
+        ],
+        "radius": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/circle/problem_0165.png",
+      "answer": "x^2+y^2=4/3",
+      "fact_expr": "G: Circle;Expression(G) = (x^2 + y^2 = 1);P: Point;TangentOfPoint(P,G) = {LineSegmentOf(P,A),LineSegmentOf(P,B)};TangentPoint(LineSegmentOf(P,A),G) = A;TangentPoint(LineSegmentOf(P,B),G) = B;A: Point;B: Point;AngleOf(A, P, B) = ApplyUnit(120, degree)",
+      "coords": {}
+    },
+    {
+      "index": 166,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 2.8284271247461903,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 8.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0166.png",
+      "answer": "16",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/16 + y^2/8 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, l) ;l: Line;Intersection(l, C) = {A, B};A: Point;B: Point",
+      "coords": {}
+    },
+    {
+      "index": 167,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0167.png",
+      "answer": "sqrt(10)/2",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;P: Point;PointOnCurve(P, C);O: Origin;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;Abs(LineSegmentOf(O, P)) = Abs(LineSegmentOf(O, F2));Tan(AngleOf(P, F2, F1)) = 3",
+      "coords": {}
+    },
+    {
+      "index": 168,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 169,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0169.png",
+      "answer": "5/7",
+      "fact_expr": "l: Line;C: Ellipse;A: Point;F2: Point;F1: Point;B: Point;LeftFocus(C) = F1;RightFocus(C)=F2;Expression(C)=(x^2/a^2 + y^2/b^2 = 1);PointOnCurve(F1, l);Intersection(l, C) = {A, B};DotProduct((VectorOf(A,F2)+VectorOf(F1,F2)),VectorOf(A,F1))=0;VectorOf(A,F1)=4/3*VectorOf(F1,B);a:Number;b:Number;a>b;b>0",
+      "coords": {}
+    },
+    {
+      "index": 170,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.125,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0170.png",
+      "answer": "y = -1/8",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = y/2)",
+      "coords": {}
+    },
+    {
+      "index": 171,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0171.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);O: Origin;Abs(VectorOf(O, P)) = Abs(VectorOf(O, F1));Area(TriangleOf(F1, F2, P)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 172,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0172.png",
+      "answer": "y^2=6*x",
+      "fact_expr": "l: Line;C: Parabola;p: Number;A: Point;B: Point;F: Point;p>0;Expression(C) = (y^2 = 2*p*x);Focus(C) = F;PointOnCurve(F, l);Slope(l) = sqrt(3);Intersection(l, C) = {A, B};YCoordinate(MidPoint(LineSegmentOf(A,B))) = sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 173,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0173.png",
+      "answer": "sqrt(15)",
+      "fact_expr": "G: Parabola;A: Point;B: Point;P: Point;Expression(G) = (y^2 = 4*x);Coordinate(P) = (1, 1);IsChordOf(LineSegmentOf(A,B),G);MidPoint(LineSegmentOf(A,B))=P",
+      "coords": {
+        "P": [
+          1.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 174,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0174.png",
+      "answer": "16/5",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 175,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 8.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 64.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0175.png",
+      "answer": "x^2/48-y^2/16=1",
+      "fact_expr": "G: Hyperbola;H: Ellipse;O:Origin;Expression(H) = (x^2 + 4*y^2 = 64);Center(G)=O;Vertex(G)=Focus(H);FocalLength(G) = 16",
+      "coords": {}
+    },
+    {
+      "index": 176,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0176.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Distance(RightFocus(G), Asymptote(G)) = sqrt(3)/4*FocalLength(G)",
+      "coords": {}
+    },
+    {
+      "index": 177,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0177.png",
+      "answer": "8",
+      "fact_expr": "l: Line;P: Point;Q: Point;Expression(G) = (y^2 = 4*x);Coordinate(P) = (x1,y1);Coordinate(Q) = (x2,y2);PointOnCurve(Focus(G),l);Intersection(l, G) = {P,Q};x1:Number;x2:Number;y1:Number;y2:Number;x1+x2=6;G:Parabola",
+      "coords": {}
+    },
+    {
+      "index": 178,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 5.656854249492381,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0178.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;A: Point;x0:Number;y0:Number;Expression(G) = (x^2/4 - y^2/32 = 1);Coordinate(A) = (x0, y0);PointOnCurve(A, RightPart(G));Distance(A,RightFocus(G)) = 2*x0",
+      "coords": {}
+    },
+    {
+      "index": 179,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "left"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0179.png",
+      "answer": "1",
+      "fact_expr": "G: Parabola;H: Line;A: Point;F: Point;B: Point;Expression(G) = (y^2 = -4*x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};Slope(H) = sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 180,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.6259966743174588,
+        "b": 2.298015331604694,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 1.93011531933813e-06,
+        "converged": true,
+        "iterations": 91
+      },
+      "output_path": "results/test/hyperbola/problem_0180.png",
+      "answer": "x^2/3-y^2/6=1",
+      "fact_expr": "G: Hyperbola;Coordinate(Focus(G)) = (pm*3, 0);Expression(Asymptote(G)) = (y=pm*(sqrt(2)*x))",
+      "coords": {}
+    },
+    {
+      "index": 181,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "down"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0181.png",
+      "answer": "y = 1",
+      "fact_expr": "C: Parabola;Expression(C) = (y = -x^2/4)",
+      "coords": {}
+    },
+    {
+      "index": 182,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 2.8284271247461903,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0182.png",
+      "answer": "12",
+      "fact_expr": "C: Hyperbola;A: Point;P: Point;F: Point;Expression(C) = (x^2 - y^2/8 = 1);Coordinate(A) = (0, 4);RightFocus(C)= F;PointOnCurve(P,LeftPart(C))",
+      "coords": {
+        "A": [
+          0.0,
+          4.0
+        ]
+      }
+    },
+    {
+      "index": 183,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 184,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.33732376966822,
+        "b": 1.7454107534785184,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 1.8920117601792354e-05,
+        "converged": true,
+        "iterations": 51
+      },
+      "output_path": "results/test/hyperbola/problem_0184.png",
+      "answer": "{5/4, 5/3}",
+      "fact_expr": "G: Hyperbola;Expression(Asymptote(G)) = (y = pm*(3/4)*x)",
+      "coords": {}
+    },
+    {
+      "index": 185,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0185.png",
+      "answer": "y^2/7-x^2/14=1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/2 - y^2 = 1);G1: Hyperbola;Asymptote(G) = Asymptote(G1);H: Point;Coordinate(H) = (-2, -3);PointOnCurve(H, G1) = True",
+      "coords": {
+        "H": [
+          -2.0,
+          -3.0
+        ]
+      }
+    },
+    {
+      "index": 186,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0186.png",
+      "answer": "10",
+      "fact_expr": "P: Point;G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);PointOnCurve(P, G) = True;F1: Point;F2: Point;Focus(G) = {F1, F2}",
+      "coords": {}
+    },
+    {
+      "index": 187,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.3645083904266357,
+        "b": 1.1801031827926636,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 1.282816697312228e-06,
+        "converged": true,
+        "iterations": 88
+      },
+      "output_path": "results/test/hyperbola/problem_0187.png",
+      "answer": "5/2",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;Omega: Circle;M: Point;F1: Point;F2: Point;N: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(Omega) = (-2*y + x^2 + y^2 = 0);LeftFocus(C) = F1;RightFocus(C) = F2;Length(RealAxis(C)) = 2;Expression(Asymptote(C)) = (y = pm*(1/2)*x);Abs(LineSegmentOf(M, F1)) - Abs(LineSegmentOf(M, F2)) = 2;PointOnCurve(N, Omega)",
+      "coords": {}
+    },
+    {
+      "index": 188,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0188.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;p: Number;H: Line;O: Origin;A: Point;F:Point;p>0;Expression(G) = (y^2 = p*x);Slope(H)=2;Focus(G)=F;PointOnCurve(F,H);Intersection(H,yAxis)=A;Area(TriangleOf(O,A,F))=1",
+      "coords": {}
+    },
+    {
+      "index": 189,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 190,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 191,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 192,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.7320508075688772,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0192.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;H: Parabola;p: Number;Expression(G) = (x^2/3 - y^2 = 1);p>0;Expression(H) = (y^2 = -2*p*x);Focus(H) = LeftFocus(G)",
+      "coords": {}
+    },
+    {
+      "index": 193,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 194,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.4149149626832767,
+        "b": 2.450642358201144,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 1.0319236343469396e-05,
+        "converged": true,
+        "iterations": 94
+      },
+      "output_path": "results/test/hyperbola/problem_0194.png",
+      "answer": "y^2/3-x^2=1",
+      "fact_expr": "G: Hyperbola;Expression(OneOf(Asymptote(G))) = (y = sqrt(3)*x);H: Point;Coordinate(H) = (0, 2);OneOf(Focus(G)) = H",
+      "coords": {
+        "H": [
+          0.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 195,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0195.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2=2*p*x);p: Number;p>0;A: Point;B: Point;IsChordOf(LineSegmentOf(A,B),G) = True;Focus(G) = F;F: Point;PointOnCurve(F,LineSegmentOf(A,B)) = True;l: Line;Directrix(G) = l;IsPerpendicular(LineSegmentOf(A,A1),l) = True;A1: Point;FootPoint(LineSegmentOf(A,A1),l) = A1;IsPerpendicular(LineSegmentOf(B,B1),l) = True;B1: Point;FootPoint(LineSegmentOf(B,B1),l) = B1;N: Point;MidPoint(LineSegmentOf(A1,B1)) = N;LineSegmentOf(A,A1) = 4;LineSegmentOf(B,B1) = 2",
+      "coords": {}
+    },
+    {
+      "index": 196,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 197,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 198,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 199,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0199.png",
+      "answer": "4*sqrt(3)/3",
+      "fact_expr": "l: Line;C: Parabola;A: Point;O: Origin;B: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;Inclination(l)=pi/3;PointOnCurve(F,l);Intersection(l,C)={A,B}",
+      "coords": {}
+    },
+    {
+      "index": 200,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 201,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0201.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;P: Point;F: Point;Expression(G) = (x^2 = 4*y);PointOnCurve(P, G);Focus(G) = F;Distance(P,F) = 3",
+      "coords": {}
+    },
+    {
+      "index": 202,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0202.png",
+      "answer": "2\nApplyUnit(120, degree)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/2 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G) = True;Abs(LineSegmentOf(P, F1)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 203,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.366584062576294,
+        "b": 2.364173412322998,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.00017838670464698225,
+        "converged": true,
+        "iterations": 89
+      },
+      "output_path": "results/test/hyperbola/problem_0203.png",
+      "answer": "x^2 - y^2/3 = 1",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;A: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2=2*p*x);Coordinate(A) = (2, 4);PointOnCurve(OneOf(Focus(G)), Directrix(H));Eccentricity(G) = 2;p:Number;PointOnCurve(A,H)",
+      "coords": {
+        "A": [
+          2.0,
+          4.0
+        ]
+      }
+    },
+    {
+      "index": 204,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 205,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.4142135623730951,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0205.png",
+      "answer": "12",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/2 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;l: Line;PointOnCurve(F1, l);P: Point;Q: Point;Intersection(l, LeftPart(G)) = {P, Q};Abs(LineSegmentOf(P, Q)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 206,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0206.png",
+      "answer": "sqrt(2)/4",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;A: Point;B: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Focus(C) ={F1,F2};IsPerpendicular(LineSegmentOf(P, F2), LineSegmentOf(F1, F2));Abs(LineSegmentOf(P, F2)) = Abs(LineSegmentOf(F1, F2));Intersection(LineSegmentOf(P, F1), C) = A;Intersection(LineSegmentOf(P, F2), C) =B;Abs(LineSegmentOf(P, A)) = Abs(LineSegmentOf(A, F1))",
+      "coords": {}
+    },
+    {
+      "index": 207,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "down"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0207.png",
+      "answer": "(-oo,-6)",
+      "fact_expr": "G: Parabola;Expression(G) = (y = -x^2/8);A: Point;Intersection(Directrix(G), SymmetryAxis(G)) = A;H: Line;PointOnCurve(A, H) = True;Intersection(H, G) = {M, N};M: Point;N: Point;PointOnCurve(B, SymmetryAxis(G)) = True;B: Point;DotProduct((VectorOf(B, M) + VectorOf(M, N)/2), VectorOf(M, N)) = 0;O: Origin",
+      "coords": {}
+    },
+    {
+      "index": 208,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0208.png",
+      "answer": "sqrt(3)-1",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, G);Focus(G) = {F1,  F2};AngleOf(P, F1, F2) = pi/6;Abs(VectorOf(P, F1) + VectorOf(P, F2)) = Abs(VectorOf(F1, F2))",
+      "coords": {}
+    },
+    {
+      "index": 209,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0209.png",
+      "answer": "{3,6}",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;L: Line;PointOnCurve(F,L);Intersection(L, G) = {A, B};A: Point;B: Point;Abs(LineSegmentOf(F, A)) = 2*Abs(LineSegmentOf(F, B));H: Circle;Expression(H) = (x^2 + y^2 = 2*p);Length(InterceptChord(LineOf(A, B), H)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 210,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0210.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;H: Circle;F: Point;A: Point;M: Point;O:Origin;Vertex(G)=O;Coordinate(F) = (0, 1);Focus(G)=F;Expression(H) = ((x + 1)^2 + (y - 4)^2 = 1);PointOnCurve(A,G);PointOnCurve(M,H)",
+      "coords": {
+        "F": [
+          0.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 211,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0211.png",
+      "answer": "{5,3}",
+      "fact_expr": "G: Ellipse;m: Number;Expression(G) = (y^2/4 + x^2/m = 1);FocalLength(G) = 2",
+      "coords": {}
+    },
+    {
+      "index": 212,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0212.png",
+      "answer": "sqrt(13)/2",
+      "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;P: Point;F: Point;O: Origin;A: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F;PointOnCurve(F,l);IsPerpendicular(l,OneOf(Asymptote(C)));Intersection(l,RightPart(C))=P;MidPoint(LineSegmentOf(P, F)) = A;Abs(LineSegmentOf(O, A)) = -a + (3*b)/2",
+      "coords": {}
+    },
+    {
+      "index": 213,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0213.png",
+      "answer": "sqrt(7)/4",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;P: Point;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G) = F2;AngleOf(F1, P, F2) = ApplyUnit(60, degree);Abs(LineSegmentOf(P, F1)) = 3*Abs(LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 214,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 215,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 216,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 217,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0217.png",
+      "answer": "1/2",
+      "fact_expr": "G: Parabola;I: Point;A: Point;B: Point;x1: Number;y1: Number;x2: Number;y2: Number;Expression(G) = (y^2 = 4*x);Coordinate(I) = (0, 2);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);IsChordOf(LineSegmentOf(A, B), G);Intersection(OverlappingLine(LineSegmentOf(A, B)), yAxis) = I",
+      "coords": {
+        "I": [
+          0.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 218,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 219,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 220,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0220.png",
+      "answer": "1",
+      "fact_expr": "G: Parabola;p: Number;H: Line;M: Point;A: Point;p>0;x1:Number;y1:Number;x2:Number;y2:Number;Expression(G) = (y^2 = 2*(p*x));Coordinate(M) = (p, 0);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);Intersection(H,G) = {A, B};y1*y2=-2;B:Point;PointOnCurve(M,H)",
+      "coords": {}
+    },
+    {
+      "index": 221,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0221.png",
+      "answer": "4*a+2*m",
+      "fact_expr": "A: Point;B: Point;G: Hyperbola;F1: Point;F2: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(A, RightPart(G));PointOnCurve(B, RightPart(G));Focus(G)={F1,F2};RightFocus(G) = F2;PointOnCurve(F2, LineSegmentOf(A, B));Abs(LineSegmentOf(A, B)) = m;m:Number;b:Number;a:Number",
+      "coords": {}
+    },
+    {
+      "index": 222,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0222.png",
+      "answer": "0.2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;F1: Point;P: Point;F2: Point;a>0;b>0;L:Line;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(F1,L);IsPerpendicular(L,xAxis);Intersection(L,G)=P;AngleOf(F1,P,F2)=ApplyUnit(45,degree)",
+      "coords": {}
+    },
+    {
+      "index": 223,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0223.png",
+      "answer": "-2",
+      "fact_expr": "l: Line;C: Parabola;A: Point;O: Origin;H: Point;B: Point;Expression(C) = (y^2 = 4*x);Coordinate(H) = (2, 0);PointOnCurve(H, l);Intersection(l, C) = {A, B}",
+      "coords": {
+        "H": [
+          2.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 224,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0224.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 225,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0225.png",
+      "answer": "4*sqrt(2)",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Focus(C)={F1,F2};PointOnCurve(P,RightPart(C));Abs(LineSegmentOf(P, F1)) = 3*Abs(LineSegmentOf(P, F2));Expression(Asymptote(C)) = (y = pm*x);Area(TriangleOf(P, F1, F2)) = 4*sqrt(2)",
+      "coords": {}
+    },
+    {
+      "index": 226,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 227,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0227.png",
+      "answer": "25*pi/8",
+      "fact_expr": "L: Line;Expression(L) = (x - y - 4 = 0);P: Point;PointOnCurve(P,L) = True;Negation(PointOnCurve(P,xAxis));Z: Parabola;Expression(Z) = (x^2 = 4*y);L1: Line;L2: Line;TangentOfPoint(P,Z) = {L1,L2};Intersection(L1,xAxis) = G;G: Point;Intersection(L2,xAxis) = H;H: Point",
+      "coords": {}
+    },
+    {
+      "index": 228,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0228.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 229,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0229.png",
+      "answer": "-1/2",
+      "fact_expr": "m: Line;G: Ellipse;O: Origin;P: Point;P1: Point;P2: Point;Expression(G) = (x^2/2 + y^2 = 1);Intersection(m, G) = {P1, P2};MidPoint(LineSegmentOf(P1, P2)) = P;k1: Number;Slope(m) = k1;Negation(k1 = 0);k2: Number;Slope(LineOf(O, P)) = k2",
+      "coords": {}
+    },
+    {
+      "index": 230,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0230.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/b^2 = 1);b: Number;0 < b;b < 2;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, l) = True;l: Line;Intersection(l, G) = {A, B};B: Point;A: Point;Max(Abs(VectorOf(A, F2)) + Abs(VectorOf(B, F2))) = 5",
+      "coords": {}
+    },
+    {
+      "index": 231,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 232,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0232.png",
+      "answer": "1/2",
+      "fact_expr": "G: Ellipse;H: Line;Expression(G) = (x^2/4 + y^2/3 = 1);Expression(H) = (y = x*(sqrt(3)/3))",
+      "coords": {}
+    },
+    {
+      "index": 233,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 234,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 235,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 236,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 237,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_ecc_mismatch;ellipse_focus_mismatch",
+      "params": {
+        "a": 1.9257268905639648,
+        "b": 1.7993911322594118,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 2.9999999999999996
+      },
+      "optimization": {
+        "final_loss": 2.5576271215184017e-15,
+        "converged": true,
+        "iterations": 2
+      },
+      "output_path": "results/test/ellipse/problem_0237.png",
+      "answer": "x^2/4+y^2/3=1",
+      "fact_expr": "C: Ellipse;F: Point;O:Origin;Coordinate(F) = (1, 0);RightFocus(C) = F;Center(C)=O;Eccentricity(C)=1/2",
+      "coords": {
+        "F": [
+          1.0,
+          0.0
+        ]
+      },
+      "validation_reasons": [
+        "ellipse_ecc_mismatch",
+        "ellipse_focus_mismatch"
+      ]
+    },
+    {
+      "index": 238,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0238.png",
+      "answer": "3",
+      "fact_expr": "G: Parabola;M: Point;F: Point;Focus(G)=F;Expression(G) = (y^2 = 4*x);PointOnCurve(M,G);Distance(M,F)=Abs(LineSegmentOf(M,F));Abs(LineSegmentOf(M,F))=4;XCoordinate(M)=x1;x1:Number",
+      "coords": {}
+    },
+    {
+      "index": 239,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0239.png",
+      "answer": "6",
+      "fact_expr": "G: Parabola;p: Number;H: Circle;p>0;Expression(G) = (x^2 = 2*(p*y));Expression(H) = (y^2 + (x - 1)^2 = 9);IsTangent(H, Directrix(G))",
+      "coords": {}
+    },
+    {
+      "index": 240,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0240.png",
+      "answer": "3",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Focus(C) = {F1, F2};PointOnCurve(P, C);IsPerpendicular(VectorOf(P, F2),VectorOf(P, F1));Area(TriangleOf(P, F1, F2)) = 9",
+      "coords": {}
+    },
+    {
+      "index": 241,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 242,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 243,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 244,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0244.png",
+      "answer": "y^2=-36*(x-5)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1);H: Parabola;RightFocus(G) = Vertex(H);LeftVertex(G) = Focus(H)",
+      "coords": {}
+    },
+    {
+      "index": 245,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.4149149626832767,
+        "b": 2.450642358201144,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 1.0319236343469396e-05,
+        "converged": true,
+        "iterations": 94
+      },
+      "output_path": "results/test/hyperbola/problem_0245.png",
+      "answer": "{2,2*sqrt(3)/3}",
+      "fact_expr": "G: Hyperbola;O:Origin;Center(G)=O;Expression(OneOf(Asymptote(G))) = (y = sqrt(3)*x)",
+      "coords": {}
+    },
+    {
+      "index": 246,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 8.0,
+        "b": 6.0,
+        "major_axis": "x",
+        "x_coef": 64.0,
+        "y_coef": 36.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0246.png",
+      "answer": "12",
+      "fact_expr": "G: Ellipse;P: Point;F1:Point;F2:Point;Expression(G) = (x^2/64 + y^2/36 = 1);PointOnCurve(P, G);OneOf(Focus(G))=F1;OneOf(Focus(G))=F2;Negation(F1=F2);Distance(P, F1) = 4",
+      "coords": {}
+    },
+    {
+      "index": 247,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 2.449489742783178,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 6.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0247.png",
+      "answer": "16",
+      "fact_expr": "G: Ellipse;B: Point;A: Point;F1: Point;F2: Point;Expression(G) = (x^2/16 + y^2/6 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2, LineOf(A, B));Intersection(LineOf(A, B), G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 248,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "y",
+        "x_coef": 1.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0248.png",
+      "answer": "[-7, 1]",
+      "fact_expr": "l: Line;G: Ellipse;C: Point;A: Point;B: Point;Expression(G) = (x^2 + y^2/9 = 1);Coordinate(C) = (1, 1);a: Number;b: Number;Expression(l) = (a*x + b*y = 0);Intersection(l, G) = {A, B}",
+      "coords": {
+        "C": [
+          1.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 249,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0249.png",
+      "answer": "{0}+[1/2, oo)",
+      "fact_expr": "G: Parabola;H: Line;k: Number;Expression(G) = (y^2 = 2*x);Expression(H) = (y = k*x + 1);NumIntersection(H, G) <= 1",
+      "coords": {}
+    },
+    {
+      "index": 250,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 251,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 252,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0252.png",
+      "answer": "-4/5",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;G: Line;Expression(G) = (y = 2*x - 4);A: Point;B: Point;Intersection(G, C) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 253,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0253.png",
+      "answer": "sqrt(3)/2",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;B1: Point;B2: Point;UpperVertex(C) = B1;LowerVertex(C) = B2;M: Point;PointOnCurve(M, C);Negation(PointOnCurve(M, axis));Slope(LineOf(M, B1))*Slope(LineOf(M, B2)) = -b/(2*a)",
+      "coords": {}
+    },
+    {
+      "index": 254,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0254.png",
+      "answer": "{(-oo,0)+[4,oo)}",
+      "fact_expr": "G: Parabola;H: Curve;a: Real;Expression(G) = (y^2 = 8*x);Expression(H) = And(x^2/a+y^2/4=1,y>=0);NumIntersection(Directrix(G), H) = 1",
+      "coords": {}
+    },
+    {
+      "index": 255,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0255.png",
+      "answer": "2*sqrt(2)/3",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;H: Line;PointOnCurve(F,H);A: Point;B: Point;Intersection(H, G) = {A, B};WhenMin(Abs(LineSegmentOf(A, F)) + 4*Abs(LineSegmentOf(B, F)))",
+      "coords": {}
+    },
+    {
+      "index": 256,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 257,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0257.png",
+      "answer": "sqrt(2)-1",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;A: Point;PointOnCurve(A, G);F: Point;OneOf(Focus(G)) = F;IsPerpendicular(LineSegmentOf(A, F), xAxis);FocalLength(G) = Abs(LineSegmentOf(A, F))",
+      "coords": {}
+    },
+    {
+      "index": 258,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0258.png",
+      "answer": "1/2",
+      "fact_expr": "G: Parabola;A: Point;F: Point;B: Point;Expression(G) = (x^2 = 4*y);Focus(G) = F;PointOnCurve(A, G);PointOnCurve(B, G);VectorOf(A, F) = lambda*VectorOf(F, B);Abs(VectorOf(A, F)) = 3/2;lambda:Number",
+      "coords": {}
+    },
+    {
+      "index": 259,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0259.png",
+      "answer": "32/3",
+      "fact_expr": "G: Parabola;p: Number;H: Line;F: Point;A: Point;B: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(F) = (1, 0);Focus(G) = F;PointOnCurve(F, H);Intersection(H, Directrix(G)) = A;OneOf(Intersection(H, G)) = B;VectorOf(F, A) = -3*VectorOf(F, B)",
+      "coords": {
+        "F": [
+          1.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 260,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0260.png",
+      "answer": "9",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2/3 = 1);F: Point;LeftFocus(C) = F;A: Point;Coordinate(A) = (-3, -3);P: Point;PointOnCurve(P, C)",
+      "coords": {
+        "A": [
+          -3.0,
+          -3.0
+        ]
+      }
+    },
+    {
+      "index": 261,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "left"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0261.png",
+      "answer": "2*sqrt(13)",
+      "fact_expr": "G: Parabola;A: Point;F: Point;P: Point;O: Origin;Expression(G) = (y^2 = -8*x);Focus(G) = F;PointOnCurve(P, Directrix(G));PointOnCurve(A, G);Abs(LineSegmentOf(A, F)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 262,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0262.png",
+      "answer": "sqrt(5)/2",
+      "fact_expr": "G: Parabola;p: Number;B: Point;A: Point;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;PointOnCurve(A, G);PointOnCurve(B, G);Quadrant(A) = 1;Quadrant(B) = 1;Abs(LineSegmentOf(A, F)) = 2;Abs(LineSegmentOf(B, F)) = 4;Abs(LineSegmentOf(A, B)) = 3",
+      "coords": {}
+    },
+    {
+      "index": 263,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0263.png",
+      "answer": "(sqrt(5)-1)/2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;O: Origin;M: Point;N: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(RightFocus(G),H);IsPerpendicular(RealAxis(G),H);Intersection(H,G) = {M,N};IsPerpendicular(LineSegmentOf(O, M), LineSegmentOf(O, N))",
+      "coords": {}
+    },
+    {
+      "index": 264,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0264.png",
+      "answer": "9",
+      "fact_expr": "G: Hyperbola;H: Circle;Z: Circle;P: Point;M: Point;N: Point;Expression(G) = (x^2/9 - y^2/16 = 1);Expression(H) = (y^2 + (x - 5)^2 = 4);Expression(Z) = (y^2 + (x + 5)^2 = 1);PointOnCurve(P, G);PointOnCurve(M, H);PointOnCurve(N, Z)",
+      "coords": {}
+    },
+    {
+      "index": 265,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0265.png",
+      "answer": "sqrt(6)/2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(OneOf(Asymptote(G))) = (y = x*(sqrt(2)/2));Eccentricity(G)=e;e:Number",
+      "coords": {}
+    },
+    {
+      "index": 266,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0266.png",
+      "answer": "x^2-4*y^2=1",
+      "fact_expr": "O: Origin;P: Point;G: Hyperbola;M: Point;Expression(G) = (x^2/4 - y^2 = 1);PointOnCurve(P, G);MidPoint(LineSegmentOf(O, P)) = M",
+      "coords": {}
+    },
+    {
+      "index": 267,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.3166247903554,
+        "b": 2.6457513110645907,
+        "major_axis": "x",
+        "x_coef": 11.0,
+        "y_coef": 7.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0267.png",
+      "answer": "4*sqrt(11)",
+      "fact_expr": "l: Line;G: Ellipse;F1: Point;A: Point;B: Point;F2: Point;Expression(G) = (x^2/11 + y^2/7 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2, l);Intersection(l, G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 268,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0268.png",
+      "answer": "sqrt(5)/3",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;c: Number;Coordinate(F) = (c, 0);RightFocus(C) = F;P: Point;PointOnCurve(P, C);G: Circle;Expression(G) = (y^2 + (-c/3 + x)^2 = b^2/9);Q: Point;TangentPoint(LineSegmentOf(P, F), G) = Q;VectorOf(P, Q) = 2*VectorOf(Q, F)",
+      "coords": {}
+    },
+    {
+      "index": 269,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0269.png",
+      "answer": "4",
+      "fact_expr": "l: Line;C: Parabola;M: Point;F: Point;A: Point;Expression(C) = (x^2 = 4*y);Focus(C)=F;PointOnCurve(F,l);Slope(l)=sqrt(3);Intersection(l,C)={A,B};Intersection(TangentOnPoint(A,C),TangentOnPoint(B,C))=M;B:Point",
+      "coords": {}
+    },
+    {
+      "index": 270,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.4641016151377544,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 12.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0270.png",
+      "answer": "sqrt(3)/4",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;M: Point;O: Origin;Expression(G) = (x^2/12 + y^2/3 = 1);OneOf(Focus(G)) = F1;PointOnCurve(P, G);MidPoint(LineSegmentOf(P, F1)) = M;PointOnCurve(M, yAxis);Distance(M, O) = Abs(LineSegmentOf(M, O))",
+      "coords": {}
+    },
+    {
+      "index": 271,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "down"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0271.png",
+      "answer": "{x=0,4*x-3*y-3=0}",
+      "fact_expr": "l: Line;G: Parabola;H: Circle;Expression(G) = (x^2 = -4*y);Expression(H) = ((x - 1)^2 + (y - 2)^2 = 1);PointOnCurve(Focus(G),l);IsTangent(l,H)",
+      "coords": {}
+    },
+    {
+      "index": 272,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 273,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 274,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.4506710767745972,
+        "b": 2.0511727333068848,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 4.657696081267204e-06,
+        "converged": true,
+        "iterations": 116
+      },
+      "output_path": "results/test/hyperbola/problem_0274.png",
+      "answer": "2*sqrt(26)",
+      "fact_expr": "C: Hyperbola;m: Number;P: Point;F1: Point;F2: Point;m>0;Expression(C) = (x^2/4 - y^2/m = 1);Expression(Asymptote(C)) = (y = pm*(sqrt(2)*x));LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, RightPart(C));Abs(LineSegmentOf(P, F1)) = m - 1",
+      "coords": {}
+    },
+    {
+      "index": 275,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0275.png",
+      "answer": "(4*x^2 - y^2 - 6*y - 1 = 0)&Negation(y = 0)\ny = -1/4",
+      "fact_expr": "G: Parabola;Expression(G) = (y = x^2);P: Point;Q: Point;PointOnCurve(P, G) = True;PointOnCurve(Q, G) = True;Negation(Vertex(G) = P);Negation(Vertex(G) = Q);O: Origin;AngleOf(P, O, Q) = pi/4;l1: Line;l2: Line;PointOnCurve(P, l1) = True;PointOnCurve(Q, l2) = True;IsTangent(l1, G) = True;IsTangent(l2, G) = True;M: Point;Intersection(l1, l2) = M;A: Point;B: Point;Intersection(l1, xAxis) = A;Intersection(l2, xAxis) = B;l3: Line;PointOnCurve(Orthocenter(TriangleOf(A, B, M)), l3) = True;I: Point;Coordinate(I) = (0, -1/4);PointOnCurve(I, l3) = True",
+      "coords": {
+        "I": [
+          0.0,
+          -0.25
+        ]
+      }
+    },
+    {
+      "index": 276,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0276.png",
+      "answer": "3/4",
+      "fact_expr": "l: Line;G: Parabola;Expression(G) = (y = x^2/2);Intersection(l, G) = {A, B};A: Point;B: Point;IsPerpendicular(TangentOnPoint(A,G),TangentOnPoint(B,G)) = True;Coordinate(A) = (2, 2)",
+      "coords": {
+        "A": [
+          2.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 277,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 278,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 279,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 0.3689577579498291,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 5.885034397579147e-07,
+        "converged": true,
+        "iterations": 122
+      },
+      "output_path": "results/test/hyperbola/problem_0279.png",
+      "answer": "3*sqrt(5)/5",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;F1: Point;F2: Point;P: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-sqrt(3)/2, 0);Coordinate(F2) = (sqrt(3)/2, 0);Focus(G) = {F1, F2};Quadrant(P) = 1;PointOnCurve(P, G);Tan(AngleOf(P, F1, F2)) = 1/2;Tan(AngleOf(P, F2, F1)) = -2",
+      "coords": {
+        "F1": [
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+          0.0
+        ],
+        "F2": [
+          0.8660254037844386,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 280,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0280.png",
+      "answer": "[sqrt(3)/2,1)",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;O: Circle;A: Point;P: Point;B: Point;l1: Line;l2: Line;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Expression(O) = (x^2 + y^2 = b^2);TangentOfPoint(P, O) = {l1, l2};TangentPoint(l1,O)=A;TangentPoint(l2,O)=B;AngleOf(A, P, B) = ApplyUnit(60, degree);PointOnCurve(P,C)",
+      "coords": {}
+    },
+    {
+      "index": 281,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0281.png",
+      "answer": "3*sqrt(3)",
+      "fact_expr": "E: Ellipse;a: Number;F1: Point;M: Point;F2: Point;Expression(E) = (y^2/3 + x^2/a^2 = 1);RightFocus(E) = F2;LeftFocus(E)=F1;PointOnCurve(M, E);AngleOf(F1, M, F2) = (2*pi)/3",
+      "coords": {}
+    },
+    {
+      "index": 282,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 283,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0283.png",
+      "answer": "y=pm*2*x",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;A: Point;F: Point;B: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = (4/3)*x);RightFocus(G) = F;Intersection(H, G) = {A, B};IsPerpendicular(LineSegmentOf(A, F), LineSegmentOf(B, F))",
+      "coords": {}
+    },
+    {
+      "index": 284,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0284.png",
+      "answer": "2",
+      "fact_expr": "C: Parabola;Expression(C) = (x^2 = 2*(p*y));p: Number;p>0;F: Point;Focus(C) = F;l: Line;Directrix(C) = l;A: Point;In(A,C);F1: Circle;Center(F1) = F;Radius(F1) = LineSegmentOf(F,A);B: Point;D: Point;Intersection(F1, l) = {B, D};AngleOf(B, F, D) = ApplyUnit(90, degree);Area(TriangleOf(A, B, D)) = 4*sqrt(2)",
+      "coords": {}
+    },
+    {
+      "index": 285,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 2.23606797749979,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 5.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0285.png",
+      "answer": "6-sqrt(2)",
+      "fact_expr": "G: Ellipse;A: Point;P: Point;F1: Point;Expression(G) = (x^2/9 + y^2/5 = 1);Coordinate(A) = (1, 1);LeftFocus(G) = F1;PointOnCurve(P, G)",
+      "coords": {
+        "A": [
+          1.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 286,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0286.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;G: Circle;A: Point;O: Origin;B: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2 + y^2 = b^2);l1: Line;l2: Line;TangentOfPoint(OneOf(Endpoint(MajorAxis(C))),G) = {l1,l2};TangentPoint(l1,G)=A;TangentPoint(l2,G)=B;AngleOf(A, O, B) = ApplyUnit(90, degree)",
+      "coords": {}
+    },
+    {
+      "index": 287,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0287.png",
+      "answer": "3",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);H: Circle;PointOnCurve(Focus(G),H) = True;PointOnCurve(Vertex(G),H) = True;IsTangent(Directrix(G),H) = True",
+      "coords": {}
+    },
+    {
+      "index": 288,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 289,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0289.png",
+      "answer": "[sqrt(5)/2, sqrt(6)/2]",
+      "fact_expr": "G: Curve;Expression(G) = (x^2/4 + y^2/m = 1);m: Number;In(m, [-2, -1])",
+      "coords": {}
+    },
+    {
+      "index": 290,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0290.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "M: Point;Coordinate(M) = (1, 1);G: Line;Slope(G) = -1/2;PointOnCurve(M, G);C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;A: Point;B: Point;Intersection(G, C) = {A, B};MidPoint(LineSegmentOf(A, B)) = M",
+      "coords": {
+        "M": [
+          1.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 291,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 292,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.33732376966822,
+        "b": 1.7454107534785184,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 1.8920117601792354e-05,
+        "converged": true,
+        "iterations": 51
+      },
+      "output_path": "results/test/hyperbola/problem_0292.png",
+      "answer": "x^2/16-y^2/9=1",
+      "fact_expr": "G: Hyperbola;Coordinate(OneOf(Focus(G))) = (-5, 0);Expression(Asymptote(G))=(pm*4*y+3*x=0)",
+      "coords": {}
+    },
+    {
+      "index": 293,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0293.png",
+      "answer": "10",
+      "fact_expr": "G: Hyperbola;F1: Point;F2: Point;Expression(G) = (x^2/16 - y^2/9 = 1);LeftFocus(G) = F1;RightFocus(G) = F2",
+      "coords": {}
+    },
+    {
+      "index": 294,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0294.png",
+      "answer": "(1, 3]",
+      "fact_expr": "F1: Point;F2: Point;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, LeftPart(G)) = True;Min(Abs(VectorOf(P, F2))^2/Abs(LineSegmentOf(P, F1))) = 8*a",
+      "coords": {}
+    },
+    {
+      "index": 295,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0295.png",
+      "answer": "sqrt(5)/2",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2/16 - y^2/b^2 = 1);b: Number;b>0;F: Point;LeftFocus(C) = F;M: Point;PointOnCurve(M, OneOf(Asymptote(C)));O: Origin;IsPerpendicular(LineSegmentOf(O, M), LineSegmentOf(M, F));Area(TriangleOf(O, M, F)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 296,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0296.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(C) = F;O: Origin;PointOnCurve(O, G) = True;Slope(G) = 4/3;G: Line;Intersection(G, RightPart(C)) = A;A: Point;Abs(LineSegmentOf(O, A)) = Abs(LineSegmentOf(O, F))",
+      "coords": {}
+    },
+    {
+      "index": 297,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0297.png",
+      "answer": "4",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 8*x);G: Line;PointOnCurve(Focus(C), G);A: Point;B: Point;Intersection(G, C) = {A, B};L: Line;PointOnCurve(A, L);IsPerpendicular(L, Directrix(C));M: Point;FootPoint(L, Directrix(C)) = M;Abs(LineSegmentOf(A, B)) = Abs(LineSegmentOf(B, M))",
+      "coords": {}
+    },
+    {
+      "index": 298,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0298.png",
+      "answer": "1/2",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;A: Point;F2: Point;D: Point;F1: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C)=F1;RightFocus(C)=F2;UpperVertex(C)=A;MidPoint(LineSegmentOf(A, F2)) = D;IsPerpendicular(VectorOf(A,F2),VectorOf(D,F1))",
+      "coords": {}
+    },
+    {
+      "index": 299,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 300,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.4142135623730951,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0300.png",
+      "answer": "(sqrt(3),2)",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2 - y^2/2 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;G: Point;PointOnCurve(G, C) ;Quadrant(G) = 1;H: Point;VectorOf(O, H) = VectorOf(O, G) + lambda*(VectorOf(G, F1)/Abs(VectorOf(G, F1)) + VectorOf(G, F2)/Abs(VectorOf(G, F2)));lambda: Number;Negation(lambda = 0);O: Origin;Intersection(LineOf(G,H), xAxis) = P;P : Point;Coordinate(P) = (sqrt(3)/3, 0)",
+      "coords": {
+        "P": [
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+          0.0
+        ]
+      }
+    },
+    {
+      "index": 301,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0301.png",
+      "answer": "2*sqrt(13)/13",
+      "fact_expr": "G: Hyperbola;a: Number;H: Ellipse;a>0;Expression(G) = (-y^2/9 + x^2/a^2 = 1);Expression(OneOf(Asymptote(G)))=(3*x-2*y=0);Vertex(G)=Focus(H);Focus(G)=Vertex(H)",
+      "coords": {}
+    },
+    {
+      "index": 302,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0302.png",
+      "answer": "sqrt(3)/3",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;A: Point;B: Point;D: Point;C: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (a, 0);Coordinate(B) = (0, b);In(A,Vertex(G));In(B,Vertex(G));L1:Line;L2:Line;PointOnCurve(A,L1);PointOnCurve(B,L2);IsPerpendicular(L1,LineSegmentOf(A,B));IsPerpendicular(L2,LineSegmentOf(A,B));Intersection(L1, G) = C;Intersection(L2, G) = D;Negation(C=D);2*Abs(LineSegmentOf(B, D)) = 3*Abs(LineSegmentOf(A, C))",
+      "coords": {}
+    },
+    {
+      "index": 303,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 304,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0304.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/3 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 305,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 8.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 64.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0305.png",
+      "answer": "x^2/36-y^2/12=1",
+      "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2 + 4*y^2 = 64);Focus(G) = Focus(H);Expression(OneOf(Asymptote(G))) = (x+sqrt(3)*y=0)",
+      "coords": {}
+    },
+    {
+      "index": 306,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0306.png",
+      "answer": "{2/3,6}",
+      "fact_expr": "W: Ellipse;m: Number;Expression(W) = (x^2/2 + y^2/m = 1);Eccentricity(W) = sqrt(6)/3",
+      "coords": {}
+    },
+    {
+      "index": 307,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0307.png",
+      "answer": "sqrt(13)/2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (9*x^2 - 4*y^2 = 36)",
+      "coords": {}
+    },
+    {
+      "index": 308,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0308.png",
+      "answer": "3/2",
+      "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;M: Point;Expression(G)=(y=x^2/2);Intersection(l,G)={A,B};Abs(LineSegmentOf(A,B))=4;IsChordOf(LineSegmentOf(A,B),G);MidPoint(LineSegmentOf(A,B))=M",
+      "coords": {}
+    },
+    {
+      "index": 309,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0309.png",
+      "answer": "4",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2 - y^2 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C);AngleOf(F1, P, F2) = ApplyUnit(60, degree)",
+      "coords": {}
+    },
+    {
+      "index": 310,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 311,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 312,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0312.png",
+      "answer": "x^2=pm*2*y",
+      "fact_expr": "C: Parabola;Expression(C) = (x^2 = 2*p*y);p: Number;F: Point;Focus(C) = F;P: Point;Coordinate(P) = (x0, p/8);x0: Number;PointOnCurve(P,C) = True;Abs(LineSegmentOf(P,F)) = 5/8",
+      "coords": {}
+    },
+    {
+      "index": 313,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.366584062576294,
+        "b": 2.364173412322998,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.00017838670464698225,
+        "converged": true,
+        "iterations": 89
+      },
+      "output_path": "results/test/hyperbola/problem_0313.png",
+      "answer": "6",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;m: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = 2;PointOnCurve(RightFocus(G), H);IsPerpendicular(xAxis, H);Length(InterceptChord(H, G)) = m",
+      "coords": {}
+    },
+    {
+      "index": 314,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.75,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0314.png",
+      "answer": "ApplyUnit(90, degree)",
+      "fact_expr": "M: Parabola;Expression(M) = (y^2 = 3*x);l: Line;H: Point;Coordinate(H) = (3, 0);PointOnCurve(H, l);A: Point;B: Point;Intersection(l, M) = {A, B};O: Origin",
+      "coords": {
+        "H": [
+          3.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 315,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.7320508075688772,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0315.png",
+      "answer": "[3+2*sqrt(3),+oo)",
+      "fact_expr": "G: Hyperbola;O: Origin;P: Point;F: Point;Expression(G) = (x^2/3 - y^2 = 1);PointOnCurve(P, RightPart(G));Center(G)=O;LeftFocus(G)=F",
+      "coords": {}
+    },
+    {
+      "index": 316,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 317,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0317.png",
+      "answer": "{1,3}",
+      "fact_expr": "G: Parabola;p: Number;H: Line;A: Point;B: Point;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (x = p/2);PointOnCurve(A,G);PointOnCurve(B,G);Focus(G)=F;Abs(LineSegmentOf(A, F)) + Abs(LineSegmentOf(B, F)) = 4;Distance(MidPoint(LineSegmentOf(A,B)),H)=1",
+      "coords": {}
+    },
+    {
+      "index": 318,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0318.png",
+      "answer": "sqrt(15)/5",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;A: Point;UpperVertex(G) = A;P: Point;Quadrant(P) = 1;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) + Abs(LineSegmentOf(P, F2)) = 4*Abs(LineSegmentOf(F1, F2));Area(TriangleOf(P, F1, A))=2*Area(TriangleOf(P, F1, F2))",
+      "coords": {}
+    },
+    {
+      "index": 319,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0319.png",
+      "answer": "5",
+      "fact_expr": "C: Parabola;p: Number;l: Line;A: Point;F: Point;M: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;Expression(l)=(x=-5/4);PointOnCurve(M, C);Directrix(C)=l;PointOnCurve(A, l);IsPerpendicular(LineSegmentOf(M, A),l);Inclination(LineOf(A,F)) = pi/3",
+      "coords": {}
+    },
+    {
+      "index": 320,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0320.png",
+      "answer": "sqrt(37)/5",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;P: Point;Q: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2, H);Intersection(H, G) = {P, Q};IsPerpendicular(LineSegmentOf(P, Q), LineSegmentOf(P, F1));Abs(LineSegmentOf(P, Q)) = (5/12)*Abs(LineSegmentOf(P, F1))",
+      "coords": {}
+    },
+    {
+      "index": 321,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0321.png",
+      "answer": "9",
+      "fact_expr": "E: Parabola;l1:Line;l2:Line;A: Point;B: Point;C: Point;D: Point;F:Point;Expression(E) = (y^2 = x);Focus(E)=F;PointOnCurve(F,l1);PointOnCurve(F,l2);IsPerpendicular(l1,l2);Intersection(l1,E)={A,B};Intersection(l2,E)={C,D}",
+      "coords": {}
+    },
+    {
+      "index": 322,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 2.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0322.png",
+      "answer": "2*x - y - 15 = 0",
+      "fact_expr": "G: Hyperbola;H: LineSegment;P: Point;Expression(G) = (x^2/16 - y^2/4 = 1);Coordinate(P) = (8, 1);IsChordOf(H, G);MidPoint(H) = P",
+      "coords": {
+        "P": [
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+          1.0
+        ]
+      }
+    },
+    {
+      "index": 323,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0323.png",
+      "answer": "2*sqrt(3)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 8*x);OneOf(Focus(G))=Focus(H);Distance(Focus(G),Asymptote(G))=1",
+      "coords": {}
+    },
+    {
+      "index": 324,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.2135674953460693,
+        "b": 2.4270472526550293,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 4.453341261978494e-06,
+        "converged": true,
+        "iterations": 125
+      },
+      "output_path": "results/test/hyperbola/problem_0324.png",
+      "answer": "y=pm*(1/2)*x",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(C) = sqrt(5)/2",
+      "coords": {}
+    },
+    {
+      "index": 325,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0325.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;P: Point;y0: Number;Coordinate(P) = (6, y0);PointOnCurve(P, G);F: Point;Focus(G) = F;Abs(LineSegmentOf(P, F)) = 8",
+      "coords": {}
+    },
+    {
+      "index": 326,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 327,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0327.png",
+      "answer": "6",
+      "fact_expr": "C: Parabola;p: Number;G: Line;A: Point;F: Point;B: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Expression(G) = (y = x - 3);Focus(C) = F;Intersection(G, C) = {A, B};Abs(LineSegmentOf(A, F)) + Abs(LineSegmentOf(B, F)) = Abs(LineSegmentOf(A, B))",
+      "coords": {}
+    },
+    {
+      "index": 328,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.4142135623730951,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0328.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;H: Parabola;p: Number;Expression(G) = (x^2 - y^2 = 2);p>0;Expression(H) = (y^2 = 2*(p*x));PointOnCurve(OneOf(Focus(G)), Directrix(H))",
+      "coords": {}
+    },
+    {
+      "index": 329,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0329.png",
+      "answer": "1/2",
+      "fact_expr": "G: Hyperbola;H: Line;A: Point;a:Number;b:Number;Expression(G) = (x^2 - y^2 = 1);Expression(H) = (y = x);Coordinate(A) = (a, b);PointOnCurve(A,RightPart(G));Distance(A, H) = sqrt(2)",
+      "coords": {}
+    },
+    {
+      "index": 330,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 331,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 3.872983346207417,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0331.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/15 = 1);e: Number;Eccentricity(G) = e",
+      "coords": {}
+    },
+    {
+      "index": 332,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0332.png",
+      "answer": "2",
+      "fact_expr": "G: Ellipse;P: Point;F1:Point;F2:Point;Expression(G) = (x^2/4 + y^2/3 = 1);PointOnCurve(P, G);Distance(P,F1)=2;OneOf(Focus(G))=F1;OneOf(Focus(G))=F2;Negation(F1=F2)",
+      "coords": {}
+    },
+    {
+      "index": 333,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 334,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 335,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0335.png",
+      "answer": "15",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);P: Point;M: Point;F1: Point;F2: Point;Coordinate(M) = (6, 4);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G)",
+      "coords": {
+        "M": [
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+          4.0
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+      }
+    },
+    {
+      "index": 336,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0336.png",
+      "answer": "(-oo, -sqrt(6)/2]+[sqrt(6)/2, +oo)",
+      "fact_expr": "C: Ellipse;l: Line;k: Number;Expression(C) = (x^2/2 + y^2 = 1);Expression(l) = (y = k*x + 2);IsIntersect(l, C)",
+      "coords": {}
+    },
+    {
+      "index": 337,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_point_off_curve",
+      "params": {
+        "a": 1.9960020780563354,
+        "b": 1.4970015287399292,
+        "orientation": "vertical"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "iterations": 2
+      },
+      "output_path": "results/test/hyperbola/problem_0337.png",
+      "answer": "2*sqrt(30)",
+      "fact_expr": "C: Hyperbola;D:Hyperbola;b: Number;a: Number;G: Point;a>0;b>0;Expression(C) = (-x^2/b^2 + y^2/a^2 = 1);Coordinate(G) = (2, 6);Expression(D)=(x^2/4-y^2/6=1);Asymptote(C)=Asymptote(D);PointOnCurve(G, C)",
+      "coords": {
+        "G": [
+          2.0,
+          6.0
+        ]
+      },
+      "validation_reasons": [
+        "hyperbola_point_off_curve"
+      ]
+    },
+    {
+      "index": 338,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0338.png",
+      "answer": "2",
+      "fact_expr": "O: Circle;Expression(O) = (x^2 + y^2 = 1);P: Point;Coordinate(P) = (2, 0);L1: Line;L2: Line;TangentOfPoint(P, O) = {L1, L2};TangentPoint(L1, O) = A;TangentPoint(L2, O) = B;A: Point;B: Point;C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Z1: Line;Z2: Line;Asymptote(C) = {Z1, Z2};PointOnCurve(A, Z1);PointOnCurve(B, Z2)",
+      "coords": {
+        "P": [
+          2.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 339,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0339.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;H: Ellipse;a: Number;Expression(G) = (x^2/9 - y^2/3 = 1);a>0;Expression(H) = (y^2/4 + x^2/a^2 = 1);Focus(H) = Focus(G)",
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+    },
+    {
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+      "error": null,
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+        "b": 1.5,
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "results/test/hyperbola/problem_0340.png",
+      "answer": "3",
+      "fact_expr": "G: Hyperbola;a: Number;H: Circle;A: Point;M: Point;F2: Point;F1: Point;N: Point;a>0;Expression(G) = (-y^2/7 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;LeftVertex(G) = A;Center(H) = F2;Radius(H) = Abs(LineSegmentOf(F2, A));Intersection(H, RightPart(G)) = {M, N};PointOnCurve(N, PerpendicularBisector(LineSegmentOf(A, M)))",
+      "coords": {}
+    },
+    {
+      "index": 341,
+      "success": true,
+      "conic_type": "ellipse",
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+        "major_axis": "x",
+        "x_coef": 5.0,
+        "y_coef": 1.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0341.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;p: Number;H: Ellipse;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (x^2/5 + y^2 = 1);Focus(G) = RightFocus(H)",
+      "coords": {}
+    },
+    {
+      "index": 342,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "converged": true,
+        "iterations": 106
+      },
+      "output_path": "results/test/hyperbola/problem_0342.png",
+      "answer": "sqrt(5)/2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Ellipse;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a > b;b > 0;Expression(H) = (y^2/b^2 + x^2/a^2 = 1);Eccentricity(H) = sqrt(3)/2",
+      "coords": {}
+    },
+    {
+      "index": 343,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 344,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0344.png",
+      "answer": "25/8",
+      "fact_expr": "l: Line;G: Parabola;B: Point;A: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (4, 4);Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B}",
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+        "A": [
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+          4.0
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+      }
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+    {
+      "index": 345,
+      "success": true,
+      "conic_type": "parabola",
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0345.png",
+      "answer": "3",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;A: Point;XCoordinate(A) = 1;PointOnCurve(A, G);Distance(A, Focus(G)) = 5/2",
+      "coords": {}
+    },
+    {
+      "index": 346,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0346.png",
+      "answer": "sqrt(41)",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);l: Line;Directrix(G) = l;C: Point;Coordinate(C) = (-3, -4);P: Point;PointOnCurve(P, G);d: Number;Distance(P, l) = d",
+      "coords": {
+        "C": [
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+          -4.0
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+      }
+    },
+    {
+      "index": 347,
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+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0347.png",
+      "answer": "(5/2,+oo)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;A: Point;F2: Point;B: Point;P: Point;Q: Point;F1: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F2;LeftFocus(G)=F1;PointOnCurve(F1, H);IsPerpendicular(H,xAxis);Intersection(H, LeftPart(G)) = {A, B};Intersection(LineSegmentOf(A,F2),yAxis)=P;Intersection(LineSegmentOf(B, F2), yAxis) = Q;Perimeter(TriangleOf(P, Q, F2)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 348,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 9.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0348.png",
+      "answer": "9*x-32*y+73=0",
+      "fact_expr": "G: Ellipse;H: LineSegment;M: Point;Expression(G) = (x^2/16 + y^2/9 = 1);Coordinate(M) = (-1, 2);IsChordOf(H, G);MidPoint(H) = M",
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+        "M": [
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+          2.0
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+    {
+      "index": 349,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 350,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 3.872983346207417,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0350.png",
+      "answer": "5",
+      "fact_expr": "G: Hyperbola;H: Circle;U: Circle;P: Point;M: Point;N: Point;Expression(G) = (x^2 - y^2/15 = 1);Expression(H) = (y^2 + (x + 4)^2 = 4);Expression(U) = (y^2 + (x - 4)^2 = 1);PointOnCurve(P, RightPart(G));PointOnCurve(M, H);PointOnCurve(N, U)",
+      "coords": {}
+    },
+    {
+      "index": 351,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0351.png",
+      "answer": "11",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;A: Point;Coordinate(A) = (1, 3);P: Point;PointOnCurve(P, RightPart(G))",
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+        "A": [
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+          3.0
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+      }
+    },
+    {
+      "index": 352,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "up"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0352.png",
+      "answer": "(0,-1/4)",
+      "fact_expr": "G: Parabola;Expression(G) = (y = x^2)",
+      "coords": {}
+    },
+    {
+      "index": 353,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.4142135623730951,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0353.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "G: Hyperbola;H: Ellipse;b: Number;Expression(G) = (x^2/2 - y^2/2 = 1);b>0;Expression(H) = (x^2/4 + y^2/b^2 = 1);PointOnCurve(Focus(H),Directrix(G))",
+      "coords": {}
+    },
+    {
+      "index": 354,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+      "optimization": {
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+        "converged": true,
+        "iterations": 175
+      },
+      "output_path": "results/test/hyperbola/problem_0354.png",
+      "answer": "-6",
+      "fact_expr": "Eccentricity(C) = 3*sqrt(5)/5;C: Hyperbola;Expression(C) = (-y^2/4 + x^2/a^2 = 1);a: Number;a>0;G: Parabola;m: Real;Expression(G) = (y^2 = 2*(m*x));LeftFocus(C) = Focus(G)",
+      "coords": {}
+    },
+    {
+      "index": 355,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 3.0,
+        "direction": "up"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0355.png",
+      "answer": "(0,3)",
+      "fact_expr": "G: Parabola;F: Point;Expression(G) = (x^2 = 12*y);Focus(G)=F",
+      "coords": {}
+    },
+    {
+      "index": 356,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0356.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;P: Point;y0: Number;Coordinate(P) = (6, y0);PointOnCurve(P, G);F: Point;Focus(G) = F;Abs(LineSegmentOf(P, F)) = 8",
+      "coords": {}
+    },
+    {
+      "index": 357,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0357.png",
+      "answer": "1/2",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;F2: Point;P: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-1, 0);Coordinate(F2) = (1, 0);Focus(G)={F1,F2};PointOnCurve(P,G);Abs(VectorOf(P,F1))+Abs(VectorOf(P,F2))=4",
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+          0.0
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+        "F2": [
+          1.0,
+          0.0
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+    {
+      "index": 358,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0358.png",
+      "answer": "7/2",
+      "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;Expression(G) = (y^2 = 6*x);Coordinate(A) = (x1, y1);Coordinate(P) = (2, 2);Focus(G) = F;x1: Number;y1: Number;PointOnCurve(A, G)",
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+        "P": [
+          2.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 359,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 360,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 36.0,
+        "y_coef": 9.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0360.png",
+      "answer": "3*sqrt(3)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/36 + y^2/9 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;F1: Point;F2: Point;P: Point;PointOnCurve(P,G) = True;AngleOf(F1, P, F2) = ApplyUnit(60, degree)",
+      "coords": {}
+    },
+    {
+      "index": 361,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_focus_mismatch",
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+        "b": 2.629857301712036,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "iterations": 175
+      },
+      "output_path": "results/test/hyperbola/problem_0361.png",
+      "answer": "x^2/2-y^2/16=1",
+      "fact_expr": "C: Hyperbola;a: Number;b: Number;F1: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-3*sqrt(2), 0);LeftFocus(C) = F1;Eccentricity(C)=3",
+      "coords": {
+        "F1": [
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+          0.0
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+      "validation_reasons": [
+        "hyperbola_focus_mismatch"
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+    },
+    {
+      "index": 362,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0362.png",
+      "answer": "Abs(a)/2",
+      "fact_expr": "G: Parabola;a: Number;Negation(a=0);Expression(G) = (y^2 = a*x)",
+      "coords": {}
+    },
+    {
+      "index": 363,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_point_off_curve",
+      "params": {
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+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 4.0
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+      "optimization": {
+        "final_loss": 8.798975889096597e-06,
+        "converged": true,
+        "iterations": 98
+      },
+      "output_path": "results/test/ellipse/problem_0363.png",
+      "answer": "{x^2/16+y^2/4=1,y^2/64+x^2/16=1}",
+      "fact_expr": "O: Origin;Center(G) = O;G: Ellipse;PointOnCurve(Focus(G), axis);Length(MajorAxis(G)) = 2*Length(MinorAxis(G));P: Point;Coordinate(P) = (4, 0);PointOnCurve(P, G)",
+      "coords": {
+        "P": [
+          4.0,
+          0.0
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+      },
+      "validation_reasons": [
+        "ellipse_point_off_curve"
+      ]
+    },
+    {
+      "index": 364,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 2.8284271247461903,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0364.png",
+      "answer": "(x^2-y^2/8=1)&(x>=1)",
+      "fact_expr": "C: Circle;Expression(C) = (y^2 + (x + 3)^2 = 4);A: Point;M: Circle;Coordinate(A) = (3, 0);PointOnCurve(A, M);IsOutTangent(C, M);M1: Point;Center(M) = M1",
+      "coords": {
+        "A": [
+          3.0,
+          0.0
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+      }
+    },
+    {
+      "index": 365,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0365.png",
+      "answer": "2*sqrt(2)/3",
+      "fact_expr": "l: Line;C: Parabola;A: Point;F: Point;B: Point;P: Point;Expression(C) = (y^2 = 4*x);Coordinate(P) = (-1, 0);Focus(C) = F;PointOnCurve(P, l);Slope(l) = k;k:Number;k>0;Intersection(l, C) = {A, B};LineSegmentOf(A, F)>LineSegmentOf(B, F);Abs(LineSegmentOf(A, F)) = 2*Abs(LineSegmentOf(B, F))",
+      "coords": {
+        "P": [
+          -1.0,
+          0.0
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+      }
+    },
+    {
+      "index": 366,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0366.png",
+      "answer": "2",
+      "fact_expr": "Q: Point;Coordinate(Q) = (2*sqrt(2), 0);G: Parabola;Expression(G) = (y = x^2/4);P: Point;Coordinate(P) = (x0, y0);x0: Number;y0: Number;PointOnCurve(P, G)",
+      "coords": {
+        "Q": [
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+          0.0
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+      }
+    },
+    {
+      "index": 367,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 368,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 369,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0369.png",
+      "answer": "25/4",
+      "fact_expr": "F: Point;C: Parabola;Expression(C) = (y^2 = 2*p*x);p: Number;p>0;Focus(C) = F;A: Point;B: Point;PointOnCurve(A, C) = True;PointOnCurve(B, C) = True;VectorOf(A, F) = 4*VectorOf(F, B);Projection(A, Directrix(C)) = M;Projection(B, Directrix(C)) = N;M: Point;N: Point;Area(TriangleOf(M, F, N)) = 5",
+      "coords": {}
+    },
+    {
+      "index": 370,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.7454107534785184,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 1.8920117601792354e-05,
+        "converged": true,
+        "iterations": 51
+      },
+      "output_path": "results/test/hyperbola/problem_0370.png",
+      "answer": "{5/3,5/4}",
+      "fact_expr": "G: Hyperbola;Expression(Asymptote(G)) = (y = pm*(3/4)*x)",
+      "coords": {}
+    },
+    {
+      "index": 371,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0371.png",
+      "answer": "y^2=4*x",
+      "fact_expr": "G: Parabola;p: Number;H: Line;A: Point;B: Point;p>0;Expression(G) = (y^2 = 2*p*x);PointOnCurve(Focus(G),H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = 6;XCoordinate(MidPoint(LineSegmentOf(A, B))) = 2",
+      "coords": {}
+    },
+    {
+      "index": 372,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0372.png",
+      "answer": "",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);c: Number;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(P, F1), xAxis) ;AngleOf(P, F2, F1) = ApplyUnit(45, degree)",
+      "coords": {}
+    },
+    {
+      "index": 373,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_point_off_curve",
+      "params": {
+        "a": 1.8993116218750659,
+        "b": 1.324128928251209,
+        "major_axis": "x",
+        "x_coef": 3.999999999999999,
+        "y_coef": 2.000000000000001
+      },
+      "optimization": {
+        "final_loss": 0.013791354113027786,
+        "converged": true,
+        "iterations": 100
+      },
+      "output_path": "results/test/ellipse/problem_0373.png",
+      "answer": "x^2/4 + y^2/2 = 1",
+      "fact_expr": "G: Ellipse;H: Point;I: Point;Coordinate(H) = (sqrt(2), 1);Coordinate(I) = (sqrt(3), sqrt(2)/2);PointOnCurve(H, G);PointOnCurve(I, G)",
+      "coords": {
+        "H": [
+          1.4142135623730951,
+          1.0
+        ],
+        "I": [
+          1.7320508075688772,
+          0.7071067811865476
+        ]
+      },
+      "validation_reasons": [
+        "ellipse_point_off_curve"
+      ]
+    },
+    {
+      "index": 374,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 4.47213595499958,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0374.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/20 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, RightPart(G)) = True;MidPoint(LineSegmentOf(P, F2)) = M;M: Point;O: Origin;Center(H) = O;Radius(H) = LineSegmentOf(O, F1);H: Circle;PointOnCurve(M, H) = True",
+      "coords": {}
+    },
+    {
+      "index": 375,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0375.png",
+      "answer": "5/3",
+      "fact_expr": "G: Parabola;p: Number;H: Line;B: Point;A: Point;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;Distance(F,Directrix(G)) = 2;PointOnCurve(F, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, F)) = 3*Abs(LineSegmentOf(F, B))",
+      "coords": {}
+    },
+    {
+      "index": 376,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 377,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0377.png",
+      "answer": "8/5",
+      "fact_expr": "C: Parabola;p: Number;A: Point;P: Point;H: Point;0 < p;p < 4;Expression(C) = (x^2 = 2*(p*y));Coordinate(A) = (0, 4);Directrix(C) = l ;PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(P, H),l);Abs(LineSegmentOf(P, H)) = Abs(LineSegmentOf(P, A));AngleOf(A, P, H) = ApplyUnit(120, degree);l:Line;FootPoint(LineSegmentOf(P, H),l)=H",
+      "coords": {
+        "A": [
+          0.0,
+          4.0
+        ]
+      }
+    },
+    {
+      "index": 378,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 379,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0379.png",
+      "answer": "3",
+      "fact_expr": "l: Line;G: Hyperbola;b: Number;a: Number;F1: Point;F2:Point;B: Point;A: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Slope(l)=sqrt(5)/2;LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(F1,l);Intersection(l,LeftPart(G))=A;Intersection(l,RightPart(G))=B;PointOnCurve(F2,PerpendicularBisector(LineSegmentOf(F1,B)))",
+      "coords": {}
+    },
+    {
+      "index": 380,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 381,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0381.png",
+      "answer": "sqrt(6)/3",
+      "fact_expr": "H: Line;Expression(H) = (y = (sqrt(3)/3)*x);G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;A: Point;B: Point;Intersection(H, G) = {A, B};F: Point;RightFocus(G) = F;IsPerpendicular(LineSegmentOf(A, F), LineSegmentOf(B, F))",
+      "coords": {}
+    },
+    {
+      "index": 382,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0382.png",
+      "answer": "4*sqrt(2)",
+      "fact_expr": "P: Point;Coordinate(P) = (2, 0);H: Line;PointOnCurve(P, H) = True;C: Parabola;Expression(C) = (y^2 = 4*x);Intersection(H, C) = {A, B};A: Point;O: Origin;B: Point;H1: Line;Expression(H1) = (x=-2);Intersection(LineOf(O,A),H1) = M;Intersection(LineOf(O,B),H1) = N;M: Point;N: Point;G: Circle;IsDiameter(LineSegmentOf(M,N),G) = True",
+      "coords": {
+        "P": [
+          2.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 383,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0383.png",
+      "answer": "y^2-4*x-y+4=0",
+      "fact_expr": "M: Point;Coordinate(M) = (1, 1);H: Line;PointOnCurve(M, H);G: Parabola;Expression(G) = (y^2 = 8*x);A: Point;B: Point;Intersection(H, G) = {A, B}",
+      "coords": {
+        "M": [
+          1.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 384,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 385,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0385.png",
+      "answer": "(sqrt(21)/7)+6",
+      "fact_expr": "H: Ellipse;Expression(H) = (x^2/9 + y^2/8 = 1);F1: Point;F2: Point;LeftFocus(H) = F1;RightFocus(H) = F2;P: Point;PointOnCurve(P, H);Q: Point;lambda: Real;VectorOf(F1, P) = lambda*VectorOf(P, Q);Abs(VectorOf(P, Q)) = Abs(VectorOf(P, F2));G: Hyperbola;Expression(G) = (x^2/4 - y^2/3 = 1);lambda>0",
+      "coords": {}
+    },
+    {
+      "index": 386,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0386.png",
+      "answer": "(1,sqrt(2))",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;H: Line;PointOnCurve(RightFocus(G), H) = True;Inclination(H) = ApplyUnit(45, degree);NumIntersection(H, RightPart(G)) = 2",
+      "coords": {}
+    },
+    {
+      "index": 387,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0387.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 + y^2/3 = 1);PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G) = F2;AngleOf(F1, P, F2) = pi/3",
+      "coords": {}
+    },
+    {
+      "index": 388,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.4641016151377544,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 12.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0388.png",
+      "answer": "-2/3",
+      "fact_expr": "M: Point;Coordinate(M) = (2, 1);PointOnCurve(M, l) ;l: Line;G: Ellipse;Expression(G) = (x^2/12 + y^2/4 = 1);Intersection(l, G) = {A, B};A: Point;B: Point;MidPoint(LineSegmentOf(A, B)) = M",
+      "coords": {
+        "M": [
+          2.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 389,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0389.png",
+      "answer": "5/3",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;H: Point;Coordinate(H) = (3, -4);PointOnCurve(H, OneOf(Asymptote(G)))",
+      "coords": {
+        "H": [
+          3.0,
+          -4.0
+        ]
+      }
+    },
+    {
+      "index": 390,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0010501146316528,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 3.455598402979376e-07,
+        "converged": true,
+        "iterations": 81
+      },
+      "output_path": "results/test/parabola/problem_0390.png",
+      "answer": "-1",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;M: Point;Coordinate(M) = (1, 2);PointOnCurve(M, G);l: Line;A: Point;B: Point;Negation(A=B);Intersection(l, G) = {A, B};t: Number;Coordinate(Center(InscribedCircle(TriangleOf(M, A, B)))) = (1, t)",
+      "coords": {
+        "M": [
+          1.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 391,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0391.png",
+      "answer": "4",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;P: Point;PointOnCurve(P, C) = True;Abs(LineSegmentOf(P, F)) = 8;M: Point;MidPoint(LineSegmentOf(P, F)) = M",
+      "coords": {}
+    },
+    {
+      "index": 392,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0392.png",
+      "answer": "(sqrt(3)+sqrt(39))/6",
+      "fact_expr": "l: Line;Gamma: Hyperbola;A: Point;O: Origin;B: Point;Expression(Gamma) = (-y^2/b^2 + x^2/a^2 = 1);a:Number;b:Number;a>0;b>0;Expression(l)=(x = sqrt(a^2 + b^2));Intersection(l, Gamma) = {A, B};DotProduct((VectorOf(A, B) + VectorOf(A, O)),VectorOf(B, O)) = 0",
+      "coords": {}
+    },
+    {
+      "index": 393,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "vertical"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0393.png",
+      "answer": "sqrt(5)/2",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-x^2/b^2 + y^2/a^2 = 1);b: Number;a: Number;a>0;b>0;G: Line;Expression(G) = (2*x - y = 0);OneOf(Asymptote(C)) = G",
+      "coords": {}
+    },
+    {
+      "index": 394,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 395,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0395.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "C: Parabola;p: Number;F: Point;l: Line;H: Line;A: Point;B: Point;M: Point;N: Point;Q: Point;p > 0;Expression(C) = (y^2 = 2*p*x);Focus(C) = F;Directrix(C) = l;PointOnCurve(F, H);Inclination(H) = ApplyUnit(60, degree);Intersection(H, C) = {A, B};IsPerpendicular(LineSegmentOf(A, M), l);IsPerpendicular(LineSegmentOf(B, N), l);FootPoint(LineSegmentOf(A, M), l) = M;FootPoint(LineSegmentOf(B, N), l) = N;MidPoint(LineSegmentOf(M, N)) = Q;Abs(LineSegmentOf(Q, F)) = 2",
+      "coords": {}
+    },
+    {
+      "index": 396,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 397,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0010501146316528,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 3.455598402979376e-07,
+        "converged": true,
+        "iterations": 81
+      },
+      "output_path": "results/test/parabola/problem_0397.png",
+      "answer": "x-y-1=0",
+      "fact_expr": "G: Parabola;p: Number;M: Point;Expression(G) = (x^2 = 2*(p*y));Coordinate(M) = (2, 1);PointOnCurve(M, G);l:Line;IsTangent(l,G);TangentPoint(l,G)=M",
+      "coords": {
+        "M": [
+          2.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 398,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 399,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0399.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Focus(C) = {F1, F2};PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2));AngleOf(P, F1, F2) = ApplyUnit(45, degree)",
+      "coords": {}
+    },
+    {
+      "index": 400,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 401,
+      "success": true,
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+      "error": null,
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+        "y_coef": 3.0
+      },
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0401.png",
+      "answer": "(-12, 0)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 + y^2/k = 1);k: Number;e: Number;Eccentricity(G) = e;In(e, (1, 2))",
+      "coords": {}
+    },
+    {
+      "index": 402,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0402.png",
+      "answer": "3",
+      "fact_expr": "G: Parabola;H:Line;A: Point;C: Point;B: Point;D: Point;F:Point;l1:Line;l2:Line;Expression(G) = (y^2 = 4*x);Focus(G)=F;PointOnCurve(F, H);Intersection(H,G)={A,B};PointOnCurve(A,l1);PointOnCurve(B,l2);IsPerpendicular(l1,xAxis);IsPerpendicular(l2,yAxis);FootPoint(l1,xAxis)=C;FootPoint(l2,yAxis)=D",
+      "coords": {}
+    },
+    {
+      "index": 403,
+      "success": true,
+      "conic_type": "ellipse",
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0403.png",
+      "answer": "{2*sqrt(41) - 2, 10}",
+      "fact_expr": "G: Curve;a: Number;F1: Point;F2: Point;P: Point;Expression(G) = (y^2/16 + x^2/a = 1);FocalLength(G) = 10;PointOnCurve(P, G);OneOf(Focus(G))=F1;OneOf(Focus(G))=F2;Negation(F1=F2);Distance(P, F1) = 2",
+      "coords": {}
+    },
+    {
+      "index": 404,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 405,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 406,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0406.png",
+      "answer": "5",
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+      "coords": {}
+    },
+    {
+      "index": 407,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0407.png",
+      "answer": "(1,0);sqrt(2)/2",
+      "fact_expr": "G: Hyperbola;H: Parabola;Expression(G) = (x^2 - y^2 = 1);Expression(H) = (y^2 = 4*x);F: Point;Focus(H) = F",
+      "coords": {}
+    },
+    {
+      "index": 408,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0408.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2/4 - y^2/b^2 = 1);b: Number;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;Abs(LineSegmentOf(F1, F2)) = 6",
+      "coords": {}
+    },
+    {
+      "index": 409,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0409.png",
+      "answer": "7",
+      "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 6*x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};XCoordinate(MidPoint(LineSegmentOf(A, B))) = 2",
+      "coords": {}
+    },
+    {
+      "index": 410,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0410.png",
+      "answer": "(pm*sqrt(5),0)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 411,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 412,
+      "success": true,
+      "conic_type": "ellipse",
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+      "params": {
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+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0412.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2/2 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 413,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0413.png",
+      "answer": "(0, 1/2)",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 2*y)",
+      "coords": {}
+    },
+    {
+      "index": 414,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.4142135623730951,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0414.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;H: Parabola;p: Number;Expression(G) = (x^2 - y^2 = 2);p>0;Expression(H) = (y^2 = 2*(p*x));Focus(H) = RightFocus(G)",
+      "coords": {}
+    },
+    {
+      "index": 415,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 416,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "up"
+      },
+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0416.png",
+      "answer": "-4",
+      "fact_expr": "G: Parabola;P: Point;Q: Point;A: Point;L1:Line;L2:Line;Expression(G) = (x^2 = 2*y);PointOnCurve(P,G);PointOnCurve(Q,G);XCoordinate(P)=4;XCoordinate(Q)=-2;TangentOfPoint(P,G)=L1;TangentOfPoint(Q,G)=L2;Intersection(L1,L2)=A",
+      "coords": {}
+    },
+    {
+      "index": 417,
+      "success": true,
+      "conic_type": "ellipse",
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+        "x_coef": 2.0,
+        "y_coef": 1.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0417.png",
+      "answer": "[2,2*sqrt(2))",
+      "fact_expr": "C: Ellipse;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/2 + y^2 = 1);Coordinate(P) = (x0, y0);Focus(C)={F1,F2};x0:Number;y0:Number;0<x0^2/2+y0^2;x0^2/2+y0^2<1",
+      "coords": {}
+    },
+    {
+      "index": 418,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0418.png",
+      "answer": "1/2",
+      "fact_expr": "G: Parabola;H: Line;A: Point;F: Point;B: Point;Expression(G) = (y^2 = 8*x);Focus(G)=F;PointOnCurve(F, H);Intersection(H, G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 419,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0419.png",
+      "answer": "sqrt(6)/2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/2 - y^2 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 420,
+      "success": true,
+      "conic_type": "parabola",
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+      "params": {
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+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0420.png",
+      "answer": "pi/3",
+      "fact_expr": "C: Parabola;G: Line;B: Point;A: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F, G);Intersection(G, C) = {A, B};Quadrant(A)=1;Abs(LineSegmentOf(A, F)) = 3*Abs(LineSegmentOf(F, B))",
+      "coords": {}
+    },
+    {
+      "index": 421,
+      "success": true,
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+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0421.png",
+      "answer": "sqrt(3) - 1",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;G: Line;c: Number;M: Point;F1: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y = sqrt(3)*(c + x));LeftFocus(C) = F1;RightFocus(C) = F2;FocalLength(C) = 2*c;OneOf(Intersection(G, C)) = M;AngleOf(M, F1, F2) = 2*AngleOf(M, F2, F1)",
+      "coords": {}
+    },
+    {
+      "index": 422,
+      "success": true,
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+        "direction": "up"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0422.png",
+      "answer": "sqrt(2)",
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+      "coords": {}
+    },
+    {
+      "index": 423,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0423.png",
+      "answer": "25/8",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, LeftPart(G));l: Line;OneOf(Asymptote(G)) = l;Q: Point;Projection(P, l) = Q;Min(Abs(LineSegmentOf(P, F2)) + Abs(LineSegmentOf(P, Q))) = 5",
+      "coords": {}
+    },
+    {
+      "index": 424,
+      "success": true,
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+        "b": 1.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0424.png",
+      "answer": "[-sqrt(5), -1) + (-1, 1) + (1, sqrt(5)]",
+      "fact_expr": "A: Point;Coordinate(A) = (0, 2);l: Line;PointOnCurve(A, l);G: Curve;Expression(G) = (x^2 - y^2 = 1);IsIntersect(l, G)",
+      "coords": {
+        "A": [
+          0.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 425,
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+        "iterations": 283
+      },
+      "output_path": "results/test/hyperbola/problem_0425.png",
+      "answer": "sqrt(10)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/b^2 = 1);b: Number;b>0;Expression(OneOf(Asymptote(G))) = (y = 3*x)",
+      "coords": {}
+    },
+    {
+      "index": 426,
+      "success": true,
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+        "x_coef": 2.0,
+        "y_coef": 1.0
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+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0426.png",
+      "answer": "2",
+      "fact_expr": "G: Ellipse;O: Point;P: Point;F: Point;Expression(G) = (x^2/2 + y^2 = 1);PointOnCurve(P, G);Center(G) = O;LeftFocus(G) = F",
+      "coords": {}
+    },
+    {
+      "index": 427,
+      "success": true,
+      "conic_type": "ellipse",
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0427.png",
+      "answer": "(sqrt(2)/2,1)",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;A: Point;Coordinate(A) = (a, 0);RightVertex(G) = A;P: Point;PointOnCurve(P, G) = True;O: Origin;AngleOf(A, P, O) = ApplyUnit(90, degree)",
+      "coords": {}
+    },
+    {
+      "index": 428,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 429,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 430,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 431,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0431.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "C: Hyperbola;a: Number;b: Number;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);G: Line;O:Origin;PointOnCurve(O,G);P: Point;F: Point;Q: Point;a>0;b>0;Intersection(G, C) = {P, Q};OneOf(Focus(C)) = F;Tan(AngleOf(P, F, Q)) = 4/3;Abs(LineSegmentOf(Q, F)) = 5*Abs(LineSegmentOf(P, F));Negation(P=Q)",
+      "coords": {}
+    },
+    {
+      "index": 432,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0432.png",
+      "answer": "x^2/2 + y^2 = 1",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;O: Circle;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Expression(O) = (x^2 + y^2 = 1);RightFocus(C) = F2;L:Line;TangentOnPoint(F2, O)=L;Length(InterceptChord(L,C))=sqrt(2)",
+      "coords": {}
+    },
+    {
+      "index": 433,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_point_off_curve",
+      "params": {
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+        "b": 1.4970015287399292,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "iterations": 2
+      },
+      "output_path": "results/test/hyperbola/problem_0433.png",
+      "answer": "1",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Point;H: Point;e: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(G) = (e, 1);Coordinate(H) = (-sqrt(3), sqrt(2));Eccentricity(C) = e;PointOnCurve(G, C);PointOnCurve(H, C)",
+      "coords": {
+        "H": [
+          -1.7320508075688772,
+          1.4142135623730951
+        ]
+      },
+      "validation_reasons": [
+        "hyperbola_point_off_curve"
+      ]
+    },
+    {
+      "index": 434,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.8284271247461903,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0434.png",
+      "answer": "(x - 3)^2 + y^2 = 1",
+      "fact_expr": "G: Hyperbola;H: Circle;Expression(G) = (x^2/8 - y^2 = 1);Center(H) = RightFocus(G);IsTangent(H, Asymptote(G))",
+      "coords": {}
+    },
+    {
+      "index": 435,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 436,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "vertical"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0436.png",
+      "answer": "y=pm*(sqrt(2)/2)*x",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-x^2/b^2 + y^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;Focus(C) = {F1, F2};O: Origin;G: Circle;Center(G) = O;Radius(G) = b;P: Point;Intersection(G, C) = P;Quadrant(P) = 1;IsPerpendicular(LineSegmentOf(O, P), LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 437,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0437.png",
+      "answer": "10",
+      "fact_expr": "G: Ellipse;a: Number;b: Number;H: Circle;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a = 1);Expression(H) = (-6*x + x^2 + y^2 + 8 = 0);OneOf(Focus(G)) = Center(H);Length(MinorAxis(G))=8",
+      "coords": {}
+    },
+    {
+      "index": 438,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0438.png",
+      "answer": "(2/3)*sqrt(3)",
+      "fact_expr": "G: Hyperbola;p: Number;H: Parabola;p>0;Expression(G) = (x^2/3 - 16*y^2/p^2 = 1);Expression(H) = (y^2 = 2*(p*x));PointOnCurve(LeftFocus(G), Directrix(H))",
+      "coords": {}
+    },
+    {
+      "index": 439,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0439.png",
+      "answer": "8",
+      "fact_expr": "E: Ellipse;Expression(E) = (x^2/25 + y^2/9 = 1);F1: Point;F2: Point;Focus(E) = {F1, F2};l: Line;PointOnCurve(F1, l);A: Point;B: Point;Intersection(l, E) = {A, B};Abs(LineSegmentOf(F2, A)) + Abs(LineSegmentOf(F2, B)) = 12",
+      "coords": {}
+    },
+    {
+      "index": 440,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0440.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 + (x - 2)^2 = 3);IsTangent(H, Asymptote(G))",
+      "coords": {}
+    },
+    {
+      "index": 441,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.8284271247461903,
+        "b": 2.23606797749979,
+        "major_axis": "x",
+        "x_coef": 8.0,
+        "y_coef": 5.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0441.png",
+      "answer": "x^2/3-y^2/5=1",
+      "fact_expr": "H: Ellipse;Expression(H) = (x^2/8 + y^2/5 = 1);Focus(H) = Vertex(C);C: Hyperbola;Focus(C) = Vertex(H)",
+      "coords": {}
+    },
+    {
+      "index": 442,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0442.png",
+      "answer": "2*x-y-15=0",
+      "fact_expr": "P: Point;Coordinate(P) = (8, 1);G: Hyperbola;Expression(G) = (x^2 - 4*y^2 = 4);H: LineSegment;IsChordOf(H,G) = True;MidPoint(H) = P",
+      "coords": {
+        "P": [
+          8.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 443,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0443.png",
+      "answer": "5/3",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;A: Point;LeftVertex(C) = A;F: Point;RightFocus(C) = F;G: Circle;Center(G) = F;B: Point;Quadrant(B) = 1;TangentPoint(G, OneOf(Asymptote(C))) = B;Slope(LineOf(A, B)) = 1/2",
+      "coords": {}
+    },
+    {
+      "index": 444,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 4.496487617492676,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 5.969127983007638e-07,
+        "converged": true,
+        "iterations": 103
+      },
+      "output_path": "results/test/parabola/problem_0444.png",
+      "answer": "13/2",
+      "fact_expr": "C: Parabola;p: Number;P: Point;F: Point;p>0;Expression(C) = (y^2 = 2*p*x);Coordinate(P) = (2, 6);PointOnCurve(P, C);Focus(C) = F",
+      "coords": {
+        "P": [
+          2.0,
+          6.0
+        ]
+      }
+    },
+    {
+      "index": 445,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0445.png",
+      "answer": "3*sqrt(10)",
+      "fact_expr": "G: Hyperbola;H: Line;P: Point;F1: Point;F2: Point;Expression(G) = (x^2 - y^2/9 = 1);Expression(H) = (x = 1);LeftFocus(G)=F1;RightFocus(G)=F2;Intersection(H,OneOf(Asymptote(G)))=P",
+      "coords": {}
+    },
+    {
+      "index": 446,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0446.png",
+      "answer": "4",
+      "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;F: Point;x1: Number;x2: Number;y1: Number;y2: Number;Expression(G) = (y^2 = 2*x);Coordinate(A) = (x1, y1) ;Coordinate(B) = (x2, y2);Focus(G) = F;PointOnCurve(F, l);x1 + x2 = 3;Intersection(l, G) = {A, B};Negation(A = B);IsChordOf(LineSegmentOf(A, B), G)",
+      "coords": {}
+    },
+    {
+      "index": 447,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0447.png",
+      "answer": "(0,1)",
+      "fact_expr": "G: Parabola;Expression(G) = (y = x^2/4)",
+      "coords": {}
+    },
+    {
+      "index": 448,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 12.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0448.png",
+      "answer": "{30,10}",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/25 - y^2/144 = 1);P: Point;PointOnCurve(P, G);A: Point;Coordinate(A) = (13, 0);Distance(P, A) = 20;B: Point;Coordinate(B) = (-13, 0)",
+      "coords": {
+        "A": [
+          13.0,
+          0.0
+        ],
+        "B": [
+          -13.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 449,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.3300262689590454,
+        "b": 2.30381441116333,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 6.658357051492203e-06,
+        "converged": true,
+        "iterations": 122
+      },
+      "output_path": "results/test/hyperbola/problem_0449.png",
+      "answer": "x^2/4-y^/12=1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Expression(OneOf(Asymptote(G))) = (y = sqrt(3)*x);H: Parabola;Expression(H) = (y^2 = 16*x);OneOf(Focus(G)) = Focus(H)",
+      "coords": {}
+    },
+    {
+      "index": 450,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0450.png",
+      "answer": "x+2y-3=0",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2 + 2*y^2 = 4);P: Point;Coordinate(P) = (1, 1);L: Line;PointOnCurve(P, L) = True;Intersection(L, C) = {A, B};A: Point;B: Point;MidPoint(LineSegmentOf(A, B)) = P",
+      "coords": {
+        "P": [
+          1.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 451,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          3.0,
+          -0.0
+        ],
+        "radius": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/circle/problem_0451.png",
+      "answer": "1/16",
+      "fact_expr": "H: Circle;Expression(H) = (-6*x + x^2 + y^2 - 7 = 0);G: Parabola;Expression(G) = (y = a*x^2);a: Number;a>0;IsTangent(H, Directrix(G))",
+      "coords": {}
+    },
+    {
+      "index": 452,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0452.png",
+      "answer": "sqrt(2) - 1",
+      "fact_expr": "E: Parabola;p: Number;C: Ellipse;a: Number;b: Number;F: Point;p>0;Expression(E) = (y^2 = 2*(p*x));Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Focus(E) = F;RightFocus(C) = F;A: Point;B: Point;Intersection(C, E) = {A, B};PointOnCurve(F, LineOf(A, B))",
+      "coords": {}
+    },
+    {
+      "index": 453,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 454,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0454.png",
+      "answer": "",
+      "fact_expr": "G: Hyperbola;H: Curve;a: Number;Expression(G) = (x^2/9 - y^2/4 = 1);Expression(H) = (y = a*x^2 + 1/3);L1:Line;L2:Line;Asymptote(G)={L1,L2};IsTangent(L1,H);IsTangent(L2,H)",
+      "coords": {}
+    },
+    {
+      "index": 455,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0455.png",
+      "answer": "x=-1",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;H: Line;PointOnCurve(Focus(G), H);Slope(H) = -1;A: Point;B: Point;Intersection(H, G) = {A, B};XCoordinate(MidPoint(LineSegmentOf(A, B))) = 3",
+      "coords": {}
+    },
+    {
+      "index": 456,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0456.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "G: Parabola;p: Number;A: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(A) = (x0, -2);x0:Number;PointOnCurve(A,G);Distance(A,Focus(G))=3*Distance(A,yAxis)",
+      "coords": {}
+    },
+    {
+      "index": 457,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 458,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.4641016151377544,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 12.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0458.png",
+      "answer": "6",
+      "fact_expr": "C: Ellipse;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/16 + y^2/12 = 1);Focus(C) = {F1,F2};PointOnCurve(P, C);Abs(LineSegmentOf(P, F2)) = 3",
+      "coords": {}
+    },
+    {
+      "index": 459,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0459.png",
+      "answer": "4/3",
+      "fact_expr": "l: Line;C: Parabola;F: Point;M: Point;P: Point;Q: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {P, Q};Intersection(l, Directrix(C)) = M;VectorOf(F, M) = 3*VectorOf(F, P)",
+      "coords": {}
+    },
+    {
+      "index": 460,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 461,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0461.png",
+      "answer": "x^2/4+y^2=1",
+      "fact_expr": "G: Hyperbola;H: Ellipse;I: Point;Expression(G) = (x^2/2 - y^2 = 1);Coordinate(I) = (1, sqrt(3)/2);PointOnCurve(I, H);Focus(G)=Focus(H)",
+      "coords": {
+        "I": [
+          1.0,
+          0.8660254037844386
+        ]
+      }
+    },
+    {
+      "index": 462,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0462.png",
+      "answer": "63/4",
+      "fact_expr": "P: Point;Coordinate(P) = (5, 0);C: Hyperbola;Expression(C) = (x^2 - y^2/3 = 1);M: Point;N: Point;PointOnCurve(M,RightPart(C)) ;PointOnCurve(N,RightPart(C)) ;IsPerpendicular(VectorOf(P, M), VectorOf(P, N))",
+      "coords": {
+        "P": [
+          5.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 463,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0463.png",
+      "answer": "4*sqrt(6)",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2 - y^2/3 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, RightPart(C)) = True;Abs(LineSegmentOf(P, F2)) = 5",
+      "coords": {}
+    },
+    {
+      "index": 464,
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+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0464.png",
+      "answer": "2",
+      "fact_expr": "l: Line;G: Hyperbola;b: Number;a: Number;M: Point;A: Point;B: Point;F: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftVertex(G) = M;RightFocus(G) = F;PointOnCurve(F, l);IsPerpendicular(xAxis, l);Intersection(l, G) = {A,B};IsPerpendicular(LineSegmentOf(M, A), LineSegmentOf(M, B))",
+      "coords": {}
+    },
+    {
+      "index": 465,
+      "success": true,
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+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0465.png",
+      "answer": "x^2-y^2/3=1",
+      "fact_expr": "G:Hyperbola;a:Number;b:Number;a>0;b>0;C:Parabola;e:Number;Expression(G)=(x^2/a^2-y^2/b^2=1);Expression(C)=(y^2=8*x);Eccentricity(G)=e;e=2;OneOf(Focus(G))=Focus(C)",
+      "coords": {}
+    },
+    {
+      "index": 466,
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+      "params": {
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0466.png",
+      "answer": "2",
+      "fact_expr": "C1: Parabola;C2: Circle;M: Point;A: Point;B: Point;Coordinate(M) = (1, 1);Expression(C1) = (y^2 = 8*x);Expression(C2) = (x^2+y^2-4*x+3=0);PointOnCurve(A, C1);PointOnCurve(B, C2)",
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+        "M": [
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+          1.0
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+      }
+    },
+    {
+      "index": 467,
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+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 468,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0468.png",
+      "answer": "(1/2, 2/3)",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;G: Line;A: Point;B: Point;F: Point;k: Number;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftVertex(C) = A;PointOnCurve(A, G);Slope(G) = k;Intersection(G, C) = {A, B};Projection(B, xAxis) = F;RightFocus(C) = F;1/3 < k;k < 1/2",
+      "coords": {}
+    },
+    {
+      "index": 469,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 470,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "left"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0470.png",
+      "answer": "(-1,0)",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = -4*x)",
+      "coords": {}
+    },
+    {
+      "index": 471,
+      "success": true,
+      "conic_type": "parabola",
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+        "direction": "right"
+      },
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0471.png",
+      "answer": "",
+      "fact_expr": "P: Point;Coordinate(P) = (2, 1);G: Parabola;Expression(G) = (y^2 = 4*x);A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G);MidPoint(LineSegmentOf(A, B)) = P",
+      "coords": {
+        "P": [
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+          1.0
+        ]
+      }
+    },
+    {
+      "index": 472,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
+      },
+      "optimization": {
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+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0472.png",
+      "answer": "{0,-1,-4/5}",
+      "fact_expr": "G: Line;Expression(G) = (y = x*(a + 1) - 1);H: Curve;Expression(H) = (y^2 = a*x);a: Real;NumIntersection(G, H)=1",
+      "coords": {}
+    },
+    {
+      "index": 473,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0473.png",
+      "answer": "14",
+      "fact_expr": "G: Hyperbola;b: Number;b>0;Expression(G) = (x^2/4 - y^2/b = 1);Expression(Asymptote(G)) = (y = pm*x)",
+      "coords": {}
+    },
+    {
+      "index": 474,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 3.4641016151377544,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0474.png",
+      "answer": "y=pm*sqrt(3)*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;Eccentricity(G) = 2;H: Ellipse;Expression(H) = (x^2/25 + y^2/9 = 1);Focus(G) = Focus(H)",
+      "coords": {}
+    },
+    {
+      "index": 475,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 476,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.125,
+        "direction": "up"
+      },
+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0476.png",
+      "answer": "17/8",
+      "fact_expr": "G: Parabola;A: Point;Expression(G) = (y = 2*x^2);Coordinate(A) = (1, 2);PointOnCurve(A, G);Focus(G) = F;F: Point",
+      "coords": {
+        "A": [
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+          2.0
+        ]
+      }
+    },
+    {
+      "index": 477,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0477.png",
+      "answer": "sqrt(6)/2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;HalfFocalLength(G) = c;c: Number;PointOnCurve(RightFocus(G), L);Slope(L) = 1;L: Line;NumIntersection(L, RightPart(G)) = 2;H: Parabola;Expression(H) = (y^2 = 4*c*x);Length(InterceptChord(Directrix(H), G)) = (2*sqrt(2)/3)*b*e^2;e: Number;Eccentricity(G) = e",
+      "coords": {}
+    },
+    {
+      "index": 478,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 5.196152422706632,
+        "major_axis": "y",
+        "x_coef": 27.0,
+        "y_coef": 36.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0478.png",
+      "answer": "y^2/4-x^2/5=1",
+      "fact_expr": "C: Hyperbola;F:Ellipse;P:Point;Expression(F) = (x^2/27 + y^2/36 = 1);Coordinate(P) = (sqrt(15), 4);Focus(C) = Focus(F);PointOnCurve(P,C)",
+      "coords": {
+        "P": [
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+          4.0
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+      }
+    },
+    {
+      "index": 479,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 4.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0479.png",
+      "answer": "4*x+9*y-13=0",
+      "fact_expr": "G: Ellipse;H: Line;A: Point;B: Point;M: Point;Expression(G) = (x^2/9 + y^2/4 = 1);Coordinate(M) = (1, 1);Intersection(G, H) = {A,B};IsChordOf(LineSegmentOf(A,B),G);MidPoint(LineSegmentOf(A, B)) = M;PointOnCurve(M,H)",
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+        "M": [
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+          1.0
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+      }
+    },
+    {
+      "index": 480,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0480.png",
+      "answer": "2*sqrt(3)/3",
+      "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;A: Point;F: Point;B: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;PointOnCurve(F,l);Slope(l)=a/b;L1:Line;L2:Line;Asymptote(C)={L1,L2};Intersection(L1,l)=A;Intersection(L2,l)=B;VectorOf(A, F) = 2*VectorOf(F, B)",
+      "coords": {}
+    },
+    {
+      "index": 481,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 5.0,
+        "y_coef": 4.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0481.png",
+      "answer": "5*sqrt(5)/3",
+      "fact_expr": "L: Line;Slope(L) = 2;G: Ellipse;Expression(G) = (x^2/5 + y^2/4 = 1);RightFocus(G) = F2;F2: Point;PointOnCurve(F2, L) = True;Intersection(L, G) = {A, B};A: Point;B: Point;IsChordOf(LineSegmentOf(A,B), G) = True",
+      "coords": {}
+    },
+    {
+      "index": 482,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0482.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;H: Line;PointOnCurve(F, H);A: Point;B: Point;Intersection(H, G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 483,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0483.png",
+      "answer": "-1",
+      "fact_expr": "C: Parabola;p: Number;Z: Line;A: Point;F: Point;M: Point;B: Point;p>0;Expression(C) = (y^2 = 2*p*x);Coordinate(M) = (-1, 4);Focus(C) = F;PointOnCurve(M, Z);IsPerpendicular(Z, yAxis);Intersection(Z, C) = A;Abs(LineSegmentOf(A, F)) = Abs(LineSegmentOf(A, M));Intersection(LineOf(A, F), C) = {A, B}",
+      "coords": {
+        "M": [
+          -1.0,
+          4.0
+        ]
+      }
+    },
+    {
+      "index": 484,
+      "success": false,
+      "error": "Circle from diameter: cannot parse diameter pattern"
+    },
+    {
+      "index": 485,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 2.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0485.png",
+      "answer": "20",
+      "fact_expr": "C: Hyperbola;A: Point;F1: Point;F2: Point;Expression(C) = (x^2/16 - y^2/4 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(A, LeftPart(C));Abs(LineSegmentOf(A, F1)) = 12",
+      "coords": {}
+    },
+    {
+      "index": 486,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 487,
+      "success": false,
+      "conic_type": "parabola",
+      "error": "validation: parabola_focus_mismatch",
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0487.png",
+      "answer": "8*sqrt(6)/9",
+      "fact_expr": "l: Line;G: Parabola;p: Number;F: Point;A: Point;B: Point;m:Number;p>0;Expression(G) = (y^2 = 2*p*x);Expression(l)=(y=-x+m);m>=0;m<1;Coordinate(F) = (-1, 0);Focus(G) = F;Intersection(l,G) = {A, B};Negation(A = B)",
+      "coords": {
+        "F": [
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+          0.0
+        ]
+      },
+      "validation_reasons": [
+        "parabola_focus_mismatch"
+      ]
+    },
+    {
+      "index": 488,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0488.png",
+      "answer": "(0,1)",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y)",
+      "coords": {}
+    },
+    {
+      "index": 489,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 490,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0490.png",
+      "answer": "7/2",
+      "fact_expr": "G: Ellipse;H: Line;P: Point;F2: Point;F1: Point;Expression(G) = (x^2/4 + y^2 = 1);Focus(G) = {F1, F2};PointOnCurve(F1, H);IsPerpendicular(H, xAxis);IsIntersect(H, G);OneOf(Intersection(H, G)) = P",
+      "coords": {}
+    },
+    {
+      "index": 491,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.4142135623730951,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0491.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;H: Line;P: Point;F1: Point;F2: Point;Expression(G) = (x^2 - y^2/2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2, H);IsPerpendicular(xAxis, H);Intersection(H, G) = P",
+      "coords": {}
+    },
+    {
+      "index": 492,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0492.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Distance(RightFocus(G), OneOf(Asymptote(G))) = 2*a",
+      "coords": {}
+    },
+    {
+      "index": 493,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0493.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/3 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 494,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0494.png",
+      "answer": "5/3",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;A: Point;B: Point;PointOnCurve(A,G) = True;PointOnCurve(B,G) = True;VectorOf(A,F)= 3*VectorOf(F,B)",
+      "coords": {}
+    },
+    {
+      "index": 495,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0495.png",
+      "answer": "4",
+      "fact_expr": "G: Ellipse;M: Point;F1: Point;O: Origin;N: Point;Expression(G) = (x^2/25 + y^2/9 = 1);PointOnCurve(M,G);LeftFocus(G)=F1;Distance(M,F1)=2;MidPoint(LineSegmentOf(M,F1))=N",
+      "coords": {}
+    },
+    {
+      "index": 496,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 497,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0497.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Slope(OneOf(Asymptote(G)))=sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 498,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 499,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
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+          0.0
+        ],
+        "radius": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/circle/problem_0499.png",
+      "answer": "x^2/4 + y^2 = 1",
+      "fact_expr": "C: Circle;N: Point;P: Point;M: Point;O: Origin;l: Line;Expression(C) = (x^2 + y^2 = 4);PointOnCurve(M, C);PointOnCurve(M, l);IsPerpendicular(l, xAxis);FootPoint(l, xAxis) = N;VectorOf(N, P) = VectorOf(P, M)",
+      "coords": {}
+    },
+    {
+      "index": 500,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0500.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;c: Number;e: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);HalfFocalLength(C) = c;Eccentricity(C) = e",
+      "coords": {}
+    },
+    {
+      "index": 501,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0501.png",
+      "answer": "(2, 1)",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);P: Point;PointOnCurve(P, G);F: Point;Focus(G) = F;A: Point;Coordinate(A) = (2, 3);WhenMin(LineSegmentOf(P, A) + LineSegmentOf(P, F))",
+      "coords": {
+        "A": [
+          2.0,
+          3.0
+        ]
+      }
+    },
+    {
+      "index": 502,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 6.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0502.png",
+      "answer": "33",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/64 - y^2/36 = 1);M: Point;PointOnCurve(M, G) = True;LeftFocus(G) = F1;RightFocus(G) = F2;F1: Point;F2: Point;Abs(LineSegmentOf(M, F1)) = 17",
+      "coords": {}
+    },
+    {
+      "index": 503,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 504,
+      "success": true,
+      "conic_type": "ellipse",
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+        "b": 2.6457513110645907,
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+        "x_coef": 16.0,
+        "y_coef": 7.0
+      },
+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0504.png",
+      "answer": "7/3",
+      "fact_expr": "C: Ellipse;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/16 + y^2/7 = 1);PointOnCurve(P, C);Focus(C) = {F1, F2};Radius(InscribedCircle(TriangleOf(P,F1,F2)))=1;Quadrant(P)=1",
+      "coords": {}
+    },
+    {
+      "index": 505,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0505.png",
+      "answer": "18",
+      "fact_expr": "C: Parabola;l1: Line;l2: Line;A: Point;B: Point;D: Point;E: Point;F: Point;Expression(C) = (x^2 = 2*y);Focus(C) = F;PointOnCurve(F,l1);PointOnCurve(F,l2);IsPerpendicular(l1,l2);Intersection(l1, C) = {A, B};Intersection(l2, C) = {D,E}",
+      "coords": {}
+    },
+    {
+      "index": 506,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
+      },
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0506.png",
+      "answer": "16",
+      "fact_expr": "G: Ellipse;H: Circle;P: Point;M: Point;F1: Point;Expression(G) = (x^2/25 + y^2/16 = 1);Expression(H) = ((x + 1)^2 + (y - 3)^2 = 1);LeftFocus(G)=F1;PointOnCurve(P,G);PointOnCurve(M, H)",
+      "coords": {}
+    },
+    {
+      "index": 507,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 2.5736796855926514,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 5.191216132516274e-06,
+        "converged": true,
+        "iterations": 118
+      },
+      "output_path": "results/test/hyperbola/problem_0507.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/2 + x^2/a^2 = 1);a: Number;F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);AngleOf(F1, P, F2) = pi/3;Eccentricity(G) = sqrt(6)/2",
+      "coords": {}
+    },
+    {
+      "index": 508,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0508.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;C: Point;Intersection(Directrix(G), xAxis) = C;H: Circle;Center(H) = F;Radius(H) = p;A: Point;B: Point;Intersection(H, G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 509,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0509.png",
+      "answer": "(0,sqrt(2)-1]",
+      "fact_expr": "E: Ellipse;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;F2: Point;F1: Point;LeftFocus(E) = F1;RightFocus(E) = F2;A: Point;B: Point;IsChordOf(LineSegmentOf(A,B),E);PointOnCurve(F1,LineSegmentOf(A,B));Abs(VectorOf(F2,A)-VectorOf(F2,B)) >= Abs(VectorOf(F2,A)+VectorOf(F2,B))",
+      "coords": {}
+    },
+    {
+      "index": 510,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 3.455598402979376e-07,
+        "converged": true,
+        "iterations": 81
+      },
+      "output_path": "results/test/parabola/problem_0510.png",
+      "answer": "pi/3",
+      "fact_expr": "P: Point;Coordinate(P) = (1, 2);E: Parabola;Expression(E) = (y^2 = 2*p*x);p: Number;p>0;PointOnCurve(P, E);M: Point;Coordinate(M) = (1, 0);PointOnCurve(M, l);Slope(l) > 0;l: Line;Intersection(l, E) = {A, B};A: Point;B: Point;VectorOf(A, M) = 3*VectorOf(M, B)",
+      "coords": {
+        "P": [
+          1.0,
+          2.0
+        ],
+        "M": [
+          1.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 511,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0511.png",
+      "answer": "(sqrt(5)-1)/2",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;A: Point;B: Point;F: Point;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftVertex(G)=A;UpperVertex(G)=B;RightFocus(G) = F;IsPerpendicular(LineSegmentOf(A, B), LineSegmentOf(B, F));a > b;b > 0",
+      "coords": {}
+    },
+    {
+      "index": 512,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0512.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Parabola;O: Origin;M: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(M, G);Abs(LineSegmentOf(M, F)) = 3",
+      "coords": {}
+    },
+    {
+      "index": 513,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 514,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0514.png",
+      "answer": "x^2/5+y^2/4=1",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;F: Point;B: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F;UpperVertex(G) = B;Length(MinorAxis(G)) = 4;Eccentricity(G)=sqrt(5)/5",
+      "coords": {}
+    },
+    {
+      "index": 515,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0515.png",
+      "answer": "y^2=4*x",
+      "fact_expr": "G: Parabola;p: Number;A: Point;F: Point;O: Origin;p>0;Expression(G) = (y^2 = 2*(p*x));PointOnCurve(A, G);Focus(G) = F;Abs(LineSegmentOf(A,F))=4;AngleOf(O, F, A) = ApplyUnit(120, degree)",
+      "coords": {}
+    },
+    {
+      "index": 516,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0516.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Hyperbola;b: Number;H: Parabola;p: Number;M: Point;F: Point;m:Number;m>0;Expression(G) = (x^2/2 - y^2/b^2 = 1);p>0;Expression(H) = (y^2 = 2*p*x);Coordinate(M) = (1, m);PointOnCurve(M, H);Focus(H) = F;Distance(M, F) = 5;Distance(F,OneOf(Asymptote(G))) = 2*sqrt(2)",
+      "coords": {}
+    },
+    {
+      "index": 517,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 518,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0518.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, G);L1: Line;L2: Line;Asymptote(G) = {L1, L2};Distance(P, L1)*Distance(P, L2) = b^2/3",
+      "coords": {}
+    },
+    {
+      "index": 519,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0519.png",
+      "answer": "2*sqrt(3)/3",
+      "fact_expr": "C: Hyperbola;a: Number;b: Number;A1: Circle;M: Point;A: Point;N: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightVertex(C) = A;Center(A1)=A;Length(Radius(A1))=b;Intersection(A1, OneOf(Asymptote(C))) = {M, N};AngleOf(M, A, N) = ApplyUnit(60, degree)",
+      "coords": {}
+    },
+    {
+      "index": 520,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 5.656854249492381,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0520.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;A: Point;x0: Number;y0:Number;Expression(G) = (x^2/4 - y^2/32 = 1);Coordinate(A) = (x0, y0);PointOnCurve(A, RightPart(G));Distance(A,RightFocus(G)) = 2*x0",
+      "coords": {}
+    },
+    {
+      "index": 521,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0521.png",
+      "answer": "{0,1}",
+      "fact_expr": "H: Line;Expression(H) = (y = k*x + 1);k: Number;G: Parabola;Expression(G) = (y^2 = 4*x);NumIntersection(H, G) = 1",
+      "coords": {}
+    },
+    {
+      "index": 522,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0522.png",
+      "answer": "3*x-y-11=0",
+      "fact_expr": "G: Parabola;H: LineSegment;P: Point;Expression(G) = (y^2 = 6*x);Coordinate(P) = (4, 1);PointOnCurve(P,H);MidPoint(H)=P;IsChordOf(H,G)",
+      "coords": {
+        "P": [
+          4.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 523,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0523.png",
+      "answer": "2/3",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;G: Line;A: Point;F: Point;B: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);RightFocus(C)=F;PointOnCurve(F, G);Slope(G) = sqrt(3);Intersection(G, C) = {A, B};VectorOf(A, F) = VectorOf(F, B)/2",
+      "coords": {}
+    },
+    {
+      "index": 524,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0524.png",
+      "answer": "9/8",
+      "fact_expr": "G: Parabola;A: Point;B: Point;M: Point;Expression(G) = (y^2 = 8*x);Length(LineSegmentOf(A, B)) = 6;MidPoint(LineSegmentOf(A, B)) = M;IsChordOf(LineSegmentOf(A,B),G)",
+      "coords": {}
+    },
+    {
+      "index": 525,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0525.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;O: Origin;F2: Point;P: Point;F1: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;IsDiameter(LineSegmentOf(O,F2),G);Intersection(G, OneOf(Asymptote(C))) = {O, P};AngleOf(O, P, F1) = ApplyUnit(45, degree)",
+      "coords": {}
+    },
+    {
+      "index": 526,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0526.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "C: Parabola;p: Number;G: Line;A: Point;O: Origin;B: Point;p>0;Expression(C) = (y^2 = 2*(p*x));PointOnCurve(Focus(C), G);Slope(G) = 1;Intersection(G, C) = {A, B};Abs(LineSegmentOf(A, B)) = 8",
+      "coords": {}
+    },
+    {
+      "index": 527,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0527.png",
+      "answer": "sqrt(14)/4",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;H: Circle;I: Line;B: Point;C: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(I) = (y = b/3);Intersection(I, G) = {B, C};IsDiameter(LineSegmentOf(B,C),H);RightFocus(G)=F;PointOnCurve(F,H);F:Point",
+      "coords": {}
+    },
+    {
+      "index": 528,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0528.png",
+      "answer": "1/3",
+      "fact_expr": "l: Line;Gamma: Parabola;O: Origin;F: Point;A: Point;B: Point;Expression(Gamma) = (y^2 = 4*x);Expression(l)=(sqrt(3)*x - y - sqrt(3) = 0);Intersection(l,Gamma) = {A, B};Intersection(l,xAxis)=F;LineSegmentOf(O,F)=lambda*LineSegmentOf(O,A)+mu*LineSegmentOf(O,B);lambda<=mu;lambda:Number;mu:Number",
+      "coords": {}
+    },
+    {
+      "index": 529,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0529.png",
+      "answer": "9/10",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, RightPart(G));Abs(LineSegmentOf(P, F2)) = Abs(LineSegmentOf(F1, F2));Expression(Asymptote(G)) = (3*x + pm*(4*y) = 0)",
+      "coords": {}
+    },
+    {
+      "index": 530,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0530.png",
+      "answer": "4",
+      "fact_expr": "l1: Line;Expression(l1) = (4*x - 3*y + 16 = 0);l2: Line;Expression(l2) = (x = -1);G: Parabola;Expression(G) = (y^2 = 4*x);P: Point;PointOnCurve(P, G);d1: Number;Distance(P, l1) = d1;d2: Number;Distance(P, l2) = d2",
+      "coords": {}
+    },
+    {
+      "index": 531,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0531.png",
+      "answer": "3",
+      "fact_expr": "G: Curve;H: Curve;Expression(G) = (y^2 = 2*x);Expression(H) = (y^2 + (x - 2)^2 = 4)",
+      "coords": {}
+    },
+    {
+      "index": 532,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0532.png",
+      "answer": "(3, 0)\ny^2 = 12*x",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);C: Parabola;F: Point;RightFocus(G) = F;O: Origin;Vertex(C) = O;Focus(C) = F",
+      "coords": {}
+    },
+    {
+      "index": 533,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 534,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 535,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 536,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.2135674953460693,
+        "b": 2.4270472526550293,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 4.453341261978494e-06,
+        "converged": true,
+        "iterations": 125
+      },
+      "output_path": "results/test/hyperbola/problem_0536.png",
+      "answer": "16",
+      "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (x^2/4 - y^2/m = 1);Eccentricity(G) = sqrt(5)",
+      "coords": {}
+    },
+    {
+      "index": 537,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0537.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "C: Ellipse;Gamma: Circle;P: Point;F1: Point;Q: Point;Expression(C) = (x^2/2 + y^2 = 1);LeftFocus(C)=F1;PointOnCurve(P, C);IsDiameter(LineSegmentOf(P,F1),Gamma);PointOnCurve(Q, Gamma);O:Origin",
+      "coords": {}
+    },
+    {
+      "index": 538,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 539,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0539.png",
+      "answer": "x^2/3-y^2=1",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;E: Circle;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(E) = (y^2 + (x - 2)^2 = 1);Center(E)=RightFocus(C);IsTangent(E, Asymptote(C))",
+      "coords": {}
+    },
+    {
+      "index": 540,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 541,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0541.png",
+      "answer": "sqrt(6)/2",
+      "fact_expr": "G: Hyperbola;b: Number;b>0;Expression(G) = (x^2/4 - y^2/b^2 = 1);Expression(OneOf(Asymptote(G))) = (y = x*(sqrt(2)/2))",
+      "coords": {}
+    },
+    {
+      "index": 542,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0542.png",
+      "answer": "[sqrt(5),sqrt(6)]",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;O: Origin;M: Point;N: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = 1 - x);Eccentricity(G)=e;Intersection(H,G)={M,N};Range(e)=[1/sqrt(3),1/sqrt(2)];IsPerpendicular(LineSegmentOf(O,M),LineSegmentOf(O,N));e:Number",
+      "coords": {}
+    },
+    {
+      "index": 543,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 544,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 545,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0545.png",
+      "answer": "12",
+      "fact_expr": "G: Hyperbola;b: Number;P: Point;F1: Point;F2: Point;b>0;Expression(G) = (x^2/9 - y^2/b^2 = 1);Focus(G)={F1,F2};PointOnCurve(P,G);Abs(LineSegmentOf(P, F1)) = 6",
+      "coords": {}
+    },
+    {
+      "index": 546,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0546.png",
+      "answer": "12",
+      "fact_expr": "G: Parabola;p: Number;H: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(H) = (2, 3);Distance(H, Focus(G)) = 5",
+      "coords": {
+        "H": [
+          2.0,
+          3.0
+        ]
+      }
+    },
+    {
+      "index": 547,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0547.png",
+      "answer": "5",
+      "fact_expr": "G:Hyperbola;a: Number;P: Point;F2: Point;F1: Point;Expression(G) = (-y^2/9 + x^2/a^2 = 1);PointOnCurve(P, RightPart(G));Expression(OneOf(Asymptote(G))) = (3*x - y = 0);LeftFocus(G) =F1;RightFocus(G)=F2;Abs(LineSegmentOf(P, F2)) = 3",
+      "coords": {}
+    },
+    {
+      "index": 548,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0548.png",
+      "answer": "5/3",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);F2: Point;F1: Point;LeftFocus(G) = F1;RightFocus(G) = F2;A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G) = True;PointOnCurve(F1, LineSegmentOf(A, B)) = True;Perimeter(InscribedCircle(TriangleOf(A, B, F2))) = pi;Coordinate(A) = (x1, y1);x1: Number;y1: Number;Coordinate(B) = (x2, y2);x2: Number;y2: Number",
+      "coords": {}
+    },
+    {
+      "index": 549,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 550,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0550.png",
+      "answer": "y = pm*sqrt(2)*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;H: Circle;Expression(H) = (x^2 + y^2 = a^2);L: Line;TangentOfPoint(F2,H) = L;Intersection(L,LeftPart(G)) = M;M: Point;AngleOf(F1, M, F2) = ApplyUnit(45, degree)",
+      "coords": {}
+    },
+    {
+      "index": 551,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0551.png",
+      "answer": "[3, 27/8]",
+      "fact_expr": "G: Ellipse;b: Number;H: Line;M: Point;N: Point;F2: Point;F1: Point;lambda: Number;Expression(G) = (x^2/4 + y^2/b^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2, H);Intersection(H, G) = {M, N};Min(Length(LineSegmentOf(M, N))) = 3;Abs(VectorOf(N, F2)) = lambda*Abs(VectorOf(F2, M));In(lambda, [1, 2]);IsChordOf(LineSegmentOf(M, N), G)",
+      "coords": {}
+    },
+    {
+      "index": 552,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0552.png",
+      "answer": "3*sqrt(2)/2",
+      "fact_expr": "G: Parabola;H: Line;A: Point;O: Origin;B: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, F)) = 3",
+      "coords": {}
+    },
+    {
+      "index": 553,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0553.png",
+      "answer": "1/9",
+      "fact_expr": "H: Parabola;Expression(H) = (y^2 = 2*p*x);M: Point;a: Real;m:Number;m>0;p: Number;p>0;Distance(M,Focus(H))=5;G: Hyperbola;Expression(G) = (-y^2 + x^2/a=1);A:Point;LeftVertex(G)=A;Coordinate(M) = (1, m);PointOnCurve(M, H);IsParallel(OneOf(Asymptote(G)),LineOf(A,M))",
+      "coords": {}
+    },
+    {
+      "index": 554,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.7320508075688772,
+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 3.0,
+        "y_coef": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0554.png",
+      "answer": "3/2",
+      "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/3 + y^2/2 = 1);PointOnCurve(P, G);Distance(P, LeftFocus(G)) = sqrt(3)/2",
+      "coords": {}
+    },
+    {
+      "index": 555,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 556,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          4.0,
+          1.0
+        ],
+        "radius": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/circle/problem_0556.png",
+      "answer": "x^2/12-y^2/4=1",
+      "fact_expr": "c: Hyperbola;Expression(Asymptote(c)) = (x + pm*sqrt(3)*y = 0);G: Circle;Expression(G) = (-2*y - 8*x + x^2 + y^2 + 16 = 0);PointOnCurve(RightFocus(c), G)",
+      "coords": {}
+    },
+    {
+      "index": 557,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "left"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0557.png",
+      "answer": "x^2/4+y^2/3=1",
+      "fact_expr": "G: Ellipse;O: Origin;Center(G) = O;Eccentricity(G) = 1/2;H: Parabola;Expression(H) = (y^2 = -4*x);l: Line;Directrix(H) = l;PointOnCurve(OneOf(Focus(G)), l)",
+      "coords": {}
+    },
+    {
+      "index": 558,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0558.png",
+      "answer": "1/4",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);F: Point;RightFocus(G) = F;l: Line;PointOnCurve(F, l);Q: Point;Intersection(l, MinorAxis(G)) = Q;M: Point;N: Point;Intersection(l, G) = {M, N};lambda: Number;mu: Number;VectorOf(F, M) = lambda*VectorOf(M, Q);VectorOf(F, N) = mu*VectorOf(N, Q)",
+      "coords": {}
+    },
+    {
+      "index": 559,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.4506710767745972,
+        "b": 2.0511727333068848,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 4.657696081267204e-06,
+        "converged": true,
+        "iterations": 116
+      },
+      "output_path": "results/test/hyperbola/problem_0559.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Expression(Asymptote(G)) = (y = pm*(sqrt(2)*x))",
+      "coords": {}
+    },
+    {
+      "index": 560,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0560.png",
+      "answer": "2",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(C) = F;P: Point;PointOnCurve(P, RightPart(C));O: Origin;Abs(LineSegmentOf(P, F)) = Abs(LineSegmentOf(O, F));Cos(AngleOf(O, F, P)) = 1/4",
+      "coords": {}
+    },
+    {
+      "index": 561,
+      "success": true,
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+        "y_coef": 3.0
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+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0561.png",
+      "answer": "6",
+      "fact_expr": "P: Point;G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);PointOnCurve(P, G) = True;H1: Circle;Expression(H1) = (y^2 + (x + 1)^2 = 1);H2: Circle;Expression(H2) = (y^2 + (x - 1)^2 = 1);Q: Point;R: Point;PointOnCurve(Q, H1) = True;PointOnCurve(R, H2) = True",
+      "coords": {}
+    },
+    {
+      "index": 562,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 563,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "iterations": 89
+      },
+      "output_path": "results/test/hyperbola/problem_0563.png",
+      "answer": "(pm*2*sqrt(2), 0)\ny = pm*(sqrt(15)/3)*x",
+      "fact_expr": "G: Hyperbola;a: Number;b: Number;H: Ellipse;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2/8 + y^2/5 = 1);Eccentricity(G) = 2*sqrt(6)/3;Vertex(G) = Focus(H)",
+      "coords": {}
+    },
+    {
+      "index": 564,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0564.png",
+      "answer": "[sqrt(6/3),1)",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;LeftVertex(C) = A1;RightVertex(C) = A2;A1: Point;A2: Point;PointOnCurve(P, C) = True;P: Point;AngleOf(A1, P, A2) = ApplyUnit(120, degree)",
+      "coords": {}
+    },
+    {
+      "index": 565,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.4142164151010692,
+        "b": 1.4142164151010692,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 2.2160979279094996e-06,
+        "converged": true,
+        "iterations": 95
+      },
+      "output_path": "results/test/hyperbola/problem_0565.png",
+      "answer": "(-4,0)+(0,4)",
+      "fact_expr": "G:Hyperbola;F1: Point;F2: Point;m:Number;P:Point;Coordinate(F1)=(-2,0);Coordinate(F2)=(2, 0);Distance(P,F1)-Distance(P,F2)=m;Locus(P)=G",
+      "coords": {
+        "F1": [
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+          0.0
+        ],
+        "F2": [
+          2.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 566,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0566.png",
+      "answer": "(0, 1)",
+      "fact_expr": "H: Ellipse;Expression(H) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);e1: Number;e2: Number;Eccentricity(H) = e1;Eccentricity(G) = e2",
+      "coords": {}
+    },
+    {
+      "index": 567,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 568,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0568.png",
+      "answer": "(16/5,-9/5)",
+      "fact_expr": "l: Line;G: Ellipse;P: Point;A: Point;B: Point;Expression(G) = (x^2/16 + y^2/9 = 1);Expression(l)=(x-y+1=0);Intersection(l,G)={A,B};PointOnCurve(P,G);WhenMax(Area(TriangleOf(P,A,B)))",
+      "coords": {}
+    },
+    {
+      "index": 569,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 5.0,
+        "major_axis": "y",
+        "x_coef": 25.0,
+        "y_coef": 75.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0569.png",
+      "answer": "25*sqrt(3)/3",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/75 = 1);P: Point;PointOnCurve(P, G);F1: Point;F2: Point;Focus(G) = {F1, F2};AngleOf(F1, P, F2) = ApplyUnit(60, degree)",
+      "coords": {}
+    },
+    {
+      "index": 570,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0570.png",
+      "answer": "4*sqrt(3)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1);H: Line;PointOnCurve(RightFocus(G), H);IsPerpendicular(H, xAxis);A: Point;B: Point;l1: Line;l2: Line;Asymptote(G) = {l1, l2};Intersection(H, l1) = A;Intersection(H, l2) = B",
+      "coords": {}
+    },
+    {
+      "index": 571,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0571.png",
+      "answer": "7",
+      "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;x1: Number;y1: Number;x2: Number;y2: Number;Expression(G) = (y = x^2/4);PointOnCurve(Focus(G), H);Intersection(H, G) = {A, B};Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);y1 + y2 = 5",
+      "coords": {}
+    },
+    {
+      "index": 572,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 573,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0573.png",
+      "answer": "5",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2/3 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;A: Point;PointOnCurve(A, C) = True;P: Point;PointOnCurve(P, OverlappingLine(LineSegmentOf(F1, A))) = True;DotProduct(VectorOf(A, F2) + VectorOf(A, P), VectorOf(F2, P)) = 0",
+      "coords": {}
+    },
+    {
+      "index": 574,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0574.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;O: Origin;A: Point;B:Point;P: Point;F:Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);l:Line;l1:Line;l2:Line;RightFocus(C)=F;PointOnCurve(F, l);IsPerpendicular(l,l1);FootPoint(l,l1)=B;Intersection(l,yAxis)=P;Intersection(l,l2)=A;Between(P,A,B)=True;Abs(LineSegmentOf(O, A)) = (5/3)*a;OneOf(Asymptote(C)) = l1;OneOf(Asymptote(C)) = l2;Negation(l1=l2)",
+      "coords": {}
+    },
+    {
+      "index": 575,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0575.png",
+      "answer": "2/3",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;G: Line;A:Point;B: Point;F: Point;e:Number;k:Number;a > b;b > 0;k>0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);RightFocus(C)=F;Slope(G)=k;LeftVertex(C)=A;Intersection(G,C)={A,B};Projection(B, xAxis) = F;Eccentricity(C) = e ;e= 1/3",
+      "coords": {}
+    },
+    {
+      "index": 576,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0576.png",
+      "answer": "(sqrt(2)-1)/4",
+      "fact_expr": "l: Line;G: Ellipse;b: Number;a: Number;A: Point;P: Point;O: Origin;c:Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftVertex(G)=A;Expression(l)=(x=2*c);PointOnCurve(P,l);AngleOf(A,P,O)=ApplyUnit(45,degree)",
+      "coords": {}
+    },
+    {
+      "index": 577,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0577.png",
+      "answer": "sqrt(30)/5",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;A: Point;RightVertex(C) = A;A1: Circle;Center(A1) = A;Radius(A1) = b;M: Point;N: Point;Intersection(A1, OneOf(Asymptote(C))) = {M, N};O: Origin;VectorOf(O, M) = (3/2)*VectorOf(O, N)",
+      "coords": {}
+    },
+    {
+      "index": 578,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0578.png",
+      "answer": "sqrt(21)/3",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;L: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2, L);IsPerpendicular(L, OneOf(Asymptote(G)));FootPoint(L, OneOf(Asymptote(G))) = P;Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 579,
+      "success": false,
+      "error": "Circle from diameter: missing point coordinates"
+    },
+    {
+      "index": 580,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0580.png",
+      "answer": "ApplyUnit(60,degree)",
+      "fact_expr": "l: Line;C: Parabola;p: Number;A: Point;F: Point;B: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};Quadrant(A)=1;Abs(LineSegmentOf(A, F)) = 3*Abs(LineSegmentOf(B, F))",
+      "coords": {}
+    },
+    {
+      "index": 581,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0581.png",
+      "answer": "5",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*x);F: Point;Focus(C) = F;P1: Point;P2: Point;PointOnCurve(P1, C) ;PointOnCurve(P2, C) ;YCoordinate(P1) = y1;YCoordinate(P2) = y2;y1^2 + y2^2 = 8;y1:Number;y2:Number",
+      "coords": {}
+    },
+    {
+      "index": 582,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0582.png",
+      "answer": "10*sqrt(5)",
+      "fact_expr": "G: Parabola;A: Point;O: Origin;B: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(A,G);PointOnCurve(B,G);Abs(LineSegmentOf(O, A)) = Abs(LineSegmentOf(O, B));Orthocenter(TriangleOf(A,O,B))=Focus(G)",
+      "coords": {}
+    },
+    {
+      "index": 583,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.0833333358168602,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0583.png",
+      "answer": "y=-1/12",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = y/3)",
+      "coords": {}
+    },
+    {
+      "index": 584,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0584.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "C1: Parabola;Expression(C1) = (y^2 = 2*p*x);p: Number;p>0;F: Point;Focus(C1) = F;C2: Hyperbola;Expression(C2) = (x^2/a^2 - y^2/b^2 = 1);a: Number;b: Number;a>0;b>0;OneOf(Intersection(C1, OneOf(Asymptote(C2)))) = A;A: Point;IsPerpendicular(LineSegmentOf(A, F), xAxis) = True",
+      "coords": {}
+    },
+    {
+      "index": 585,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0585.png",
+      "answer": "9*sqrt(3)",
+      "fact_expr": "G: Hyperbola;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/16 - y^2/9 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(60, degree)",
+      "coords": {}
+    },
+    {
+      "index": 586,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.0625,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0586.png",
+      "answer": "y=-1/16",
+      "fact_expr": "G: Parabola;Expression(G) = (y = 4*x^2)",
+      "coords": {}
+    },
+    {
+      "index": 587,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "left"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0587.png",
+      "answer": "(x+1)^2+y^2=7",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = -4*x);H: Circle;Focus(G) = Center(H);Radius(H) = sqrt(7)",
+      "coords": {}
+    },
+    {
+      "index": 588,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0588.png",
+      "answer": "(2 - sqrt(2))/2",
+      "fact_expr": "l: Line;G: Parabola;P: Point;Expression(G) = (y^2 = 4*x);Expression(l) = (x + y + sqrt(2) = 0);PointOnCurve(P, G)",
+      "coords": {}
+    },
+    {
+      "index": 589,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.4142135623730951,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0589.png",
+      "answer": "sqrt(6)/2",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2/4 - y^2/2 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 590,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0590.png",
+      "answer": "3*x+4*y-7=0",
+      "fact_expr": "G: Ellipse;H: LineSegment;P:Point;Expression(G) = (x^2/4 + y^2/3=1);MidPoint(H)=P;Coordinate(P) = (1,1);IsChordOf(H,G)",
+      "coords": {
+        "P": [
+          1.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 591,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0591.png",
+      "answer": "sqrt(13)/2",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;F: Point;P: Point;H: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F;l:Line;PointOnCurve(F, l);IsPerpendicular(l, OneOf(Asymptote(C)));FootPoint(l, OneOf(Asymptote(C)))=H;PointOnCurve(P, C);VectorOf(F, P) = 3*VectorOf(F, H)",
+      "coords": {}
+    },
+    {
+      "index": 592,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.366584062576294,
+        "b": 2.364173412322998,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.00017838670464698225,
+        "converged": true,
+        "iterations": 89
+      },
+      "output_path": "results/test/hyperbola/problem_0592.png",
+      "answer": "y=pm*(sqrt(3)*x)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/3 + x^2/a^2 = 1);a: Number;a>0;Eccentricity(G) = 2",
+      "coords": {}
+    },
+    {
+      "index": 593,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0593.png",
+      "answer": "5*x^2/4-5*y^2/16=1",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;F: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = 2*x - 4);RightFocus(G)=F;PointOnCurve(F, H);NumIntersection(H,G)=1",
+      "coords": {}
+    },
+    {
+      "index": 594,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 595,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0595.png",
+      "answer": "(4,pm*4)",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);D: Point;PointOnCurve(D, G) = True;Distance(D, Focus(G)) = 5",
+      "coords": {}
+    },
+    {
+      "index": 596,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0596.png",
+      "answer": "8",
+      "fact_expr": "l: Line;E: Parabola;C: Point;D: Point;A: Point;B: Point;Expression(E) = (y^2 = 4*x);PointOnCurve(Focus(E), l);Intersection(l, E) = {A, B};Intersection(TangentOnPoint(A,E), yAxis) = C;Intersection(TangentOnPoint(B,E),yAxis)=D",
+      "coords": {}
+    },
+    {
+      "index": 597,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0597.png",
+      "answer": "3*sqrt(2)-5",
+      "fact_expr": "C: Ellipse;E: Circle;M: Point;N: Point;F1:Point;F2:Point;LeftFocus(C)=F1;RightFocus(C)=F2;Expression(C) = (x^2/4+y^2/3=1);Expression(E)=((x-4)^2+(y-3)^2=1);PointOnCurve(M,C);PointOnCurve(N,E)",
+      "coords": {}
+    },
+    {
+      "index": 598,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0598.png",
+      "answer": "3*sqrt(2)/2",
+      "fact_expr": "G: Parabola;H: Line;P0: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (x - y + 4 = 0);PointOnCurve(P0, G)",
+      "coords": {}
+    },
+    {
+      "index": 599,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0599.png",
+      "answer": "sqrt(3) - 1",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;F: Point;LeftFocus(C) = F;O: Origin;l: Line;PointOnCurve(O, l);Intersection(l, C) = {A, B};A: Point;B: Point;2*Abs(LineSegmentOf(F, O)) = Abs(LineSegmentOf(A, B));AngleOf(B, A, F) = pi/6",
+      "coords": {}
+    },
+    {
+      "index": 600,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0600.png",
+      "answer": "2",
+      "fact_expr": "l: Line;C: Parabola;p: Number;A: Point;B: Point;p>0;Expression(C) = (y^2 = 2*(p*x));PointOnCurve(Focus(C), l);Intersection(l, C) = {A, B};Abs(LineSegmentOf(A, B)) = 4;Distance(MidPoint(LineSegmentOf(A, B)), yAxis) = 1",
+      "coords": {}
+    },
+    {
+      "index": 601,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 602,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0602.png",
+      "answer": "ApplyUnit(90, degree)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/4 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G) = True;Abs(LineSegmentOf(P, F1)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 603,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 604,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 605,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0605.png",
+      "answer": "y^2=20*x",
+      "fact_expr": "H: Parabola;Expression(H) = (y^2 = 4*m*x);m: Number;m>0;G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1);Distance(Focus(H),OneOf(Asymptote(G))) = 3",
+      "coords": {}
+    },
+    {
+      "index": 606,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 4.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0606.png",
+      "answer": "y=-4",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 16*y)",
+      "coords": {}
+    },
+    {
+      "index": 607,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0607.png",
+      "answer": "5/4",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;O: Origin;M: Point;F: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Length(RealAxis(C)) = 8;RightFocus(C)=F;PointOnCurve(M,OneOf(Asymptote(C)));IsPerpendicular(LineSegmentOf(O, M), LineSegmentOf(M, F));Area(TriangleOf(O,M,F)) = 6",
+      "coords": {}
+    },
+    {
+      "index": 608,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 609,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0609.png",
+      "answer": "28/3",
+      "fact_expr": "l: Line;G: Ellipse;B: Point;F2: Point;A: Point;F1: Point;Expression(G) = (x^2/9 + y^2/4 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, l);Intersection(l, G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 610,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0610.png",
+      "answer": "5",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;B: Point;F: Point;O: Origin;A: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F;PointOnCurve(O, G);Intersection(G, C) = {A, B};Abs(LineSegmentOf(O, A)) = Abs(LineSegmentOf(O, F));Abs(LineSegmentOf(O, F)) = 5;Abs(LineSegmentOf(B, F)) = 8",
+      "coords": {}
+    },
+    {
+      "index": 611,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0611.png",
+      "answer": "{sqrt(2),1}",
+      "fact_expr": "G: Hyperbola;H: Line;k: Number;Expression(G) = (x^2 - y^2 = 1);Expression(H) = (y = k*x + 1);NumIntersection(H, G) = 1",
+      "coords": {}
+    },
+    {
+      "index": 612,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0612.png",
+      "answer": "x^2/4+y^2=1",
+      "fact_expr": "E: Ellipse;b: Number;a: Number;A: Point;F: Point;a > b;b > 0;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (0,-2);Eccentricity(E) = sqrt(3)/2;Focus(E) = F;Slope(LineOf(A,F))=2*sqrt(3)/3",
+      "coords": {
+        "A": [
+          0.0,
+          -2.0
+        ]
+      }
+    },
+    {
+      "index": 613,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 614,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0614.png",
+      "answer": "4",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;P: Point;F: Point;A: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F;RightVertex(C) = A;PointOnCurve(P, C);Max(Abs(LineSegmentOf(P, F))) = 3;Min(Abs(LineSegmentOf(P, F))) = 1",
+      "coords": {}
+    },
+    {
+      "index": 615,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0615.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;p: Number;H: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(H) = (-2, 3);Distance(H, Focus(G)) = 5",
+      "coords": {
+        "H": [
+          -2.0,
+          3.0
+        ]
+      }
+    },
+    {
+      "index": 616,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 3.4641016151377544,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0616.png",
+      "answer": "3",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/12 = 1);F1: Point;F2: Point;LeftFocus(G)=F1;RightFocus(G)=F2;P: Point;PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(90, degree)",
+      "coords": {}
+    },
+    {
+      "index": 617,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 618,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 3.4641016151377544,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0618.png",
+      "answer": "x^2/4 - y^2/12=1",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Ellipse;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2/25 + y^2/9 = 1);Eccentricity(G) = 2;Focus(G) = Focus(H)",
+      "coords": {}
+    },
+    {
+      "index": 619,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0619.png",
+      "answer": "",
+      "fact_expr": "C: Parabola;p: Number;M: Point;N: Point;A: Point;F: Point;l: Line;p>0;Expression(C) = (y^2 = 2*p*x);Coordinate(A) = (0, 3);Focus(C) = F;Directrix(C) = l;PointOnCurve(M, C);PointOnCurve(M,l);IsPerpendicular(LineSegmentOf(M,N),l);Min(Abs(LineSegmentOf(M, A)) + Abs(LineSegmentOf(M, N))) = 2;FootPoint(LineSegmentOf(M,N),l)=N",
+      "coords": {
+        "A": [
+          0.0,
+          3.0
+        ]
+      }
+    },
+    {
+      "index": 620,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 621,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0621.png",
+      "answer": "8",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);l: Line;PointOnCurve(Focus(G), l);Slope(l) = 1;Intersection(l, G) = {A, B};A: Point;B: Point",
+      "coords": {}
+    },
+    {
+      "index": 622,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 623,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0623.png",
+      "answer": "4*sqrt(2)",
+      "fact_expr": "G: Parabola;H: Line;F: Point;B: Point;A: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, H);Inclination(H) = pi/4;Intersection(H, G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 624,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0624.png",
+      "answer": "[2, +oo)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(G) = F;H: Line;PointOnCurve(F, H);Inclination(H) = ApplyUnit(60, degree);NumIntersection(H, RightPart(G)) = 1",
+      "coords": {}
+    },
+    {
+      "index": 625,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0625.png",
+      "answer": "{y^2 = 8*x,y^2 = -16*x}",
+      "fact_expr": "G: Parabola;m: Number;H: Line;Expression(G) = (y^2 = m*x);Expression(H) = (x = 1);Distance(Directrix(G), H) = 3",
+      "coords": {}
+    },
+    {
+      "index": 626,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 627,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.23606797749979,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 5.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0627.png",
+      "answer": "x=2",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = m*x);m: Number;E: Ellipse;Expression(E) = (x^2/9 + y^2/5 = 1);Focus(C) = LeftFocus(E)",
+      "coords": {}
+    },
+    {
+      "index": 628,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 629,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 630,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.366584062576294,
+        "b": 2.364173412322998,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.00017838670464698225,
+        "converged": true,
+        "iterations": 89
+      },
+      "output_path": "results/test/hyperbola/problem_0630.png",
+      "answer": "12",
+      "fact_expr": "G: Hyperbola;m: Real;Expression(G) = (x^2/4 - y^2/m = 1);Eccentricity(G) = 2",
+      "coords": {}
+    },
+    {
+      "index": 631,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0631.png",
+      "answer": "8",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 8*x);F: Point;Focus(C) = F;K: Point;Intersection(Directrix(C), xAxis) = K;A: Point;PointOnCurve(A, C);Abs(LineSegmentOf(A, K)) = sqrt(2)*Abs(LineSegmentOf(A, F))",
+      "coords": {}
+    },
+    {
+      "index": 632,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0632.png",
+      "answer": "x=-1",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;H: Line;PointOnCurve(Focus(G), H);Slope(H) = 1;A: Point;B: Point;Intersection(H, G) = {A, B};YCoordinate(MidPoint(LineSegmentOf(A, B))) = 2",
+      "coords": {}
+    },
+    {
+      "index": 633,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0633.png",
+      "answer": "3",
+      "fact_expr": "C: Parabola;l: Line;A: Point;F: Point;B: Point;Expression(C) = (y^2 = 4*x);Focus(C)=F;PointOnCurve(F, l);Inclination(l)=ApplyUnit(60,degree);Intersection(l,C) = {A, B};Abs(LineSegmentOf(A, F)) > Abs(LineSegmentOf(B, F))",
+      "coords": {}
+    },
+    {
+      "index": 634,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.23606797749979,
+        "b": 2.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0634.png",
+      "answer": "4/3",
+      "fact_expr": "G: Hyperbola;M: Point;F1: Point;F2: Point;Expression(G) = (x^2/5 - y^2/4 = 1);Focus(G) = {F1,F2};PointOnCurve(M, G);IsPerpendicular(LineSegmentOf(M, F1), LineSegmentOf(M, F2))",
+      "coords": {}
+    },
+    {
+      "index": 635,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 636,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 637,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0637.png",
+      "answer": "8 - 4*sqrt(2)",
+      "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 8*x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = 16;Abs(LineSegmentOf(A, F)) < Abs(LineSegmentOf(B, F))",
+      "coords": {}
+    },
+    {
+      "index": 638,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0638.png",
+      "answer": "y=pm*(3/4)*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 639,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0639.png",
+      "answer": "12",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G);PointOnCurve(Center(G), LineSegmentOf(A, B));F1: Point;OneOf(Focus(G) )= F1",
+      "coords": {}
+    },
+    {
+      "index": 640,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0640.png",
+      "answer": "y=(15/8)*x+1",
+      "fact_expr": "C: Parabola;a: Number;P: Point;F: Point;a>0;Expression(C) = (x^2 = a*y);Coordinate(P) = (2*a, 4*a);PointOnCurve(P, C);Focus(C) = F;Distance(P, F) = 17",
+      "coords": {}
+    },
+    {
+      "index": 641,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 642,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 0.7071067811865476,
+        "b": 0.5773502691896257,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0642.png",
+      "answer": "y=pm*(sqrt(6)/3)*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (2*x^2 - 3*y^2 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 643,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 4.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0643.png",
+      "answer": "3*sqrt(5)",
+      "fact_expr": "C: Parabola;A: Point;P: Point;F: Point;l: Line;d: Number;Expression(C) = (y^2 = 16*x);Coordinate(A) = (1, 6);Focus(C) = F;PointOnCurve(P, C);Directrix(C) = l;Distance(P, l) = d",
+      "coords": {
+        "A": [
+          1.0,
+          6.0
+        ]
+      }
+    },
+    {
+      "index": 644,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 645,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0645.png",
+      "answer": "sqrt(3)-1",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Focus(G)={F1,F2};PointOnCurve(P, G);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0;Tan(AngleOf(P, F1, F2)) = sqrt(3)/3",
+      "coords": {}
+    },
+    {
+      "index": 646,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0646.png",
+      "answer": "2*sqrt(7)",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;A: Point;LeftVertex(C) = A;F: Point;RightFocus(C) = F;G: Circle;Diameter(G) = Length(RealAxis(C));l1: Line;l2: Line;Intersection(G,l1) = P;P: Point;Quadrant(P) = 2;IsParallel(LineSegmentOf(P,A) , l2) = True;O: Origin;Area(TriangleOf(O,P,A)) = sqrt(3);OneOf(Asymptote(C)) = l1;OneOf(Asymptote(C)) = l2;Negation(l1=l2)",
+      "coords": {}
+    },
+    {
+      "index": 647,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 648,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0648.png",
+      "answer": "3",
+      "fact_expr": "F: Point;G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;Focus(G) = F;PointOnCurve(F, H) = True;H: Line;OneOf(Intersection(H, G)) = A;A: Point;Intersection(H, Directrix(G)) = B;B: Point;VectorOf(B, A) = 2*VectorOf(A, F);Abs(VectorOf(B, A)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 649,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.3300262689590454,
+        "b": 2.30381441116333,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 6.658357051492203e-06,
+        "converged": true,
+        "iterations": 122
+      },
+      "output_path": "results/test/hyperbola/problem_0649.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(OneOf(Asymptote(G))) = (y = sqrt(3)*x)",
+      "coords": {}
+    },
+    {
+      "index": 650,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0650.png",
+      "answer": "-3/4",
+      "fact_expr": "l: Line;C: Ellipse;A: Point;B: Point;P: Point;Expression(C) = (x^2/4 + y^2/3 = 1);Coordinate(P) = (1, 1);PointOnCurve(P, l);Intersection(l, C) = {A, B};MidPoint(LineSegmentOf(A,B))=P;IsChordOf(LineSegmentOf(A,B),C)",
+      "coords": {
+        "P": [
+          1.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 651,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "vertical"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0651.png",
+      "answer": "(0, 4)",
+      "fact_expr": "H: Parabola;Expression(H) = (x^2 = 2*(p*y));p: Number;p>0;G: Hyperbola;Expression(G) = (-x^2/16 + y^2/9 = 1);PointOnCurve(OneOf(Vertex(G)), Directrix(H)) = True",
+      "coords": {}
+    },
+    {
+      "index": 652,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0652.png",
+      "answer": "3",
+      "fact_expr": "G: Parabola;p: Number;A: Point;N: Point;M: Point;F: Point;B: Point;l: Line;l1: Line;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(A) = ((7/2)*p, 0);Focus(G) = F;Directrix(G) = l;PointOnCurve(M, l1);IsPerpendicular(l, l1);FootPoint(l, l1) = N;Intersection(LineSegmentOf(A, N), LineSegmentOf(M, F)) = B;Abs(LineSegmentOf(A, F)) = 2*Abs(LineSegmentOf(M, F));Area(TriangleOf(A, B, M)) = (9*sqrt(2))/2;PointOnCurve(M, G)",
+      "coords": {}
+    },
+    {
+      "index": 653,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0653.png",
+      "answer": "y = -1/4",
+      "fact_expr": "G: Parabola;Expression(G) = (y = x^2)",
+      "coords": {}
+    },
+    {
+      "index": 654,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 655,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 656,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.1622776601683795,
+        "b": 2.449489742783178,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0656.png",
+      "answer": "y^2=-16*x",
+      "fact_expr": "C: Hyperbola;G: Parabola;O: Origin;Expression(C) = (x^2/10 - y^2/6 = 1);Vertex(G) = O;SymmetryAxis(G)=xAxis;Focus(G)=LeftFocus(C)",
+      "coords": {}
+    },
+    {
+      "index": 657,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0657.png",
+      "answer": "8",
+      "fact_expr": "G: Hyperbola;H: Line;M: Point;F2: Point;N: Point;F1: Point;Expression(G) = (x^2/4 - y^2/3 = 1);LeftFocus(G) =F1;RightFocus(G)=F2;PointOnCurve(F1, H);Intersection(H, LeftPart(G)) = {M, N}",
+      "coords": {}
+    },
+    {
+      "index": 658,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0658.png",
+      "answer": "1",
+      "fact_expr": "G: Hyperbola;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 - y^2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(90, degree)",
+      "coords": {}
+    },
+    {
+      "index": 659,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0659.png",
+      "answer": "2*sqrt(5)/5",
+      "fact_expr": "l: Line;C: Ellipse;b: Number;a: Number;A: Point;P: Point;Q: Point;O:Origin;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);RightVertex(C) = A;PointOnCurve(O, l);Intersection(l, C) = {P, Q};Abs(LineSegmentOf(P, Q)) = a;IsPerpendicular(LineSegmentOf(A, P), LineSegmentOf(P, Q))",
+      "coords": {}
+    },
+    {
+      "index": 660,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0660.png",
+      "answer": "2",
+      "fact_expr": "G: Ellipse;H: Circle;l: Line;m: Number;n: Number;J: Point;Expression(G) = (x^2/9 + y^2/4 = 1);Expression(H) = (x^2 + y^2 = 4);Expression(l) = (m*x + n*y = 4);Coordinate(J) = (m, n);NumIntersection(l, H)=0;PointOnCurve(J, L);L:Line",
+      "coords": {}
+    },
+    {
+      "index": 661,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0661.png",
+      "answer": "11*sqrt(5)/5",
+      "fact_expr": "G: Parabola;H: Line;P: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (x + 2*y - 12 = 0);PointOnCurve(P, G);d1: Number;d2: Number;Distance(P, Directrix(G)) = d1;Distance(P, H) = d2",
+      "coords": {}
+    },
+    {
+      "index": 662,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.75,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0662.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;A: Point;Expression(G) = (x^2 = 3*y);PointOnCurve(A,G);YCoordinate(A)=5/4",
+      "coords": {}
+    },
+    {
+      "index": 663,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 664,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 665,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 666,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0666.png",
+      "answer": "2",
+      "fact_expr": "C: Parabola;p: Number;G: Line;A: Point;B: Point;D: Point;F: Point;p>0;Expression(C) = (x^2 = 2*(p*y));Focus(C) = F;Intersection(Directrix(C), yAxis) = D;PointOnCurve(F, G);Intersection(G, C) = {A, B};IsPerpendicular(LineSegmentOf(A, B), LineSegmentOf(A, D));Abs(LineSegmentOf(B, F)) = Abs(LineSegmentOf(A, F)) + 4",
+      "coords": {}
+    },
+    {
+      "index": 667,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0667.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;F2: Point;A: Point;F1: Point;c:Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y =-(b/a)*x);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(A, RightPart(G));Abs(LineSegmentOf(A,F1))=(11/2)*a-c;HalfFocalLength(G)=c;IsParallel(LineOf(A,F2),H)",
+      "coords": {}
+    },
+    {
+      "index": 668,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0668.png",
+      "answer": "pm*2*sqrt(2)",
+      "fact_expr": "l: Line;C: Parabola;G: Line;A: Point;E: Point;B: Point;H: Point;F: Point;k: Number;Expression(C) = (y^2 = 8*x);Expression(G) = (x = -2);Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};L1:Line;L2:Line;PointOnCurve(A,L1);PointOnCurve(B,L2);IsPerpendicular(L1,G);IsPerpendicular(L2,G);FootPoint(L1,G)=E;FootPoint(L2,G)=H;Abs(LineSegmentOf(A, E)) = 2*Abs(LineSegmentOf(B, H));Slope(l) = k",
+      "coords": {}
+    },
+    {
+      "index": 669,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 670,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0670.png",
+      "answer": "(1,sqrt(2)]",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;c: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);HalfFocalLength(G) = c",
+      "coords": {}
+    },
+    {
+      "index": 671,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0671.png",
+      "answer": "32/3",
+      "fact_expr": "C: Hyperbola;G: Line;M: Point;F1: Point;F2: Point;Expression(C) = (x^2/9 - y^2/16 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(Focus(C),G);IsParallel(G,Asymptote(C));Intersection(G,C)=M",
+      "coords": {}
+    },
+    {
+      "index": 672,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0672.png",
+      "answer": "1/2",
+      "fact_expr": "C: Ellipse;a: Number;G: Point;Expression(C) = (y^2/3 + x^2/a^2 = 1);Coordinate(G) = (1, 0);OneOf(Focus(C)) = G",
+      "coords": {
+        "G": [
+          1.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 673,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          1.0,
+          -3.0
+        ],
+        "radius": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/circle/problem_0673.png",
+      "answer": "{y^2=9*x, x^2=(-1/3)*y}",
+      "fact_expr": "G: Parabola;SymmetryAxis(G) = axis;O: Origin;Vertex(G) = O;H: Circle;Expression(H) = (6*y - 2*x + x^2 + y^2 + 9 = 0);PointOnCurve(Center(H), G)",
+      "coords": {}
+    },
+    {
+      "index": 674,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0674.png",
+      "answer": "1+(3*sqrt(2))/4",
+      "fact_expr": "C: Ellipse;G: Line;A: Point;F: Point;B: Point;Expression(C) = (x^2/2 + y^2 = 1);LeftFocus(C) = F;PointOnCurve(F, G);Intersection(G, C) = {A, B};k: Number;Abs(LineSegmentOf(A,F))+2*Abs(LineSegmentOf(B,F)) >= k",
+      "coords": {}
+    },
+    {
+      "index": 675,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 676,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0676.png",
+      "answer": "-4",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p > 0;PointOnCurve(Focus(G), l) = True;l: Line;Intersection(l, G) = {A, B};A: Point;Coordinate(A) = (x1, y1);x1: Number;y1: Number;B: Point;Coordinate(B) = (x2, y2);x2: Number;y2: Number",
+      "coords": {}
+    },
+    {
+      "index": 677,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 678,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0678.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;F1: Point;F2: Point;Focus(G) = {F1,F2};c: Number;FocalLength(G) = 2*c;P: Point;PointOnCurve(P, G) = True;M: Point;H: Circle;Expression(H) = (x^2 + y^2 = a^2);TangentPoint(LineOf(P,F1),H) = M;MidPoint(LineSegmentOf(P,F1)) = M",
+      "coords": {}
+    },
+    {
+      "index": 679,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 680,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_ecc_mismatch",
+      "params": {
+        "a": 1.9257268905639648,
+        "b": 1.7993911322594118,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 2.5576271215184017e-15,
+        "converged": true,
+        "iterations": 2
+      },
+      "output_path": "results/test/ellipse/problem_0680.png",
+      "answer": "{9/4,4}",
+      "fact_expr": "G: Ellipse;m: Real;Expression(G) = (y^2/3 + x^2/m = 1);Eccentricity(G) = 1/2",
+      "coords": {},
+      "validation_reasons": [
+        "ellipse_ecc_mismatch"
+      ]
+    },
+    {
+      "index": 681,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 682,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 683,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.8284271247461903,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 8.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0683.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "C: Ellipse;E: Circle;N: Point;M: Point;Expression(C) = (x^2/9 + y^2/8 = 1);Expression(E) = (y^2 + (x - 1)^2 = 1);PointOnCurve(M, C);PointOnCurve(N, E);IsTangent(LineOf(M,N),E)",
+      "coords": {}
+    },
+    {
+      "index": 684,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0684.png",
+      "answer": "[1, +oo)",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;Coordinate(Focus(G)) = (1, 0);H: Point;PointOnCurve(H, G)",
+      "coords": {}
+    },
+    {
+      "index": 685,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0685.png",
+      "answer": "y=pm*2*x",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;I: Line;k: Number;A: Point;B: Point;F:Point;a>0;b>0;Negation(k=0);Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(I) = (y = k*x);Intersection(I, G) = {A, B};IsDiameter(LineSegmentOf(A,B),H);PointOnCurve(F,H);RightFocus(G)=F;Area(TriangleOf(A,B,F))=4*a^2",
+      "coords": {}
+    },
+    {
+      "index": 686,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.7320508075688772,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 3.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0686.png",
+      "answer": "4*sqrt(3)",
+      "fact_expr": "E: Ellipse;Expression(E) = (x^2/3 + y^2 = 1);F1: Point;Focus(E) = {F1, F2};l: Line;PointOnCurve(F1, l);A: Point;B: Point;Intersection(l, E) = {A, B};F2: Point",
+      "coords": {}
+    },
+    {
+      "index": 687,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0687.png",
+      "answer": "sqrt(6)/2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;k: Number;O: Origin;M: Point;F1:Point;F2: Point;N: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);k>0;Expression(H) = (y = k*x);LeftFocus(G) =F1;RightFocus(G)=F2;Intersection(H, G) = {M, N};Abs(LineSegmentOf(O, M)) = Abs(LineSegmentOf(O, F2));Area(TriangleOf(M,F2,N))=(Abs(LineSegmentOf(M,F2))+Abs(LineSegmentOf(N,F2)))^2/16",
+      "coords": {}
+    },
+    {
+      "index": 688,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0688.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;P: Point;PointOnCurve(P, G);Q: Point;Coordinate(Q) = (9*p/2, 0);Abs(LineSegmentOf(Q, F)) = 2*Abs(LineSegmentOf(P, F));Area(TriangleOf(P, Q, F)) = 8*sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 689,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 4.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0689.png",
+      "answer": "6",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 16*x);F: Point;Focus(C) = F;l: Line;Directrix(C) = l;P: Point;PointOnCurve(P, l);Q: Point;OneOf(Intersection(LineOf(P, F), C)) = Q;VectorOf(F, P) = 4*VectorOf(F, Q)",
+      "coords": {}
+    },
+    {
+      "index": 690,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 691,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "left"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0691.png",
+      "answer": "(-2, 0)",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = -8*x)",
+      "coords": {}
+    },
+    {
+      "index": 692,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.7320508075688772,
+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 3.0,
+        "y_coef": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0692.png",
+      "answer": "(3/5,-2/5)",
+      "fact_expr": "H: Line;Expression(H) = (x - y = 1);G: Ellipse;Expression(G) = (x^2/3 + y^2/2 = 1);Intersection(H,G) = {A,B};A: Point;B: Point",
+      "coords": {}
+    },
+    {
+      "index": 693,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 694,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0694.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;P: Point;Expression(G) = (y^2 = 4*x);Coordinate(P) = (3, y1);y1:Number;PointOnCurve(P,G)",
+      "coords": {}
+    },
+    {
+      "index": 695,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0695.png",
+      "answer": "25",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);P: Point;PointOnCurve(P, G) = True;Focus(G) = {F1,F2};F1: Point;F2: Point;Distance(P,F1) * Distance(P,F2) = m;m:Number",
+      "coords": {}
+    },
+    {
+      "index": 696,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0696.png",
+      "answer": "sqrt(5)+4",
+      "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;P: Point;F: Point;M:Number;N:Number;Expression(G) = (y^2 = 4*x);Expression(H) = (x = -1);Coordinate(A) = (-1, 1);Coordinate(B) = (3, 2);PointOnCurve(P, G);Focus(G) = F;Min(Distance(P,A)+Distance(P,H))=M;Min(Abs(LineSegmentOf(P, B)) + Abs(LineSegmentOf(P, F))) = N",
+      "coords": {
+        "A": [
+          -1.0,
+          1.0
+        ],
+        "B": [
+          3.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 697,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 36.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0697.png",
+      "answer": "12+6*sqrt(3)",
+      "fact_expr": "C: Ellipse;A: Point;F1: Point;F2: Point;Expression(C) = (x^2/36 + y^2/9 = 1);LeftFocus(C)=F1;RightFocus(C)=F2;PointOnCurve(A, C)",
+      "coords": {}
+    },
+    {
+      "index": 698,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
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+          0.0
+        ],
+        "radius": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/circle/problem_0698.png",
+      "answer": "{(y^2=3*x), (y^2=-3*x)}",
+      "fact_expr": "G: Parabola;O: Origin;H: Circle;Vertex(G) = O;SymmetryAxis(G) = xAxis;Expression(H) = (x^2 + y^2 = 4);Length(InterceptChord(G, H)) = 2*sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 699,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
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+          -0.0
+        ],
+        "radius": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/circle/problem_0699.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "G: Hyperbola;H: Circle;a: Number;a>0;Expression(G) = (-y^2 + x^2/a^2 = 1);Expression(H) = (-4*x + x^2 + y^2 = 0);RightFocus(G) = Center(H)",
+      "coords": {}
+    },
+    {
+      "index": 700,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0700.png",
+      "answer": "1/2",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;B: Point;A: Point;F: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (-a, 0);Coordinate(B) = (0, b);LeftFocus(G) = F;In(A, Vertex(G));In(B, Vertex(G));Distance(F, LineOf(A, B)) = b/sqrt(7)",
+      "coords": {}
+    },
+    {
+      "index": 701,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 702,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 703,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 704,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0704.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "H: Parabola;Expression(H) = (x^2 = 2*p*y);p: Number;p>0;G: Hyperbola;Expression(G) = (-x^2 + y^2 = 1);PointOnCurve(OneOf(Focus(G)), Directrix(H)) = True",
+      "coords": {}
+    },
+    {
+      "index": 705,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 10.0,
+        "b": 8.0,
+        "major_axis": "x",
+        "x_coef": 100.0,
+        "y_coef": 64.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0705.png",
+      "answer": "64*sqrt(3)/3",
+      "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/100 + y^2/64 = 1);PointOnCurve(P, G);Focus(G)={F1,F2};AngleOf(F1, P, F2) = ApplyUnit(60, degree)",
+      "coords": {}
+    },
+    {
+      "index": 706,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0706.png",
+      "answer": "sqrt(3)/4",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = a*x);a: Number;P: Point;Coordinate(P) = (3/2, y0);y0: Number;Focus(G) = F;F: Point;Distance(P, F) = 2;O: Origin;PointOnCurve(P, G) = True",
+      "coords": {}
+    },
+    {
+      "index": 707,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0707.png",
+      "answer": "(3,2*sqrt(3))",
+      "fact_expr": "m: Line;G: Parabola;n: Line;P: Point;M: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;Quadrant(P) = 1;TangentOnPoint(P, G) = m;PointOnCurve(P,n);IsParallel(xAxis, n);C:Line;PointOnCurve(F,C);IsParallel(C,m);Intersection(C,n)=M;Abs(LineSegmentOf(P,M))=4",
+      "coords": {}
+    },
+    {
+      "index": 708,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 709,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 2.8284271247461903,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 8.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0709.png",
+      "answer": "8*sqrt(3)/3",
+      "fact_expr": "C: Ellipse;F1: Point;P: Point;F2: Point;Expression(C) = (x^2/16+y^2/8=1);Focus(C)={F1,F2};PointOnCurve(P, C);AngleOf(F1, P, F2) = pi/3",
+      "coords": {}
+    },
+    {
+      "index": 710,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.4142135623730951,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0710.png",
+      "answer": "-1/3",
+      "fact_expr": "G: Hyperbola;H: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2 - y^2/2 = 1);Expression(H) = (x^2/4 + y^2 = 1);LeftFocus(H) = F1;RightFocus(H) = F2;OneOf(Intersection(H, G)) = P",
+      "coords": {}
+    },
+    {
+      "index": 711,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0711.png",
+      "answer": "(0,2)",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 8*y)",
+      "coords": {}
+    },
+    {
+      "index": 712,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0712.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "C: Hyperbola;a: Number;b: Number;c:Number;G: Circle;H: Line;F1: Point;O: Origin;N: Point;M: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2 + y^2 = a^2);Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);LeftFocus(C)=F1;RightFocus(C) = F2;PointOnCurve(F1, H);TangentPoint(H, G) = M;Intersection(H,RightPart(C))=N;Abs(LineSegmentOf(O, N)) = c;F2:Point;c>0",
+      "coords": {}
+    },
+    {
+      "index": 713,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0713.png",
+      "answer": "18",
+      "fact_expr": "G: Parabola;H: Line;F: Point;A: Point;B: Point;Expression(G) = (y^2 = 8*x);Focus(G) = F;PointOnCurve(F,H);Intersection(H,G)={A,B}",
+      "coords": {}
+    },
+    {
+      "index": 714,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0714.png",
+      "answer": "4/3",
+      "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Focus(G)={F1,F2};PointOnCurve(P, G);Cos(AngleOf(F1, P, F2)) = 1/2",
+      "coords": {}
+    },
+    {
+      "index": 715,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0715.png",
+      "answer": "[sqrt(3)/3,1)",
+      "fact_expr": "C: Ellipse;P: Point;F1: Point;F2: Point;I: Point;O: Origin;Expression(C) = (x^2/4 + y^2/3 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);Incenter(TriangleOf(P, F1, F2)) = I",
+      "coords": {}
+    },
+    {
+      "index": 716,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.23606797749979,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0716.png",
+      "answer": "{x^2/10-y^2/6=1, x^2/10-y^2/6=-1}",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/5 - y^2/3 = 1);Z: Hyperbola;Asymptote(G) = Asymptote(Z);FocalLength(Z) = 8",
+      "coords": {}
+    },
+    {
+      "index": 717,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0717.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;l1: Line;l2: Line;Asymptote(G) = {l1, l2};P: Point;Quadrant(P) = 1;PointOnCurve(P, l1);IsPerpendicular(l2, LineSegmentOf(P, F1));IsParallel(l2, LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 718,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0718.png",
+      "answer": "[sqrt(5),oo)",
+      "fact_expr": "l: Line;G: Hyperbola;b: Number;a: Number;F: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F;Slope(l)=2;PointOnCurve(F,l);NumIntersection(l, RightPart(G)) = 1",
+      "coords": {}
+    },
+    {
+      "index": 719,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0719.png",
+      "answer": "1/2",
+      "fact_expr": "G: Parabola;H: Line;P: Point;Expression(G) = (y = x^2);Expression(H) = (x + y + 2 = 0);PointOnCurve(P, G);WhenMin(Distance(P, H))",
+      "coords": {}
+    },
+    {
+      "index": 720,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 721,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 2.0,
+        "major_axis": "y",
+        "x_coef": 4.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0721.png",
+      "answer": "[sqrt(2),sqrt(21)/3)",
+      "fact_expr": "G: Hyperbola;H: Ellipse;D:Ellipse;a: Number;a<=1;Expression(H) = (x^2/4 + y^2/16 = 1);Expression(D) = (y^2/4 + (a*x^2)/16 = 1);PointOnCurve(Focus(G),xAxis);PointOnCurve(Intersection(D,H),Asymptote(G))",
+      "coords": {}
+    },
+    {
+      "index": 722,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "y",
+        "x_coef": 1.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0722.png",
+      "answer": "y^2=4*x",
+      "fact_expr": "G: Parabola;H: Ellipse;Expression(H) = (x^2 + y^2/5 = 1);Vertex(G) = Center(H);Focus(G) = RightVertex(H)",
+      "coords": {}
+    },
+    {
+      "index": 723,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0723.png",
+      "answer": "2/3",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>b;b>0;F2: Point;F1: Point;LeftFocus(G) = F1;RightFocus(G) = F2;L: Line;x1: Number;x2: Number;y1: Number;y2: Number;PointOnCurve(F1,L) = True;Intersection(L,G) = {A,B};Coordinate(A) = (x1,y1);Coordinate(B) = (x2,y2);Radius(InscribedCircle(TriangleOf(A,B,F2))) = 1;Abs(LineSegmentOf(F1,F2)) = 2;Abs(y1-y2) = 3;A: Point;B: Point",
+      "coords": {}
+    },
+    {
+      "index": 724,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0724.png",
+      "answer": "x+4*y-5=0",
+      "fact_expr": "G: Ellipse;A: Point;B: Point;M: Point;Expression(G) = (x^2/16 + y^2/4 = 1);Coordinate(M) = (1, 1);IsChordOf(LineSegmentOf(A,B),G);MidPoint(LineSegmentOf(A,B))=M",
+      "coords": {
+        "M": [
+          1.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 725,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0725.png",
+      "answer": "6",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, C) = {A, B};Negation(A=B);M: Point;Intersection(Directrix(C), xAxis) = M;Tan(AngleOf(A, M, B)) = 2*sqrt(6)",
+      "coords": {}
+    },
+    {
+      "index": 726,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0726.png",
+      "answer": "4",
+      "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/25 + y^2/16 = 1);PointOnCurve(P, G);F1:Point;F2:Point;OneOf(Focus(G))=F1;OneOf(Focus(G))=F2;Negation(F1=F2);Distance(P, F1) = 6",
+      "coords": {}
+    },
+    {
+      "index": 727,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0727.png",
+      "answer": "5",
+      "fact_expr": "G: Parabola;P: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(P, G);Distance(P, yAxis) = 4",
+      "coords": {}
+    },
+    {
+      "index": 728,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_focus_mismatch",
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0728.png",
+      "answer": "3/2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/5 + x^2/a^2 = 1);a: Number;H: Point;Coordinate(H) = (3, 0);RightFocus(G) = H",
+      "coords": {
+        "H": [
+          3.0,
+          0.0
+        ]
+      },
+      "validation_reasons": [
+        "hyperbola_focus_mismatch"
+      ]
+    },
+    {
+      "index": 729,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0729.png",
+      "answer": "8",
+      "fact_expr": "G: Parabola;H: Curve;k: Number;M: Point;F: Point;Expression(G) = (y^2 = 8*x);k>0;Expression(H) = (x*y = k);Intersection(G, H) = M;Focus(G) = F;Distance(M, F) = 4",
+      "coords": {}
+    },
+    {
+      "index": 730,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 6.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0730.png",
+      "answer": "y=pm*3*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/36 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 731,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 732,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 733,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 734,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0734.png",
+      "answer": "[sqrt(3)/3,sqrt(2)/2]",
+      "fact_expr": "M: Ellipse;b: Number;a: Number;c:Number;P: Point;F1: Point;F2: Point;e: Number;a > b;b > 0;Expression(M) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(M)=F1;RightFocus(M)=F2;PointOnCurve(P,M);Range(Abs(LineSegmentOf(P,F1))*Abs(LineSegmentOf(P,F2)))=[2*c^2,3*c^2];c=sqrt(a^2-b^2);Eccentricity(M)=e",
+      "coords": {}
+    },
+    {
+      "index": 735,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 736,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.463192105293274,
+        "b": 2.070901870727539,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 1.4661259228887502e-06,
+        "converged": true,
+        "iterations": 106
+      },
+      "output_path": "results/test/hyperbola/problem_0736.png",
+      "answer": "y=pm*sqrt(2)*x",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 4*x);Eccentricity(G) = sqrt(3);OneOf(Directrix(G))=Directrix(H)",
+      "coords": {}
+    },
+    {
+      "index": 737,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0737.png",
+      "answer": "(1,sqrt(2))",
+      "fact_expr": "E: Hyperbola;b: Number;a: Number;P: Point;F2: Point;F1: Point;a>0;b>0;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(E) = F1;RightFocus(E) = F2;PointOnCurve(P,Asymptote(E));Abs(Abs(LineSegmentOf(P, F1)) - Abs(LineSegmentOf(P, F2))) = 2*b",
+      "coords": {}
+    },
+    {
+      "index": 738,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0738.png",
+      "answer": "8",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);H: Circle;Expression(H) = (m + 6*x + x^2 + y^2 = 0);m: Real;IsTangent(Directrix(G), H)",
+      "coords": {}
+    },
+    {
+      "index": 739,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0739.png",
+      "answer": "8",
+      "fact_expr": "M: Point;Coordinate(M) = (sqrt(3), 0);G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);H: Line;Expression(H) = (y = k*(x + sqrt(3)));A: Point;B: Point;Intersection(G,H) = {A,B};k:Number",
+      "coords": {
+        "M": [
+          1.7320508075688772,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 740,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0740.png",
+      "answer": "7",
+      "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/25 + y^2/16 = 1);PointOnCurve(P, G);F1: Point;F2: Point;OneOf(Focus(G)) = F1;OneOf(Focus(G)) = F2;Negation(F1 = F2);Distance(P, F1) = 3",
+      "coords": {}
+    },
+    {
+      "index": 741,
+      "success": false,
+      "conic_type": "parabola",
+      "error": "validation: parabola_point_off_curve",
+      "params": {
+        "p": 0.9927864074707031,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.00015326803259085864,
+        "converged": true,
+        "iterations": 44
+      },
+      "output_path": "results/test/parabola/problem_0741.png",
+      "answer": "5",
+      "fact_expr": "G: Parabola;p: Number;P: Point;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(P) = (4, 4);PointOnCurve(P, G);Focus(G) = F",
+      "coords": {
+        "P": [
+          4.0,
+          4.0
+        ]
+      },
+      "validation_reasons": [
+        "parabola_point_off_curve"
+      ]
+    },
+    {
+      "index": 742,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 743,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0743.png",
+      "answer": "9/4",
+      "fact_expr": "G: Parabola;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(A, G);PointOnCurve(B, G);VectorOf(A, F) = 2*(VectorOf(F, B));IsChordOf(LineSegmentOf(A, B), G)",
+      "coords": {}
+    },
+    {
+      "index": 744,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 2.6457513110645907,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 7.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0744.png",
+      "answer": "2*sqrt(3)/3",
+      "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/16 + y^2/7 = 1);Focus(G) = Endpoint(MajorAxis(H));PointOnCurve(Focus(H),Directrix(G))",
+      "coords": {}
+    },
+    {
+      "index": 745,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 746,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0746.png",
+      "answer": "y = pm*(2/3)*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/4 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 747,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0747.png",
+      "answer": "4*sqrt(3)/3",
+      "fact_expr": "r1: Hyperbola;r2:Hyperbola;O: Origin;A: Point;B: Point;x1:Number;x2:Number;y1:Number;y2:Number;a:Number;b:Number;a>0 ;b>0;Expression(r1) = (x^2/a^2-y^2/b^2=1);Expression(r2) = (y^2/b^2-x^2/a^2=1);PointOnCurve(A,r1);PointOnCurve(B,r2);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);x1^2 - x2^2 = a^2;Slope(LineOf(O, A))*Slope(LineOf(O,B)) = 1/3",
+      "coords": {}
+    },
+    {
+      "index": 748,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.2135674953460693,
+        "b": 2.4270472526550293,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 4.453341261978494e-06,
+        "converged": true,
+        "iterations": 125
+      },
+      "output_path": "results/test/hyperbola/problem_0748.png",
+      "answer": "2*x+pm*y=0",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = sqrt(5)",
+      "coords": {}
+    },
+    {
+      "index": 749,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.8284271247461903,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 8.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0749.png",
+      "answer": "(x^2+(y-1)^2=3/4)&(x>0,y>0)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/8 + y^2/4 = 1);A: Point;UpperVertex(G) = A;F: Point;LeftFocus(G) = F;C: Point;MidPoint(LineSegmentOf(O, A)) = C;O: Origin;P: Point;Quadrant(P) = 1;Abs(LineSegmentOf(C, P)) = (sqrt(3)/2)*Abs(LineSegmentOf(O, C))",
+      "coords": {}
+    },
+    {
+      "index": 750,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0750.png",
+      "answer": "y=pm*(sqrt(3)/3)*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;A1: Point;A2: Point;LeftVertex(G) = A1;RightVertex(G) = A2;P: Point;PointOnCurve(P, LeftPart(G));Slope(LineOf(P, A1)) = -1;Slope(LineOf(P, A2)) = -1/3",
+      "coords": {}
+    },
+    {
+      "index": 751,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0751.png",
+      "answer": "2*sqrt(5)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Length(RealAxis(G))=2;Length(ImageinaryAxis(G)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 752,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0752.png",
+      "answer": "y = pm*2*x",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;B: Point;P: Point;F: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;OneOf(Endpoint(ImageinaryAxis(C))) = B;PointOnCurve(P, LeftPart(C));Min(Perimeter(TriangleOf(B, P, F))) = 4*Length(RealAxis(C))",
+      "coords": {}
+    },
+    {
+      "index": 753,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0753.png",
+      "answer": "20",
+      "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;Expression(G) = (x^2 = 4*y);Slope(H)=2;PointOnCurve(Focus(G),H);Intersection(H,G)={A,B}",
+      "coords": {}
+    },
+    {
+      "index": 754,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0754.png",
+      "answer": "2",
+      "fact_expr": "M: Point;PointOnCurve(M, G) = True;G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;P: Point;MidPoint(LineSegmentOf(M,F)) = P;Distance(P, yAxis) = 2",
+      "coords": {}
+    },
+    {
+      "index": 755,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0755.png",
+      "answer": "3",
+      "fact_expr": "G: Parabola;P: Point;A: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(P) = (x1, y1);Coordinate(A) = (2, 2);Focus(G) = F;x1:Number;y1:Number;PointOnCurve(P, G)",
+      "coords": {
+        "A": [
+          2.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 756,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          -4.0,
+          0.0
+        ],
+        "radius": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/circle/problem_0756.png",
+      "answer": "x=13/8",
+      "fact_expr": "O1: Circle;Expression(O1) = (x^2 + y^2 = 1);O2: Circle;Expression(O2) = (y^2 + (x - 4)^2 = 4);P: Point;Coordinate(P) = (x1, y1);x1: Number;y1: Number;TangentOfPoint(P, O1) = LineOf(P, A);TangentOfPoint(P, O2) = LineOf(P, B);TangentPoint(LineOf(P, A), O1) = A;TangentPoint(LineOf(P, B), O2) = B;A: Point;B: Point;Abs(LineSegmentOf(P, A)) = Abs(LineSegmentOf(P, B))",
+      "coords": {}
+    },
+    {
+      "index": 757,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0757.png",
+      "answer": "2/7",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);P: Point;m: Number;n: Number;Coordinate(P) = (m, n);PointOnCurve(P, G);Distance(P, Focus(G)) = 8*m",
+      "coords": {}
+    },
+    {
+      "index": 758,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.4142135623730951,
+        "major_axis": "x",
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+        "y_coef": 2.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0758.png",
+      "answer": "4*sqrt(5)/3",
+      "fact_expr": "H: Line;Expression(H) = (y = x - 1);G: Ellipse;Expression(G) = (x^2/4 + y^2/2 = 1);A: Point;B: Point;Intersection(H, G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 759,
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+      "conic_type": "hyperbola",
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0759.png",
+      "answer": "(0, sqrt(3))",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;e: Number;Eccentricity(G) = e;In(e, (1, 2))",
+      "coords": {}
+    },
+    {
+      "index": 760,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0760.png",
+      "answer": "2*sqrt(3)/3",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;A: Point;Q: Point;H:Circle;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightVertex(C) = A;Center(H)=A;Radius(H)=sqrt(2)*a;Intersection(H,RightPart(C))={P,Q};AngleOf(P, A, Q) = pi/2",
+      "coords": {}
+    },
+    {
+      "index": 761,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0761.png",
+      "answer": "8",
+      "fact_expr": "l: Line;C: Parabola;A: Point;F: Point;M: Point;B: Point;Expression(C) = (y^2 = 6*x);Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};Intersection(Directrix(C), xAxis) = M;Abs(LineSegmentOf(A, F)) = 2*Abs(LineSegmentOf(M, F))",
+      "coords": {}
+    },
+    {
+      "index": 762,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 763,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0763.png",
+      "answer": "(2,0)",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x)",
+      "coords": {}
+    },
+    {
+      "index": 764,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0764.png",
+      "answer": "1",
+      "fact_expr": "G: Hyperbola;a: Number;H: Ellipse;Expression(G) = (-y^2/2 + x^2/a = 1);Expression(H) = (x^2/4 + y^2/a^2 = 1);Focus(G)=Focus(H)",
+      "coords": {}
+    },
+    {
+      "index": 765,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.4142135623730951,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0765.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Hyperbola;B: Point;A: Point;O: Origin;Expression(G) = (x^2 - y^2/2 = 1);PointOnCurve(A, G);PointOnCurve(B, G);DotProduct(VectorOf(O, A), VectorOf(O, B)) = 0",
+      "coords": {}
+    },
+    {
+      "index": 766,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 2.5275303393962245,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 4.481942266905428e-06,
+        "converged": true,
+        "iterations": 95
+      },
+      "output_path": "results/test/hyperbola/problem_0766.png",
+      "answer": "y^2/4-x^2=1",
+      "fact_expr": "E: Hyperbola;Expression(Asymptote(E)) = (y = pm*2*x);Coordinate(Focus(E)) = (0, pm*sqrt(5))",
+      "coords": {}
+    },
+    {
+      "index": 767,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 1.4142135623730951,
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0767.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 2)",
+      "coords": {}
+    },
+    {
+      "index": 768,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "vertical"
+      },
+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0768.png",
+      "answer": "sqrt(5)/2",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-x^2/b^2 + y^2/a^2 = 1);b: Number;a: Number;a>0;b>0;P: Point;PointOnCurve(P, xAxis) = True;G: Line;Expression(G) = (y = m + 2*x);m: Number;Negation(m = 0);NumIntersection(G, C) = 1;PointOnCurve(P, G) = True",
+      "coords": {}
+    },
+    {
+      "index": 769,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
+      },
+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0769.png",
+      "answer": "6",
+      "fact_expr": "C: Parabola;G: Line;P: Point;B: Point;A: Point;F: Point;Expression(C) = (y^2 = 4*x);Coordinate(P) = (1, 0);Focus(C) = F;PointOnCurve(P, G);Intersection(G, C) = {A, B};2*VectorOf(B, P) = VectorOf(P, A)",
+      "coords": {
+        "P": [
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+          0.0
+        ]
+      }
+    },
+    {
+      "index": 770,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0770.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 771,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0771.png",
+      "answer": "(1+sqrt(5))/2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;A: Point;B: Point;F1: Point;F2: Point;a>0;b>0;l:Line;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Focus(G) = {F1,F2};PointOnCurve(F2,l);IsPerpendicular(l,xAxis);Intersection(l,G)={A,B};Radius(InscribedCircle(TriangleOf(A,B,F1)))=a",
+      "coords": {}
+    },
+    {
+      "index": 772,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 773,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 3.4641016151377544,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 12.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0773.png",
+      "answer": "10",
+      "fact_expr": "F: Point;LeftFocus(E) = F;E: Ellipse;Expression(E) = (x^2/16 + y^2/12 = 1);P: Point;PointOnCurve(P, E) = True;A: Point;Coordinate(A) = (1, sqrt(3))",
+      "coords": {
+        "A": [
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+          1.7320508075688772
+        ]
+      }
+    },
+    {
+      "index": 774,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0774.png",
+      "answer": "2",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 775,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_point_off_curve",
+      "params": {
+        "a": 2.2628633975982666,
+        "b": 1.8162860737044515,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 2.6997004191253788e-06,
+        "converged": true,
+        "iterations": 101
+      },
+      "output_path": "results/test/ellipse/problem_0775.png",
+      "answer": "",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);A: Point;Coordinate(A)=(1, 2);PointOnCurve(A, C)",
+      "coords": {
+        "A": [
+          1.0,
+          2.0
+        ]
+      },
+      "validation_reasons": [
+        "ellipse_point_off_curve"
+      ]
+    },
+    {
+      "index": 776,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 777,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_focus_mismatch",
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0777.png",
+      "answer": "9",
+      "fact_expr": "G: Ellipse;m: Number;F1: Point;F2: Point;Expression(G) = (x^2/25 + y^2/m = 1);Coordinate(F1) = (-4, 0);Coordinate(F2) = (4, 0);Focus(G) = {F1,F2}",
+      "coords": {
+        "F1": [
+          -4.0,
+          0.0
+        ],
+        "F2": [
+          4.0,
+          0.0
+        ]
+      },
+      "validation_reasons": [
+        "ellipse_focus_mismatch"
+      ]
+    },
+    {
+      "index": 778,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0778.png",
+      "answer": "20",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;A: Point;B: Point;PointOnCurve(A, G);PointOnCurve(B, G);PointOnCurve(F2, LineSegmentOf(A, B))",
+      "coords": {}
+    },
+    {
+      "index": 779,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0779.png",
+      "answer": "16/3",
+      "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;F: Point;Expression(G) = (y = x^2/4);Focus(G) = F;PointOnCurve(F, H);Inclination(H) = ApplyUnit(30, degree);Intersection(H, G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 780,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 781,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 782,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 783,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0783.png",
+      "answer": "4",
+      "fact_expr": "C: Parabola;p: Number;G: Line;A: Point;F: Point;B: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;PointOnCurve(F, G);Intersection(G, C) = {A, B};Abs(LineSegmentOf(A, F)) = 6;Abs(LineSegmentOf(B, F)) = 3",
+      "coords": {}
+    },
+    {
+      "index": 784,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 785,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "y",
+        "x_coef": 1.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0785.png",
+      "answer": "5/2",
+      "fact_expr": "G: Ellipse;a: Number;B: Point;P: Point;e:Number;a>1;Expression(G) = (x^2 + y^2/a^2 = 1);Coordinate(B) = (-1, 0);Eccentricity(G)=e;e = (2*sqrt(5))/5;PointOnCurve(P, G)",
+      "coords": {
+        "B": [
+          -1.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 786,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.4641016151377544,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 12.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0786.png",
+      "answer": "{(2,3),(-2,3)}",
+      "fact_expr": "C: Ellipse;l: Line;P: Point;B: Point;A: Point;Expression(C) = (x^2/16 + y^2/12 = 1);Coordinate(P) = (0, 6);PointOnCurve(P, l);Intersection(l, C) = {A, B};MidPoint(LineSegmentOf(P,B)) = A",
+      "coords": {
+        "P": [
+          0.0,
+          6.0
+        ]
+      }
+    },
+    {
+      "index": 787,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0787.png",
+      "answer": "y^2=12*x",
+      "fact_expr": "C: Parabola;p: Number;G: Line;F: Point;A: Point;B: Point;H:Point;p>0;Expression(C) = (y^2 = 2*(p*x));Coordinate(H) = (-1, 0);Focus(C) = F;PointOnCurve(H, G);Intersection(G, C) = {A, B};Min(4*Abs(LineSegmentOf(F, A)) + Abs(LineSegmentOf(F, B))) = 19",
+      "coords": {
+        "H": [
+          -1.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 788,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0788.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;G: Circle;A: Point;B: Point;F2: Point;F1: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, l);Intersection(l, LeftPart(C)) =A;Intersection(l,RightPart(C))=B;IsDiameter(LineSegmentOf(A,B),G);PointOnCurve(F2,G);Abs(LineSegmentOf(A,F2))=Abs(LineSegmentOf(B,F2))",
+      "coords": {}
+    },
+    {
+      "index": 789,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0789.png",
+      "answer": "pm*sqrt(6)",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/2 + y^2 = 1);A: Point;B: Point;LeftVertex(C) = A;RightVertex(C) = B;P: Point;PointOnCurve(P, C);Negation(P=A);Negation(P=B);M: Point;Intersection(LineOf(A, P), yAxis) = M;N: Point;PointOnCurve(A, LineOf(A, N));IsParallel(LineOf(A, N), LineOf(B, P));Intersection(LineOf(A, N), yAxis) = N;S: Point;t: Number;Coordinate(S) = (t, 0);DotProduct(VectorOf(S, M), VectorOf(S, N)) = 5",
+      "coords": {}
+    },
+    {
+      "index": 790,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0790.png",
+      "answer": "1",
+      "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 + y^2 = 1);Focus(G) = {F1,F2};PointOnCurve(P,G);AngleOf(F1,P,F2)=ApplyUnit(90,degree)",
+      "coords": {}
+    },
+    {
+      "index": 791,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0791.png",
+      "answer": "2/5",
+      "fact_expr": "C: Ellipse;l: Line;l1:Line;A: Point;B: Point;D: Point;F: Point;Expression(C) = (x^2/25 + y^2/9 = 1);RightFocus(C) = F;PointOnCurve(F,l);Negation(IsParallel(l,yAxis));Intersection(l,C)={A,B};PerpendicularBisector(LineSegmentOf(A, B)) = l1;Intersection(MajorAxis(C),l1)=D",
+      "coords": {}
+    },
+    {
+      "index": 792,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0792.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "E: Hyperbola;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;RightFocus(E) = F;F: Point;M: Point;N: Point;J: Line;OneOf(Asymptote(E)) = J;PointOnCurve(M, J) = True;PointOnCurve(N, J) = True;Abs(LineSegmentOf(F, M)) = Abs(LineSegmentOf(F, N));Abs(LineSegmentOf(F, N)) = sqrt(3)*a;DotProduct(VectorOf(F, M), VectorOf(F, N)) = a^2",
+      "coords": {}
+    },
+    {
+      "index": 793,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0793.png",
+      "answer": "sqrt(13)-2",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;Q: Point;P: Point;F2: Point;F1: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) =F1;RightFocus(C)=F2;PointOnCurve(P, C);PointOnCurve(Q,Asymptote(C));Quadrant(P)=1;Quadrant(Q)=1;2*VectorOf(Q, P) = VectorOf(P, F2);DotProduct(VectorOf(Q, F1), VectorOf(Q, F2)) = 0",
+      "coords": {}
+    },
+    {
+      "index": 794,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.4641016151377544,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 12.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0794.png",
+      "answer": "[0,2]",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/16 + y^2/12 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G)",
+      "coords": {}
+    },
+    {
+      "index": 795,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0795.png",
+      "answer": "0",
+      "fact_expr": "G: Hyperbola;b: Number;P: Point;F1: Point;F2: Point;y0:Number;b>0;Expression(G) = (x^2/2 - y^2/b^2 = 1);Coordinate(P) = (sqrt(3), y0);LeftFocus(G) = F1;RightFocus(G) = F2;Expression(OneOf(Asymptote(G)))=(y=x);PointOnCurve(P, G)",
+      "coords": {}
+    },
+    {
+      "index": 796,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0796.png",
+      "answer": "sqrt(5)/3",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0;Tan(AngleOf(P, F1, F2)) = 2",
+      "coords": {}
+    },
+    {
+      "index": 797,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.9960020780563354,
+        "b": 1.4970015287399292,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "iterations": 2
+      },
+      "output_path": "results/test/hyperbola/problem_0797.png",
+      "answer": "7",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;P: Point;PointOnCurve(P, G);F1: Point;F2: Point;Focus(G) = {F1, F2};Eccentricity(G) = 5/4;DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0;Area(TriangleOf(F1, P, F2)) = 9",
+      "coords": {}
+    },
+    {
+      "index": 798,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.4641016151377544,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0798.png",
+      "answer": "11",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/12 = 1);P: Point;PointOnCurve(P, G);XCoordinate(P) = XCoordinate(RightFocus(G))",
+      "coords": {}
+    },
+    {
+      "index": 799,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0799.png",
+      "answer": "(1, 3]",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, RightPart(G));Min(Abs(LineSegmentOf(P, F1))^2/Abs(LineSegmentOf(P, F2))) = 8*a;e: Number;Eccentricity(G) = e",
+      "coords": {}
+    },
+    {
+      "index": 800,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0800.png",
+      "answer": "pi/3",
+      "fact_expr": "G: Parabola;M: Point;N: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G)=F;Intersection(Directrix(G),xAxis)=M;PointOnCurve(N,G);Abs(LineSegmentOf(M,N))=2*Abs(LineSegmentOf(N,F))",
+      "coords": {}
+    },
+    {
+      "index": 801,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0801.png",
+      "answer": "2",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;O: Origin;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;L:Line;PointOnCurve(F2,L);IsPerpendicular(Asymptote(C),L);Center(G)=F1;Radius(G)=Abs(LineSegmentOf(O,F1));IsTangent(L,G)",
+      "coords": {}
+    },
+    {
+      "index": 802,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 803,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_ecc_mismatch",
+      "params": {
+        "a": 1.9257268905639648,
+        "b": 1.7993911322594118,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 2.5576271215184017e-15,
+        "converged": true,
+        "iterations": 2
+      },
+      "output_path": "results/test/ellipse/problem_0803.png",
+      "answer": "2/7",
+      "fact_expr": "C: Ellipse;a: Number;G: Line;a>sqrt(2);Expression(C) = (y^2/3 + x^2/a^2 = 1);Expression(G) = (y = 6*x);Eccentricity(C) = 1/2",
+      "coords": {},
+      "validation_reasons": [
+        "ellipse_ecc_mismatch"
+      ]
+    },
+    {
+      "index": 804,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0804.png",
+      "answer": "9",
+      "fact_expr": "G: Parabola;A: Point;Expression(G) = (x^2 = 4*y);PointOnCurve(A, G);Distance(A, Focus(G)) = 10",
+      "coords": {}
+    },
+    {
+      "index": 805,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0805.png",
+      "answer": "(sqrt(5)-1)/2",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;Q: Point;F1: Point;O: Origin;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G)=F1;PointOnCurve(P,G);PointOnCurve(Q, RightDirectrix(G));VectorOf(P, Q) = 2*VectorOf(F1, O);lambda:Number;VectorOf(F1, Q) = lambda*(VectorOf(F1, O)/Abs(VectorOf(F1, O)) + VectorOf(F1, P)/Abs(VectorOf(F1, P)));lambda>0",
+      "coords": {}
+    },
+    {
+      "index": 806,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0806.png",
+      "answer": "sqrt(5)-1",
+      "fact_expr": "G: Parabola;P: Point;A: Point;M: Point;Expression(G) = (x^2 = 4*y);Coordinate(A) = (2, 0);PointOnCurve(P,G);Projection(P, xAxis) = M",
+      "coords": {
+        "A": [
+          2.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 807,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0807.png",
+      "answer": "8",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = sqrt(5)/3;PointOnCurve(P, G);F1: Point;F2: Point;Focus(G) = {F1, F2};Distance(P, F1) + Distance(P, F2) = 12",
+      "coords": {}
+    },
+    {
+      "index": 808,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.898979485566356,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0808.png",
+      "answer": "21",
+      "fact_expr": "G: Hyperbola;P: Point;Expression(G) = (x^2/25 - y^2/24 = 1);PointOnCurve(P, G);F1:Point;F2:Point;OneOf(Focus(G))=F1;OneOf(Focus(G))=F2;Negation(F1=F2);Distance(P, F1) = 11",
+      "coords": {}
+    },
+    {
+      "index": 809,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0809.png",
+      "answer": "5/3",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(P) = (1, 4/3);PointOnCurve(P,Asymptote(G))",
+      "coords": {
+        "P": [
+          1.0,
+          1.3333333333333333
+        ]
+      }
+    },
+    {
+      "index": 810,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0810.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "E: Parabola;p: Number;A: Point;F: Point;O: Origin;p>0;Expression(E) = (y^2 = 2*(p*x));Focus(E) = F;PointOnCurve(A, E);Abs(LineSegmentOf(A, F)) = 2*Abs(LineSegmentOf(O, F));Abs(LineSegmentOf(O, A)) = sqrt(10)",
+      "coords": {}
+    },
+    {
+      "index": 811,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 812,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 813,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0813.png",
+      "answer": "y=pm*(sqrt(2)/2)*x",
+      "fact_expr": "l: Line;C: Hyperbola;a: Number;A: Point;B: Point;F2: Point;F1: Point;a>0;Expression(C) = (-y^2/2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, l);IsPerpendicular(xAxis,l);Intersection(l, C) = {A, B};Area(TriangleOf(A, B, F2)) = 2*sqrt(6)",
+      "coords": {}
+    },
+    {
+      "index": 814,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 815,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0815.png",
+      "answer": "6",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);M: Point;PointOnCurve(M, G);F1: Point;F2: Point;Focus(G) = {F1, F2};Distance(M, F1) = 4",
+      "coords": {}
+    },
+    {
+      "index": 816,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0816.png",
+      "answer": "y^2 = 8*x",
+      "fact_expr": "l: Line;G: Parabola;p: Number;H: Circle;O: Origin;F:Point;p>0;Expression(G) = (y^2 = 2*p*x);Directrix(G)=l;Radius(H)=3;Vertex(G)=O;Focus(G)=F;PointOnCurve(O,H);PointOnCurve(F,H);IsTangent(H,l)",
+      "coords": {}
+    },
+    {
+      "index": 817,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 818,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0818.png",
+      "answer": "sqrt(6)/3",
+      "fact_expr": "M: Point;Coordinate(M) = (1, 1);Slope(l) = -1/3;PointOnCurve(M, l) = True;l: Line;C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Intersection(l, C) = {A, B};MidPoint(LineSegmentOf(A, B)) = M;B: Point;A: Point",
+      "coords": {
+        "M": [
+          1.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 819,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0819.png",
+      "answer": "2",
+      "fact_expr": "E: Parabola;p: Number;A: Point;F: Point;B: Point;G: Circle;p>0;Expression(E) = (y^2 = 2*(p*x));Focus(E) = F;PointOnCurve(A, E);Center(G) = A;IsTangent(G, yAxis);Intersection(G, LineSegmentOf(A, F)) = B;Abs(LineSegmentOf(A, B)) = 2*Abs(LineSegmentOf(B, F));Length(InterceptChord(PerpendicularBisector(LineSegmentOf(A, F)), G)) = sqrt(7)",
+      "coords": {}
+    },
+    {
+      "index": 820,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0820.png",
+      "answer": "{3,1/3}",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*p*x);p: Number;p>0;Focus(C) = F;F: Point;Coordinate(F) = (1, 0);PointOnCurve(F, G) = True;G: Line;Intersection(G, C) = {A, B};A: Point;B: Point;Intersection(Directrix(C), xAxis) = M;M: Point;Area(TriangleOf(M, A, B)) = 8*sqrt(3)/3",
+      "coords": {
+        "F": [
+          1.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 821,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 822,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 823,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0823.png",
+      "answer": "{ApplyUnit(45, degree), pi/4}",
+      "fact_expr": "C: Parabola;p: Number;M: Point;F: Point;K: Point;p>0;Expression(C) = (y^2 = 2*(p*x));PointOnCurve(M, C);Focus(C) = F;Abs(LineSegmentOf(M, F)) = p;Intersection(Directrix(C), xAxis) = K",
+      "coords": {}
+    },
+    {
+      "index": 824,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0824.png",
+      "answer": "1+2*sqrt(2)",
+      "fact_expr": "l: Line;C: Hyperbola;P: Point;F1: Point;Q: Point;Expression(C) = (x^2/2 - y^2 = 1);PointOnCurve(P, RightPart(C));OneOf(Asymptote(C))=l;Projection(P,l)=Q;LeftFocus(C)=F1",
+      "coords": {}
+    },
+    {
+      "index": 825,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0825.png",
+      "answer": "[sqrt(2)/2,1)",
+      "fact_expr": "G: Ellipse;m: Number;n: Number;H: Curve;e: Number;m > n;n > 0;Expression(G) = (y^2/n + x^2/m = 1);Expression(H) = (x^2 + y^2 = Abs(m - n));IsIntersect(G, H);Eccentricity(G) = e",
+      "coords": {}
+    },
+    {
+      "index": 826,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0826.png",
+      "answer": "3",
+      "fact_expr": "A: Point;B: Point;l: Line;G: Parabola;Expression(G) = (y^2 = 4*x);PointOnCurve(Focus(G), l);Intersection(l, G) = {A, B};Abs(LineSegmentOf(A, B)) = 8",
+      "coords": {}
+    },
+    {
+      "index": 827,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 828,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 829,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 830,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0830.png",
+      "answer": "x^2-y^2/24=1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;LeftFocus(G) = F1;RightFocus(G) = F2;F1: Point;F2: Point;O: Origin;Abs(LineSegmentOf(F1, F2)) = 10;M: Point;PointOnCurve(M, LeftPart(G)) = True;Abs(LineSegmentOf(O, M)) = sqrt(a^2 + b^2);4*Abs(LineSegmentOf(M, F1)) = 3*Abs(LineSegmentOf(M, F2))",
+      "coords": {}
+    },
+    {
+      "index": 831,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0831.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);P: Point;PointOnCurve(P, G) = True;Distance(P, Directrix(G)) = d1;d1: Number;l: Line;Expression(l) = (4*x - 3*y + 16 = 0);Distance(P, l) = d2;d2: Number",
+      "coords": {}
+    },
+    {
+      "index": 832,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0832.png",
+      "answer": "(0,20)",
+      "fact_expr": "C: Hyperbola;P: Point;F1: Point;F2: Point;I: Point;Expression(C) = (x^2/9 - y^2/16 = 1);LeftFocus(C)=F1;RightFocus(C) = F2;Quadrant(P) = 1;PointOnCurve(P, C);Incenter(TriangleOf(P, F1, F2)) = I",
+      "coords": {}
+    },
+    {
+      "index": 833,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0833.png",
+      "answer": "pm*2*sqrt(2)",
+      "fact_expr": "l: Line;Expression(l)=(x=m*y+1);m: Number;C: Parabola;Expression(C) = (y^2 = 2*p*x);p: Number;F: Point;Focus(C)=F;PointOnCurve(F,l);A: Point;B: Point;Intersection(l,C)={A,B};VectorOf(A, F) = 2*VectorOf(F, B)",
+      "coords": {}
+    },
+    {
+      "index": 834,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0834.png",
+      "answer": "16/3",
+      "fact_expr": "G: Parabola;A: Point;B: Point;C: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, LineOf(A, B));Intersection(LineOf(A, B), G) = {A, B};Intersection(LineOf(A, B), Directrix(G)) = C;Abs(LineSegmentOf(B, C)) = 2*Abs(LineSegmentOf(B, F))",
+      "coords": {}
+    },
+    {
+      "index": 835,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0835.png",
+      "answer": "4*sqrt(5)",
+      "fact_expr": "G: Hyperbola;H: Line;A: Point;B: Point;Expression(G) = (x^2/2 - y^2/3 = 1);Expression(H) = (y = x + 1);Intersection(H, G) = {A,B}",
+      "coords": {}
+    },
+    {
+      "index": 836,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.4142135623730951,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0836.png",
+      "answer": "ApplyUnit(90, degree)",
+      "fact_expr": "H: Ellipse;Expression(H) = (x^2/8 + y^2/2 = 1);G: Hyperbola;Expression(G) = (x^2/4 - y^2/2 = 1);F1: Point;F2: Point;Focus(H) = {F1, F2};Focus(G) = {F1, F2};P: Point;OneOf(Intersection(H, G)) = P",
+      "coords": {}
+    },
+    {
+      "index": 837,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 13.0,
+        "b": 12.0,
+        "major_axis": "x",
+        "x_coef": 169.0,
+        "y_coef": 144.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0837.png",
+      "answer": "22",
+      "fact_expr": "G: Ellipse;P: Point;F2: Point;F1: Point;Expression(G) = (x^2/169 + y^2/144 = 1);PointOnCurve(P, G);Focus(G)={F1,F2};Abs(LineSegmentOf(P, F1)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 838,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "down"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0838.png",
+      "answer": "y=2",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = -8*y)",
+      "coords": {}
+    },
+    {
+      "index": 839,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0839.png",
+      "answer": "2*sqrt(5)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 840,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0840.png",
+      "answer": "x=-2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x)",
+      "coords": {}
+    },
+    {
+      "index": 841,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0841.png",
+      "answer": "",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(OneOf(Asymptote(G))) = (4*x - 3*y = 0)",
+      "coords": {}
+    },
+    {
+      "index": 842,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "vertical"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0842.png",
+      "answer": "sqrt(34)/3",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-x^2/b^2 + y^2/a^2 = 1);Expression(OneOf(Asymptote(G))) = (y = (3/5)*x)",
+      "coords": {}
+    },
+    {
+      "index": 843,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 844,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0844.png",
+      "answer": "(sqrt(5)-1)/2",
+      "fact_expr": "C:Ellipse;a:Number;b:Number;lambda:Number;F1:Point;P:Point ;Q:Point;O:Origin;a>b ;b>0;Expression(C)=(x^2/a^2+y^2/b^2=1);LeftFocus(C)=F1;PointOnCurve(P,C);PointOnCurve(Q,RightDirectrix(C));VectorOf(P,Q)=2*VectorOf(F1,O);VectorOf(F1,Q) = lambda * (VectorOf(F1,P)/Abs(VectorOf(F1,P))+VectorOf(F1,O)/Abs(VectorOf(F1,O)));lambda>0;Negation(RightVertex(C)=P)",
+      "coords": {}
+    },
+    {
+      "index": 845,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_point_off_curve",
+      "params": {
+        "a": 1.4149149626832767,
+        "b": 2.450642358201144,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 1.0319236343469396e-05,
+        "converged": true,
+        "iterations": 94
+      },
+      "output_path": "results/test/hyperbola/problem_0845.png",
+      "answer": "y^2/3-x^2=1",
+      "fact_expr": "G: Hyperbola;Expression(Asymptote(G)) = (y = pm*sqrt(3)*x);P: Point;Coordinate(P) = (sqrt(3), 2*sqrt(3));PointOnCurve(P, G) = True",
+      "coords": {
+        "P": [
+          1.7320508075688772,
+          3.4641016151377544
+        ]
+      },
+      "validation_reasons": [
+        "hyperbola_point_off_curve"
+      ]
+    },
+    {
+      "index": 846,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0846.png",
+      "answer": "1",
+      "fact_expr": "C: Parabola;p: Number;G: Circle;H: Line;M: Point;F: Point;A: Point;p>0;x0: Number;x0 > p/2;Expression(C) = (y^2 = 2*(p*x));Expression(H) = (x = p/2);Coordinate(M) = (x0, 2*sqrt(2));Focus(C) = F;PointOnCurve(M, C);Center(G) = M;Intersection(G, LineSegmentOf(M, F)) = A;Length(InterceptChord(H, G)) = sqrt(3)*Abs(LineSegmentOf(M, A));Abs(LineSegmentOf(M, A)) = 2",
+      "coords": {}
+    },
+    {
+      "index": 847,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.23606797749979,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 5.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0847.png",
+      "answer": "-1/3",
+      "fact_expr": "G: Hyperbola;m: Real;n: Real;H: Ellipse;Expression(G) = (y^2/n + x^2/m = 1);Expression(H) = (x^2/5 + y^2/4 = 1);Eccentricity(G)=2;Focus(G)=Focus(H)",
+      "coords": {}
+    },
+    {
+      "index": 848,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 4.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0848.png",
+      "answer": "{(1,-4),(1,4)}",
+      "fact_expr": "M: Parabola;P: Point;F: Point;Expression(M) = (y^2 = 16*x);Focus(M)=F;PointOnCurve(P, M);Abs(LineSegmentOf(P, F)) = 5",
+      "coords": {}
+    },
+    {
+      "index": 849,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 3.4641016151377544,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0849.png",
+      "answer": "2\ny=pm*sqrt(3)*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/12 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 850,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0850.png",
+      "answer": "x^2/9-y^2/16=1",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(H) = (5, 0);OneOf(Focus(G))=H;Expression(OneOf(Asymptote(G)))=(3*y-4*x=0)",
+      "coords": {
+        "H": [
+          5.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 851,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0851.png",
+      "answer": "20",
+      "fact_expr": "l1: Line;Expression(l1) = (2*x -y + 6 = 0);l2: Line;Expression(l2) = (x = -1);C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;P: Point;PointOnCurve(P, C);WhenMin(Distance(P, l1) + Distance(P, l2))",
+      "coords": {}
+    },
+    {
+      "index": 852,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_point_off_curve",
+      "params": {
+        "a": 1.9960020780563354,
+        "b": 1.4970015287399292,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "iterations": 2
+      },
+      "output_path": "results/test/hyperbola/problem_0852.png",
+      "answer": "y = pm*(4/3)*x",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;I: Point;F: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 20*x);Coordinate(I) = (15/4, 3);Focus(H) = F;RightFocus(G) = F;PointOnCurve(I, G)",
+      "coords": {
+        "I": [
+          3.75,
+          3.0
+        ]
+      },
+      "validation_reasons": [
+        "hyperbola_point_off_curve"
+      ]
+    },
+    {
+      "index": 853,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 4.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0853.png",
+      "answer": "10",
+      "fact_expr": "G: Parabola;P: Point;Expression(G) = (y^2 = 16*x);PointOnCurve(P, G);Distance(P, yAxis) = 6",
+      "coords": {}
+    },
+    {
+      "index": 854,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 855,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0855.png",
+      "answer": "x=-1\n(1,0)",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x)",
+      "coords": {}
+    },
+    {
+      "index": 856,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 857,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.449489742783178,
+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 6.0,
+        "y_coef": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0857.png",
+      "answer": "6-2*sqrt(6)",
+      "fact_expr": "G: Ellipse;H: Circle;P: Point;A: Point;Q: Point;F: Point;Expression(G) = (x^2/6 + y^2/2 = 1);Expression(H) = (x^2 + y^2 = 8);RightFocus(G) = F;UpperVertex(G) = A;PointOnCurve(P, H);PointOnCurve(Q, G)",
+      "coords": {}
+    },
+    {
+      "index": 858,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0858.png",
+      "answer": "[0,sqrt(2)]",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/2 + y^2 = 1);P: Point;PointOnCurve(P,C) = True;DotProduct(VectorOf(O,P),VectorOf(P,F2)) >= -1;O: Origin;F2: Point;RightFocus(C) = F2",
+      "coords": {}
+    },
+    {
+      "index": 859,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0859.png",
+      "answer": "sqrt(2)\n0",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/2 - y^2/b^2 = 1);b: Number;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;Expression(OneOf(Asymptote(G))) = (y = x);y0: Number;P: Point;Coordinate(P) = (sqrt(3), y0);PointOnCurve(P, G)",
+      "coords": {}
+    },
+    {
+      "index": 860,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0860.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;p: Number;H: LineSegment;P: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(P) = (3, 1);IsChordOf(H,G);MidPoint(H)=P;Slope(OverlappingLine(H))=2",
+      "coords": {
+        "P": [
+          3.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 861,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 862,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0862.png",
+      "answer": "y=pm*(sqrt(10)/4)*x",
+      "fact_expr": "H: Parabola;Expression(H) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(H) = F;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;RightFocus(G) = F;I: Point;Coordinate(I) = ((3*a^2)/p, (2*b^2)/p);PointOnCurve(I, G)",
+      "coords": {}
+    },
+    {
+      "index": 863,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 864,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0864.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;K: Point;Coordinate(K) = (2, 0);H: Line;PointOnCurve(K, H);A: Point;B: Point;Intersection(H, G) = {A, B};C: Point;Intersection(H, Directrix(G)) = C;Area(TriangleOf(A, C, F))/Area(TriangleOf(B, C, F)) = 2/5",
+      "coords": {
+        "K": [
+          2.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 865,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0865.png",
+      "answer": "sqrt(2)+1",
+      "fact_expr": "B: Point;F: Point;C: Hyperbola;a: Number;b: Number;A: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightVertex(C) = A;RightFocus(C)=F;Intersection(LeftDirectrix(C),xAxis)=B;MidPoint(LineSegmentOf(B, F)) = A",
+      "coords": {}
+    },
+    {
+      "index": 866,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0866.png",
+      "answer": "[1/2,1)",
+      "fact_expr": "A: Point;P: Point;G: Ellipse;b: Number;a: Number;F: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F;Intersection(RightDirectrix(G), xAxis) = A;PointOnCurve(P,G);PointOnCurve(F,PerpendicularBisector(LineSegmentOf(A,P)))",
+      "coords": {}
+    },
+    {
+      "index": 867,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 868,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0868.png",
+      "answer": "4",
+      "fact_expr": "F1:Point;F2:Point;C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C)=F1;RightFocus(C)=F2;b: Number;a: Number;a>0;b>0;e:Number;Eccentricity(C)=e;P: Point;PointOnCurve(P,RightPart(C));DotProduct(VectorOf(P,F2),VectorOf(F1,F2)) =0;Abs(VectorOf(P, F2)) = (2*sqrt(3))/3;e=2*sqrt(3)/3",
+      "coords": {}
+    },
+    {
+      "index": 869,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.0625,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0869.png",
+      "answer": "2+sqrt(2)",
+      "fact_expr": "G: Parabola;Expression(G) = (y = 4*x^2);M: Point;N: Point;PointOnCurve(M, G);PointOnCurve(N, G);Negation(M=N);F: Point;Focus(G) = F;AngleOf(M, F, N) = ApplyUnit(135, degree);IsChordOf(LineSegmentOf(M, N), G);P: Point;MidPoint(LineSegmentOf(M, N)) = P;l: Line;Expression(l) = (y=-1/16);d: Number;Distance(P, l) = d;lambda: Number;Abs(LineSegmentOf(M, N))^2 = (d^2)*lambda",
+      "coords": {}
+    },
+    {
+      "index": 870,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0870.png",
+      "answer": "{[-1-sqrt(3),0)+(0,1+sqrt(3)]}",
+      "fact_expr": "l: Line;C: Curve;M: Point;Q: Point;N: Point;n:Number;Expression(C) = (x^2/2 + y^2 = 1);Expression(l)=(y=1);Intersection(l,yAxis)=M;PointOnCurve(Q,l);Negation(Q=M);XCoordinate(MidPoint(Q)) = n;PointOnCurve(N, C);AngleOf(M, Q, N) = ApplyUnit(45, degree)",
+      "coords": {}
+    },
+    {
+      "index": 871,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0871.png",
+      "answer": "sqrt(3) + 1",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;O: Origin;M: Point;F2: Point;F1: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(M, RightPart(G));DotProduct((VectorOf(O, F2) + VectorOf(O, M)),VectorOf(F2, M)) = 0;Abs(VectorOf(M, F1)) = sqrt(3)*Abs(VectorOf(M, F2))",
+      "coords": {}
+    },
+    {
+      "index": 872,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          -0.0,
+          1.0
+        ],
+        "radius": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/circle/problem_0872.png",
+      "answer": "16",
+      "fact_expr": "H: Line;b: Number;a: Number;c: Number;b*c>0;Expression(H) = (a*x + b*y + c - 1 = 0);G: Circle;Expression(G) = (-2*y + x^2 + y^2 - 15 = 0);PointOnCurve(Center(G),H) = True",
+      "coords": {}
+    },
+    {
+      "index": 873,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 874,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.366584062576294,
+        "b": 2.364173412322998,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.00017838670464698225,
+        "converged": true,
+        "iterations": 89
+      },
+      "output_path": "results/test/hyperbola/problem_0874.png",
+      "answer": "(pm*2, 0)\ny=pm*sqrt(3)*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;Eccentricity(G) = 2;H: Parabola;Expression(H) = (y^2 = 8*x);OneOf(Focus(G)) = Focus(H)",
+      "coords": {}
+    },
+    {
+      "index": 875,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 876,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          0.0,
+          0.0
+        ],
+        "radius": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/circle/problem_0876.png",
+      "answer": "[-1/3, 0)+(0, 1/3]",
+      "fact_expr": "G: Hyperbola;k: Real;H: Circle;Expression(G) = (x^2/9 - y^2/4 = k^2);Expression(H) = (x^2 + y^2 = 1);IsIntersect(G, H)",
+      "coords": {}
+    },
+    {
+      "index": 877,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 8.0,
+        "b": 5.291502622129181,
+        "major_axis": "x",
+        "x_coef": 64.0,
+        "y_coef": 28.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0877.png",
+      "answer": "5",
+      "fact_expr": "C: Ellipse;P: Point;F1: Point;O: Origin;M: Point;Expression(C) = (x^2/64 + y^2/28 = 1);LeftFocus(C) = F1;PointOnCurve(P, C);Distance(P,LeftFocus(C))=6;MidPoint(LineSegmentOf(P,F1)) = M",
+      "coords": {}
+    },
+    {
+      "index": 878,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.4142135623730951,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0878.png",
+      "answer": "ApplyUnit(90,degree)",
+      "fact_expr": "G: Hyperbola;H: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/2 - y^2/2 = 1);Expression(H) = (x^2/6 + y^2/2 = 1);OneOf(Intersection(G,H))=P;Focus(G)={F1,F2};Focus(H)={F1,F2}",
+      "coords": {}
+    },
+    {
+      "index": 879,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0879.png",
+      "answer": "(-2, 0)+(0, 2)",
+      "fact_expr": "H: Circle;Expression(H) = (x^2 + y^2 = R^2);R: Number;G: Hyperbola;Expression(G) = (x^2/4 - y^2/9 = 1);NumIntersection(H, G) = 0",
+      "coords": {}
+    },
+    {
+      "index": 880,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 3.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0880.png",
+      "answer": "x=-3",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 12*x)",
+      "coords": {}
+    },
+    {
+      "index": 881,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 2.23606797749979,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0881.png",
+      "answer": "sqrt(5)-2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/5 = 1);F1: Point;LeftFocus(G) = F1;H: Circle;Expression(H) = (x^2 + y^2 = 4);L1: Line;TangentOfPoint(F1, H) = L1;Intersection(L1, RightPart(G)) = P;P: Point;TangentPoint(L1, H) = T;T: Point;MidPoint(LineSegmentOf(P, F1)) = M;M: Point;O: Origin",
+      "coords": {}
+    },
+    {
+      "index": 882,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0882.png",
+      "answer": "y=-1/2",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 2*y)",
+      "coords": {}
+    },
+    {
+      "index": 883,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0883.png",
+      "answer": "[sqrt(2), +oo)",
+      "fact_expr": "G: Hyperbola;a: Number;b: Number;P: Point;F1: Point;F2: Point;I: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, RightPart(G));LeftFocus(G) = F1;RightFocus(G) = F2;Incenter(TriangleOf(P, F1, F2)) = I;Center(InscribedCircle(TriangleOf(P, F1, F2))) = I;Area(TriangleOf(I, P, F1)) - Area(TriangleOf(I, P, F2)) <= (sqrt(2)/2)*Area(TriangleOf(I, F1, F2))",
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+    {
+      "index": 884,
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0884.png",
+      "answer": "9",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/5 + y^2/m = 1);m: Real;Coordinate(OneOf(Focus(G))) = (0, 2)",
+      "coords": {}
+    },
+    {
+      "index": 885,
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+      "optimization": {
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+        "iterations": 98
+      },
+      "output_path": "results/test/hyperbola/problem_0885.png",
+      "answer": "x^2/4-y^2/9=1",
+      "fact_expr": "G: Hyperbola;Expression(Asymptote(G)) = (y = pm*(3/2)*x)",
+      "coords": {}
+    },
+    {
+      "index": 886,
+      "success": true,
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+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0886.png",
+      "answer": "4/5",
+      "fact_expr": "alpha: Number;Inclination(l) = alpha;l: Line;Slope(l) = Eccentricity(G);G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 887,
+      "success": true,
+      "conic_type": "ellipse",
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0887.png",
+      "answer": "3",
+      "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/4 + y^2/3 = 1);PointOnCurve(P, G);Distance(P, LeftFocus(G)) = 5/2",
+      "coords": {}
+    },
+    {
+      "index": 888,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 9.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0888.png",
+      "answer": "12",
+      "fact_expr": "C1: Ellipse;C2: Ellipse;O: Origin;M: Point;N: Point;Expression(C1) = (x^2/16+y^2/9=1);Expression(C2) = (x^2/64+y^2/36=1);PointOnCurve(M,C1);PointOnCurve(N,C2)",
+      "coords": {}
+    },
+    {
+      "index": 889,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0889.png",
+      "answer": "(1,sqrt(5))",
+      "fact_expr": "E: Hyperbola;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;G: Line;PointOnCurve(RightFocus(E), G);Slope(G) = 2;NumIntersection(G, RightPart(E)) = 2",
+      "coords": {}
+    },
+    {
+      "index": 890,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 2.30381441116333,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 6.658357051492203e-06,
+        "converged": true,
+        "iterations": 122
+      },
+      "output_path": "results/test/hyperbola/problem_0890.png",
+      "answer": "x^2/4-y^2/12 = 1",
+      "fact_expr": "E: Hyperbola;a: Number;b: Number;a > 0;b > 0;Expression(E) = (x^2/a^2 - y^2/b^2 = 1);Expression(OneOf(Asymptote(E))) = (y = sqrt(3)*x);F: Parabola;Expression(F) = (y^2 = 16*x);OneOf(Focus(E)) = Focus(F)",
+      "coords": {}
+    },
+    {
+      "index": 891,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0891.png",
+      "answer": "1/4",
+      "fact_expr": "C: Parabola;l1:Line;l2:Line;A: Point;B: Point;D: Point;E: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F,l1);PointOnCurve(F,l2);IsPerpendicular(l1,l2);Intersection(l1,C)={A, B};Intersection(l2,C)={D, E}",
+      "coords": {}
+    },
+    {
+      "index": 892,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_ecc_mismatch",
+      "params": {
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+        "b": 1.7993911322594118,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 2.5576271215184017e-15,
+        "converged": true,
+        "iterations": 2
+      },
+      "output_path": "results/test/ellipse/problem_0892.png",
+      "answer": "12",
+      "fact_expr": "G: Parabola;H: Ellipse;m: Number;n: Number;Expression(G) = (y^2 = 4*x);m > n;n > 0;Expression(H) = (y^2/n + x^2/m = 1);Eccentricity(H) = 1/2;OneOf(Focus(H))=Focus(G)",
+      "coords": {},
+      "validation_reasons": [
+        "ellipse_ecc_mismatch"
+      ]
+    },
+    {
+      "index": 893,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0893.png",
+      "answer": "6",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;O: Origin;H: Line;PointOnCurve(Focus(G), H);IsPerpendicular(H, xAxis);P: Point;Intersection(H, G) = P;LineSegmentOf(P, O) = 3*sqrt(5)",
+      "coords": {}
+    },
+    {
+      "index": 894,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "vertical"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0894.png",
+      "answer": "y = pm*2*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/4 + y^2/16 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 895,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.2175992727279663,
+        "b": 2.434081792831421,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 3.452532837400213e-06,
+        "converged": true,
+        "iterations": 124
+      },
+      "output_path": "results/test/hyperbola/problem_0895.png",
+      "answer": "2*sqrt(5)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(OneOf(Asymptote(G))) = (y = 2*x);Expression(RightDirectrix(G)) = (x = sqrt(5)/5)",
+      "coords": {}
+    },
+    {
+      "index": 896,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.2135674953460693,
+        "b": 2.4270472526550293,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 4.453341261978494e-06,
+        "converged": true,
+        "iterations": 125
+      },
+      "output_path": "results/test/hyperbola/problem_0896.png",
+      "answer": "2",
+      "fact_expr": "C: Hyperbola;m: Number;Expression(C) = (-y^2/4 + x^2/m = 1);Eccentricity(C) = sqrt(5)",
+      "coords": {}
+    },
+    {
+      "index": 897,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0897.png",
+      "answer": "2*sqrt(3)/3",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;OneOf(Focus(C)) = F;G: Circle;Expression(G) = (x^2 + y^2 = a^2);TangentOfPoint(F, G) = {l1, l2};l1: Line;l2: Line;TangentPoint(l1,G) = A;TangentPoint(l2,G) = B;A: Point;B: Point;AngleOf(A, F, B) = ApplyUnit(120, degree)",
+      "coords": {}
+    },
+    {
+      "index": 898,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0898.png",
+      "answer": "(1,3]",
+      "fact_expr": "F2: Point;F1: Point;LeftFocus(G) = F1;RightFocus(G) = F2;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;P: Point;PointOnCurve(P,LeftPart(G)) = True;Abs(LineSegmentOf(P,F2))^2/Abs(LineSegmentOf(P,F1)) = 8*a",
+      "coords": {}
+    },
+    {
+      "index": 899,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 900,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0900.png",
+      "answer": "2",
+      "fact_expr": "G: Ellipse;m: Number;m>0;Expression(G) = (y^2 + x^2/m = 1);PointOnCurve(Focus(G), xAxis);FocalLength(G) = 2",
+      "coords": {}
+    },
+    {
+      "index": 901,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.9960020780563354,
+        "b": 1.4970015287399292,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "iterations": 2
+      },
+      "output_path": "results/test/hyperbola/problem_0901.png",
+      "answer": "5/4",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(Asymptote(G)) = (y = pm*(3/4)*x)",
+      "coords": {}
+    },
+    {
+      "index": 902,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0902.png",
+      "answer": "[1, 4]",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Length(MinorAxis(G)) = 2;A: Point;UpperVertex(G) = A;B: Point;LeftVertex(G) = B;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;Area(TriangleOf(F1, A, B)) = (2 - sqrt(3))/2;P: Point;PointOnCurve(P, G)",
+      "coords": {}
+    },
+    {
+      "index": 903,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0903.png",
+      "answer": "4",
+      "fact_expr": "P: Point;Coordinate(P) = (x1, y1);x1: Number;y1: Number;PointOnCurve(P, G);G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);F: Point;Coordinate(F) = (3, 0);Abs(LineSegmentOf(P, F)) = 2;O: Origin;M: Point;MidPoint(LineSegmentOf(P, F)) = M",
+      "coords": {
+        "F": [
+          3.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 904,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 905,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0905.png",
+      "answer": "0",
+      "fact_expr": "C: Parabola;p:Number;p>0;A: Point;B: Point;P: Point;F: Point;Expression(C) = (x^2 = 2*(p*y));Focus(C) = F;IsChordOf(LineSegmentOf(A,B),C);PointOnCurve(F,LineSegmentOf(A,B));l1:Line;l2:Line;TangentOfPoint(A,C)=l1;TangentOfPoint(B,C)=l2;Intersection(l1,l2)=P",
+      "coords": {}
+    },
+    {
+      "index": 906,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0906.png",
+      "answer": "2*sqrt(3)",
+      "fact_expr": "l: Line;C: Parabola;A: Point;O: Origin;B: Point;M: Point;Expression(C) = (y^2 = 4*x);Coordinate(M) = (4, 0);DotProduct(VectorOf(O, A), VectorOf(O, B)) = -4;Intersection(l,C)={A,B};Quadrant(A) = 1;WhenMin(Area(TriangleOf(A,B,M)))",
+      "coords": {
+        "M": [
+          4.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 907,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0907.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "C: Hyperbola;G: Line;a: Number;b: Number;P: Point;F1: Point;F2: Point;Q: Point;Expression(G) = (y = x*(b/a));LeftFocus(C) = F1;RightFocus(C) = F2;Expression(C) = (x^2/a^2 - y^2/b^2 = 1);PointOnCurve(P, C);Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2));MidPoint(LineSegmentOf(P, F1)) = Q;PointOnCurve(Q, G);a > 0;b > 0",
+      "coords": {}
+    },
+    {
+      "index": 908,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0908.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P,G);Focus(G)={F1,F2};IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2));Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 909,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 2.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0909.png",
+      "answer": "3*x-2*y-16=0",
+      "fact_expr": "G: Hyperbola;H: Line;B: Point;C: Point;A: Point;Expression(G) = (x^2 - 4*y^2 = 16);Coordinate(A) = (6, 1);PointOnCurve(A, H);Intersection(H, G) = {B, C};MidPoint(LineSegmentOf(B,C))=A",
+      "coords": {
+        "A": [
+          6.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 910,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0910.png",
+      "answer": "2*sqrt(2)-2",
+      "fact_expr": "l: Line;G: Parabola;A: Point;F: Point;B: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 911,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 912,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_point_off_curve",
+      "params": {
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+      "optimization": {
+        "final_loss": 4.481942266905428e-06,
+        "converged": true,
+        "iterations": 95
+      },
+      "output_path": "results/test/hyperbola/problem_0912.png",
+      "answer": "x^2-y^2/4=1",
+      "fact_expr": "G: Hyperbola;H: Point;Coordinate(H) = (sqrt(2), 2);PointOnCurve(H,G) = True;Expression(Asymptote(G)) = (y = pm*2*x)",
+      "coords": {
+        "H": [
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+          2.0
+        ]
+      },
+      "validation_reasons": [
+        "hyperbola_point_off_curve"
+      ]
+    },
+    {
+      "index": 913,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 914,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0914.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "G: Parabola;A: Point;M: Point;Expression(G) = (x^2 = 4*y);Coordinate(A) = (2, 0);PointOnCurve(M, G)",
+      "coords": {
+        "A": [
+          2.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 915,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0915.png",
+      "answer": "4",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);P: Point;PointOnCurve(P, G);F1: Point;F2: Point;Focus(G) = {F1, F2};Distance(P, F1) = 6",
+      "coords": {}
+    },
+    {
+      "index": 916,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0916.png",
+      "answer": "2",
+      "fact_expr": "E: Parabola;Expression(E) = (y^2 = 4*x);M: Point;Coordinate(M) = (2, 0);l: Line;B: Point;A: Point;PointOnCurve(M,l) = True;Intersection(l, E) = {A, B};C: Point;Intersection(Directrix(E) , l) = C;Quadrant(A) = 1 ;Abs(LineSegmentOf(C, B))/Abs(LineSegmentOf(A, B)) = 2/3",
+      "coords": {
+        "M": [
+          2.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 917,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0917.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Focus(G)={F1,F2};PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2));Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 918,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 919,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_point_off_curve",
+      "params": {
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+        "b": 1.4970015287399292,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "iterations": 2
+      },
+      "output_path": "results/test/hyperbola/problem_0919.png",
+      "answer": "9*sqrt(2)/4",
+      "fact_expr": "C: Hyperbola;a, b: Number;b > a;a > 0;Expression(C) = (x^2/a^2 - y^2/b^2 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;Coordinate(P) = (2, sqrt(2));PointOnCurve(P, C);Abs(LineSegmentOf(P, F1))/Abs(LineSegmentOf(P, F2)) = 3;A: Point;Intersection(LineSegmentOf(P, F1), C) = {P, A}",
+      "coords": {
+        "P": [
+          2.0,
+          1.4142135623730951
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+      },
+      "validation_reasons": [
+        "hyperbola_point_off_curve"
+      ]
+    },
+    {
+      "index": 920,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 921,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0921.png",
+      "answer": "(1,sqrt(2)+1)",
+      "fact_expr": "G: Hyperbola;a: Number;b: Number;c:Number;F1: Point;F2: Point;P: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);Sin(AngleOf(P, F1, F2))/Sin(AngleOf(P, F2, F1)) = a/c",
+      "coords": {}
+    },
+    {
+      "index": 922,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 2.8284271247461903,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0922.png",
+      "answer": "4",
+      "fact_expr": "l: Line;C: Hyperbola;A: Point;F1: Point;B: Point;F2: Point;Expression(C) = (x^2 - y^2/8 = 1);LeftFocus(C)=F1;RightFocus(C)=F2;PointOnCurve(F2, l);Intersection(l, LeftPart(C)) = A;Abs(LineSegmentOf(A, F1)) = Abs(LineSegmentOf(B, F1));Intersection(l, RightPart(C)) = B",
+      "coords": {}
+    },
+    {
+      "index": 923,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0923.png",
+      "answer": "y^2=16*x",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*p*x);p: Number;F: Point;Focus(C) = F;M: Point;PointOnCurve(M, C) = True;Abs(LineSegmentOf(M, F)) = 5;IsDiameter(LineSegmentOf(M, F), G) = True;G: Circle;PointOnCurve(D, G) = True;D: Point;Coordinate(D) = (0,2);p > Abs(LineSegmentOf(M, F))",
+      "coords": {
+        "D": [
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+          2.0
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+      }
+    },
+    {
+      "index": 924,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0924.png",
+      "answer": "1/2",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);M: Point;Coordinate(M) = (-1, 1/2);H: LineSegment;IsChordOf(H, G) = True;MidPoint(H) = M",
+      "coords": {
+        "M": [
+          -1.0,
+          0.5
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+      }
+    },
+    {
+      "index": 925,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.9960020780563354,
+        "b": 1.4970015287399292,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "iterations": 2
+      },
+      "output_path": "results/test/hyperbola/problem_0925.png",
+      "answer": "16",
+      "fact_expr": "C: Hyperbola;a: Number;b: Number;G: Line;H: Point;A: Point;B: Point;F2: Point;F1:Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(H) = (-2, 0);PointOnCurve(H, C);LeftFocus(C) =F1;RightFocus(C)=F2;PointOnCurve(F1, G);Intersection(G, LeftPart(C)) = {A, B};AngleOf(A, F2, B) = ApplyUnit(45, degree);Abs(LineSegmentOf(A, F2)) = 8",
+      "coords": {
+        "H": [
+          -2.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 926,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0926.png",
+      "answer": "(sqrt(7) - 1)/3",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;A: Point;F: Point;P: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F;RightVertex(G) = A;Z: Line;PointOnCurve(F, Z);IsPerpendicular(Z, xAxis);Intersection(Z, G) = P;Radius(InscribedCircle(TriangleOf(A, F, P))) = sqrt(a^2 - b^2)/2",
+      "coords": {}
+    },
+    {
+      "index": 927,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0927.png",
+      "answer": "x^2/4-y^2/5=1",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;Expression(OneOf(Asymptote(C))) = (y = (sqrt(5)/2)*x);G: Ellipse;Expression(G) = (x^2/12 + y^2/3 = 1);Focus(C) = Focus(G)",
+      "coords": {}
+    },
+    {
+      "index": 928,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 929,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0929.png",
+      "answer": "36",
+      "fact_expr": "C: Ellipse;a: Number;F1: Point;P: Point;F2: Point;a>3;Expression(C) = (y^2/9 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);AngleOf(F1, P, F2) = ApplyUnit(120, degree)",
+      "coords": {}
+    },
+    {
+      "index": 930,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_point_off_curve",
+      "params": {
+        "a": 2.6638543605804443,
+        "b": 2.2922656641629255,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 2.1016254280344046e-06,
+        "converged": true,
+        "iterations": 106
+      },
+      "output_path": "results/test/ellipse/problem_0930.png",
+      "answer": "(-1,1)",
+      "fact_expr": "G: Ellipse;n: Number;m: Number;P: Point;A: Point;B: Point;m>0;n>0;Expression(G) = (y^2/n + x^2/m = 1);Coordinate(A) = (-1, -3);Coordinate(B) = (2, 0);Coordinate(P) = (x1, y1);x1:Number;y1:Number;-1<x1;x1<2;y1<0;PointOnCurve(P, G);PointOnCurve(A,G);PointOnCurve(B,G)",
+      "coords": {
+        "A": [
+          -1.0,
+          -3.0
+        ],
+        "B": [
+          2.0,
+          0.0
+        ]
+      },
+      "validation_reasons": [
+        "ellipse_point_off_curve"
+      ]
+    },
+    {
+      "index": 931,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.4641016151377544,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 12.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0931.png",
+      "answer": "4",
+      "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/16 + y^2/12 = 1);Coordinate(P) = (x1, y1);Quadrant(P) = 1;PointOnCurve(P, G);x1:Number;y1:Number",
+      "coords": {}
+    },
+    {
+      "index": 932,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0932.png",
+      "answer": "5",
+      "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2/4 + y^2/m = 1);Expression(OneOf(Directrix(G))) = (y = m)",
+      "coords": {}
+    },
+    {
+      "index": 933,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 934,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.463192105293274,
+        "b": 2.070901870727539,
+        "orientation": "vertical"
+      },
+      "optimization": {
+        "final_loss": 1.4661259228887502e-06,
+        "converged": true,
+        "iterations": 106
+      },
+      "output_path": "results/test/hyperbola/problem_0934.png",
+      "answer": "sqrt(2)*x+pm*2*y=0",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-x^2/b^2 + y^2/a^2 = 1);Eccentricity(G) = sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 935,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0935.png",
+      "answer": "5*x+y-1=0",
+      "fact_expr": "l: Line;Expression(l) = (k*x - y + 1 = 0);k: Number;C: Parabola;Expression(C) = (y^2 = 4*x);A: Point;B: Point;Intersection(l, C) = {A, B};O: Origin;P: Point;Intersection(Directrix(C), xAxis) = P;DotProduct(VectorOf(O, A), VectorOf(O, B)) < 0;WhenMin(DotProduct(VectorOf(P, A), VectorOf(P, B)))",
+      "coords": {}
+    },
+    {
+      "index": 936,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0936.png",
+      "answer": "sqrt(13)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Z1: Line;Z2: Line;Asymptote(G) = {Z1, Z2};H: Parabola;Expression(H) = (y^2 = 4*x);A: Point;Intersection(Z1, Directrix(H)) = A;B: Point;Intersection(Z2, Directrix(H)) = B;O: Origin;Area(TriangleOf(A, O, B)) = 2*sqrt(3);e: Number;Eccentricity(G) = e",
+      "coords": {}
+    },
+    {
+      "index": 937,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0937.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;a: Number;Expression(G) = (y^2 = 2*(a*x));Expression(Directrix(G)) = (x = -2)",
+      "coords": {}
+    },
+    {
+      "index": 938,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          0.0,
+          0.0
+        ],
+        "radius": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/circle/problem_0938.png",
+      "answer": "(y^2=2*x+1)&(-1/2<=x<=1/2)",
+      "fact_expr": "C: Circle;P: Point;Q: Point;x0:Number;y0:Number;Expression(C) = (x^2 + y^2 = 1);Coordinate(P) = (x0, y0);Coordinate(Q) = (x0*y0, x0 + y0);PointOnCurve(P, C)",
+      "coords": {}
+    },
+    {
+      "index": 939,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0939.png",
+      "answer": "2*pi/3",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;F: Point;Focus(G) = F;A: Point;B: Point;x1: Number;x2: Number;y1: Number;y2: Number;Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);PointOnCurve(A, G);PointOnCurve(B, G);p + x1 + x2 = ((2*sqrt(3))/3)*Abs(LineSegmentOf(A, B))",
+      "coords": {}
+    },
+    {
+      "index": 940,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0940.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;F: Point;M: Point;E: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);OneOf(Focus(G))=F;L:Line;PointOnCurve(F,L);IsPerpendicular(L,Asymptote(G));FootPoint(L,Asymptote(G))=M;Intersection(OverlappingLine(LineSegmentOf(F, M)), yAxis) = E;LineSegmentOf(F, M) = LineSegmentOf(M, E)",
+      "coords": {}
+    },
+    {
+      "index": 941,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 942,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0942.png",
+      "answer": "3*sqrt(3)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);P: Point;PointOnCurve(P, G);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;AngleOf(F1, P, F2) = ApplyUnit(60, degree)",
+      "coords": {}
+    },
+    {
+      "index": 943,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.23606797749979,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 5.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0943.png",
+      "answer": "8*sqrt(5)/5",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/5 + y^2/4 = 1);F1: Point;LeftFocus(G) = F1;F2: Point;RightFocus(G) = F2;L: Line;PointOnCurve(F1, L)=  True;IsPerpendicular(L, xAxis) = True;Intersection(L, G) = {A, B};A: Point;B: Point",
+      "coords": {}
+    },
+    {
+      "index": 944,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0944.png",
+      "answer": "(-1+sqrt(5))/2",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;O: Origin;A: Point;B: Point;F: Point;a > b;b > 0;RightFocus(G) = F;L:Line;IsPerpendicular(L,xAxis);PointOnCurve(F,L);Intersection(L, G) = {A, B};DotProduct(VectorOf(O, A), VectorOf(O, B)) = 0",
+      "coords": {}
+    },
+    {
+      "index": 945,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0945.png",
+      "answer": "[2,+oo)",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;P: Point;PointOnCurve(P, C) = True;L1: Line;L2: Line;Asymptote(C) = {L1,L2};Distance(P,L1) * Distance(P,L2) = 1;c: Number;HalfFocalLength(C) = c",
+      "coords": {}
+    },
+    {
+      "index": 946,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0946.png",
+      "answer": "10",
+      "fact_expr": "l: Line;C: Parabola;p: Number;A: Point;B: Point;Expression(C) = (y^2 = 2*p*x);Expression(l)=(y=2*x-4);Focus(C)=F;PointOnCurve(F,l);Intersection(l,C)={A,B};F:Point",
+      "coords": {}
+    },
+    {
+      "index": 947,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0947.png",
+      "answer": "x^2/4-y^2/12=1",
+      "fact_expr": "H: Ellipse;Expression(H) = (x^2/25 + y^2/9 = 1);G: Hyperbola;Focus(H) = Focus(G);Eccentricity(G) = 2",
+      "coords": {}
+    },
+    {
+      "index": 948,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 2.23606797749979,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0948.png",
+      "answer": "16/3",
+      "fact_expr": "C1: Hyperbola;P: Point;F: Point;Expression(C1) = (x^2/4 - y^2/5 = 1);PointOnCurve(P, C1);RightFocus(C1) = F;LineSegmentOf(P, F) = 4",
+      "coords": {}
+    },
+    {
+      "index": 949,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0949.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;b/a = 2;e: Number;Eccentricity(G) = e",
+      "coords": {}
+    },
+    {
+      "index": 950,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0950.png",
+      "answer": "y = pm*x",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;A: Point;B: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftVertex(C) = A;RightVertex(C) = B;PointOnCurve(P, C);AngleOf(P, B, A) = pi/2 + AngleOf(P, A, B)",
+      "coords": {}
+    },
+    {
+      "index": 951,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0951.png",
+      "answer": "sqrt(3)*x + pm*y=0",
+      "fact_expr": "P: Point;Coordinate(P) = (-2, 0);PointOnCurve(P, C);C: Hyperbola;G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);Focus(C) = Focus(G)",
+      "coords": {
+        "P": [
+          -2.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 952,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0952.png",
+      "answer": "y = -1/2",
+      "fact_expr": "C: Parabola;Expression(C) = (x^2 = 2*y)",
+      "coords": {}
+    },
+    {
+      "index": 953,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0953.png",
+      "answer": "1",
+      "fact_expr": "C1: Hyperbola;C: Ellipse;b: Number;a: Number;M: Point;F1: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;Expression(C1)=(2*x^2/a^2-y^2=1);Focus(C)=Focus(C1);OneOf(Intersection(C,C1))=M;Abs(LineSegmentOf(M,F1))*Abs(LineSegmentOf(M,F2))=2",
+      "coords": {}
+    },
+    {
+      "index": 954,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0954.png",
+      "answer": "4*sqrt(3)/3",
+      "fact_expr": "H: Line;Expression(H) = (y = sqrt(3)*(x - 1));G: Parabola;Expression(G) = (y^2 = 4*x);A: Point;B: Point;Intersection(H, G) = {A, B};O: Origin",
+      "coords": {}
+    },
+    {
+      "index": 955,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0955.png",
+      "answer": "sqrt(5) - 1",
+      "fact_expr": "l: Line;G: Parabola;P: Point;Expression(G) = (y^2 = 4*x);Expression(l) = (2*x - y + 3 = 0);PointOnCurve(P, G)",
+      "coords": {}
+    },
+    {
+      "index": 956,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.4149149626832767,
+        "b": 2.450642358201144,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 1.0319236343469396e-05,
+        "converged": true,
+        "iterations": 94
+      },
+      "output_path": "results/test/hyperbola/problem_0956.png",
+      "answer": "y = pm*(sqrt(3)/2)*x",
+      "fact_expr": "C1:Hyperbola;C2:Hyperbola;P:Point;Q:Point;XCoordinate(P)=XCoordinate(Q);YCoordinate(P)=2*YCoordinate(Q);Locus(P)=C1;Locus(Q)=C2;Expression(Asymptote(C1)) = (y = pm*(sqrt(3)*x))",
+      "coords": {}
+    },
+    {
+      "index": 957,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0957.png",
+      "answer": "2*sqrt(21)",
+      "fact_expr": "C: Parabola;G: Circle;A: Point;B: Point;D: Point;E: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;Center(G) = F;Intersection(G, C) = {A, B};Intersection(G, Directrix(C)) = {D, E};Abs(LineSegmentOf(A, B)) = 8",
+      "coords": {}
+    },
+    {
+      "index": 958,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 4.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0958.png",
+      "answer": "(9/2)*sqrt(2)",
+      "fact_expr": "G: Parabola;l1: Line;l2: Line;P:Point;Expression(G) = (x^2 = 16*y);Expression(l1) = (x - y - 5 = 0);Expression(l2) = (y=-4);PointOnCurve(P,G)",
+      "coords": {}
+    },
+    {
+      "index": 959,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0959.png",
+      "answer": "5/3",
+      "fact_expr": "G: Hyperbola;a: Number;a>0;Expression(G) = (-y^2/16 + x^2/a^2 = 1);Coordinate(OneOf(Focus(G))) = (5, 0)",
+      "coords": {}
+    },
+    {
+      "index": 960,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0960.png",
+      "answer": "5",
+      "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/25 + y^2/9 = 1);PointOnCurve(P, G);F1:Point;F2:Point;OneOf(Focus(G))=F1;OneOf(Focus(G))=F2;Negation(F1=F2);Distance(P, F1) = 5",
+      "coords": {}
+    },
+    {
+      "index": 961,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0961.png",
+      "answer": "6",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 962,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0962.png",
+      "answer": "y^2=10*x",
+      "fact_expr": "G: Parabola;p: Number;H: Line;O:Origin;p>0;Expression(G) = (y^2 = 2*(p*x));L:Line;PointOnCurve(Focus(G),H);PointOnCurve(O,L);IsPerpendicular(H,L);Coordinate(FootPoint(H,L))=(2,1)",
+      "coords": {}
+    },
+    {
+      "index": 963,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.4149149626832767,
+        "b": 2.450642358201144,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 1.0319236343469396e-05,
+        "converged": true,
+        "iterations": 94
+      },
+      "output_path": "results/test/hyperbola/problem_0963.png",
+      "answer": "y^2/3-x^2=1",
+      "fact_expr": "G: Hyperbola;PointOnCurve(Focus(G),yAxis);FocalLength(G) = 4;Expression(OneOf(Asymptote(G))) = (y = sqrt(3)*x)",
+      "coords": {}
+    },
+    {
+      "index": 964,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.2135674953460693,
+        "b": 2.4270472526550293,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 4.453341261978494e-06,
+        "converged": true,
+        "iterations": 125
+      },
+      "output_path": "results/test/hyperbola/problem_0964.png",
+      "answer": "{4,1/4}",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;Expression(G) = (-y^2/b + x^2/a = 1);Eccentricity(G) = sqrt(5)",
+      "coords": {}
+    },
+    {
+      "index": 965,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0965.png",
+      "answer": "{x=0, y=-3, x+3*y+9=0}",
+      "fact_expr": "H: Point;Coordinate(H) = (0, -3);l: Line;PointOnCurve(H, l);G: Parabola;Expression(G) = (y^2 = 4*x);NumIntersection(l, G) = 1",
+      "coords": {
+        "H": [
+          0.0,
+          -3.0
+        ]
+      }
+    },
+    {
+      "index": 966,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 967,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_ecc_mismatch",
+      "params": {
+        "a": 1.9257268905639648,
+        "b": 1.7993911322594118,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 2.5576271215184017e-15,
+        "converged": true,
+        "iterations": 2
+      },
+      "output_path": "results/test/ellipse/problem_0967.png",
+      "answer": "3/2",
+      "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2/2 + y^2/m = 1);PointOnCurve(Focus(G), xAxis);Eccentricity(G) = 1/2",
+      "coords": {},
+      "validation_reasons": [
+        "ellipse_ecc_mismatch"
+      ]
+    },
+    {
+      "index": 968,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 2.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0968.png",
+      "answer": "8/3",
+      "fact_expr": "G: Hyperbola;P: Point;A: Point;B: Point;O: Origin;m: Number;n: Number;Expression(G) = (x^2 - y^2/4 = 1);PointOnCurve(A, G);PointOnCurve(B, G);PointOnCurve(P, G);Negation(A = B);Negation(A = P);Negation(P = B);VectorOf(P, A) + VectorOf(P, B) = 2*VectorOf(P, O);Slope(LineSegmentOf(P, A)) = m;Slope(LineSegmentOf(P, B)) = n",
+      "coords": {}
+    },
+    {
+      "index": 969,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0969.png",
+      "answer": "sqrt(3)/2",
+      "fact_expr": "G: Hyperbola;H: Ellipse;a2: Number;b2: Number;a1: Number;b1: Number;F1: Point;P: Point;F2: Point;a2>0;b2>0;Expression(G) = (-y^2/b2^2 + x^2/a2^2 = 1);a1>b1;b1>0;Expression(H) = (y^2/b1^2 + x^2/a1^2 = 1);OneOf(Intersection(H, G)) = P;Focus(G) = {F1, F2};Focus(H) = {F1, F2};Focus(G) = Focus(H);e1: Number;e2: Number;Eccentricity(H) = e1;Eccentricity(G) = e2;AngleOf(F1, P, F2) = pi/3",
+      "coords": {}
+    },
+    {
+      "index": 970,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_point_off_curve",
+      "params": {
+        "a": 1.6259966743174588,
+        "b": 2.298015331604694,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 1.93011531933813e-06,
+        "converged": true,
+        "iterations": 91
+      },
+      "output_path": "results/test/hyperbola/problem_0970.png",
+      "answer": "2*sqrt(3)",
+      "fact_expr": "G: Hyperbola;H: Point;Coordinate(H) = (1, 2);Expression(Asymptote(G)) = (y = pm*sqrt(2)*x);PointOnCurve(H,G)",
+      "coords": {
+        "H": [
+          1.0,
+          2.0
+        ]
+      },
+      "validation_reasons": [
+        "hyperbola_point_off_curve"
+      ]
+    },
+    {
+      "index": 971,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0971.png",
+      "answer": "-5",
+      "fact_expr": "G: Hyperbola;H: Ellipse;b: Number;a: Number;A: Point;P: Point;B: Point;M: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a1:Number;b1:Number;a1 > b1;b1 > 0;Expression(H) = (y^2/b1^2 + x^2/a1^2 = 1);Vertex(G)={A,B};In(A,Vertex(H));In(B,Vertex(H));PointOnCurve(P,G);PointOnCurve(M,H);Negation(P=A);Negation(P=B);Negation(M=A);Negation(M=B);LineSegmentOf(A, P) + LineSegmentOf(B, P) = lambda*(LineSegmentOf(A, M) + LineSegmentOf(B, M));lambda:Real;Slope(LineOf(A,P))=k1;Slope(LineOf(B,P))=k2;Slope(LineOf(A,M))=k3;Slope(LineOf(B,M))=k4;k1:Number;k2:Number;k3:Number;k4:Number;k1+k2=5",
+      "coords": {}
+    },
+    {
+      "index": 972,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0972.png",
+      "answer": "sqrt(3)-1",
+      "fact_expr": "r: Ellipse;b: Number;a: Number;c: Number;M: Point;F1: Point;F2: Point;G:Line;a > b;b > 0;Expression(r) = (y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y = sqrt(3)*(c + x));LeftFocus(r)=F1;RightFocus(r)=F2;FocalLength(r)=2*c;OneOf(Intersection(G,r))=M;AngleOf(M,F1,F2)=2*AngleOf(M,F2,F1)",
+      "coords": {}
+    },
+    {
+      "index": 973,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 4.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0973.png",
+      "answer": "20",
+      "fact_expr": "O: Origin;G: Parabola;Expression(G) = (y^2 = 16*x);P: Point;PointOnCurve(P, G);F: Point;Focus(G) = F;Area(TriangleOf(O, P, F)) = 32",
+      "coords": {}
+    },
+    {
+      "index": 974,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0974.png",
+      "answer": "8",
+      "fact_expr": "C: Parabola;A: Point;F: Point;K: Point;Expression(C) = (y^2 = 8*x);Focus(C) = F;Intersection(Directrix(C), xAxis) = K;PointOnCurve(A, C);Abs(LineSegmentOf(A, K)) = sqrt(2)*Abs(LineSegmentOf(A, F))",
+      "coords": {}
+    },
+    {
+      "index": 975,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0975.png",
+      "answer": "y^2=4*x",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;K: Point;PointOnCurve(K, C);F: Point;Focus(C) = F;G: Point;Coordinate(G) = (4, 0);Min(Distance(K, F) + Distance(K, G)) = 5",
+      "coords": {
+        "G": [
+          4.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 976,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 977,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 978,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0978.png",
+      "answer": "8",
+      "fact_expr": "C: Parabola;G: Line;A: Point;B: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F, G);Slope(G) = 1;Intersection(G, C) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 979,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 980,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0980.png",
+      "answer": "-4",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;A: Point;B: Point;Endpoint(LineSegmentOf(A, B)) = {A, B};Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);IsChordOf(LineSegmentOf(A,B),G) = True;PointOnCurve(Focus(G),LineSegmentOf(A,B)) = True;x1: Number;y1: Number;x2: Number;y2: Number",
+      "coords": {}
+    },
+    {
+      "index": 981,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0981.png",
+      "answer": "(2*sqrt(6))/5",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);H: Line;Slope(H) = 1;PointOnCurve(Focus(G), H);A: Point;B: Point;Intersection(H, G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 982,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0982.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "C: Ellipse;a: Number;b: Number;G: Line;A: Point;F: Point;B: Point;a > b;b > 0;k:Number;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Eccentricity(C) = sqrt(3)/2;RightFocus(C)=F;PointOnCurve(F, G);Slope(G)=k;k>0;Intersection(G, C) = {A,B};VectorOf(A, F) = 3*VectorOf(F, B)",
+      "coords": {}
+    },
+    {
+      "index": 983,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0983.png",
+      "answer": "12",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/9 + x^2/m = 1);e: Number;Eccentricity(G) = e;e = 1/2;m: Real;PointOnCurve(Focus(G), xAxis)",
+      "coords": {}
+    },
+    {
+      "index": 984,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 3.4641016151377544,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0984.png",
+      "answer": "2*sqrt(3)",
+      "fact_expr": "C: Hyperbola;G: Line;F: Point;P: Point;Q: Point;Expression(C) = (x^2/4 - y^2/12 = 1);Expression(G) = (x = 2);RightFocus(C) = F;Quadrant(P) = 1;PointOnCurve(P, G);Quadrant(Q) = 1;Intersection(LineOf(P, F), OneOf(Asymptote(C))) = Q;VectorOf(F, P) = 2*VectorOf(P, Q)",
+      "coords": {}
+    },
+    {
+      "index": 985,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 986,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0986.png",
+      "answer": "sqrt(3)/2",
+      "fact_expr": "G: Parabola;a: Number;M: Ellipse;b: Number;H: Line;B: Point;A: Point;P: Point;Expression(G) = (y^2 = -a*x);a > b;b > 0;Expression(M) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = -x);RightVertex(M) = A;UpperVertex(M) = B;Intersection(LineOf(A,B), H) = P;PointOnCurve(P, G)",
+      "coords": {}
+    },
+    {
+      "index": 987,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0987.png",
+      "answer": "2*sqrt(3)",
+      "fact_expr": "A: Point;C: Point;N: Point;M: Point;Coordinate(A) = (0, 1);Coordinate(C) = (0, 5);E: Curve;Expression(E) = (x^2 = 4*y);Locus(M) = E;Abs(LineSegmentOf(N, C)) = 2",
+      "coords": {
+        "A": [
+          0.0,
+          1.0
+        ],
+        "C": [
+          0.0,
+          5.0
+        ]
+      }
+    },
+    {
+      "index": 988,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0988.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;l: Line;PointOnCurve(F, l) = True;Intersection(l,G) = {A,B};B: Point;A: Point;P: Point;Coordinate(P) = (-5/3, 0);IsPerpendicular(LineSegmentOf(P, B), LineSegmentOf(A, B))",
+      "coords": {
+        "P": [
+          -1.6666666666666667,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 989,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0989.png",
+      "answer": "x^2+(y-2)^2=16",
+      "fact_expr": "G: Parabola;H: Circle;L: Line;P:Point;D:Point;Expression(G) = (x^2 = 8*y);Expression(L) = (y = 2);Intersection(G,L)={P,D};PointOnCurve(P,H);PointOnCurve(D,H);IsTangent(Directrix(G),H)",
+      "coords": {}
+    },
+    {
+      "index": 990,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0990.png",
+      "answer": "3/4",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F: Point;A:Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G)=F;RightVertex(G)=A;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P,F),xAxis);Abs(LineSegmentOf(P,F))=(1/4)*Abs(LineSegmentOf(A,F))",
+      "coords": {}
+    },
+    {
+      "index": 991,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0991.png",
+      "answer": "4",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/4 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);M: Point;MidPoint(LineSegmentOf(F1, P)) = M;Abs(LineSegmentOf(O, M)) = 1;O: Origin",
+      "coords": {}
+    },
+    {
+      "index": 992,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_0992.png",
+      "answer": "12",
+      "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2/8 + y^2/m = 1);OverlappingLine(MajorAxis(G),yAxis);FocalLength(G) = 4",
+      "coords": {}
+    },
+    {
+      "index": 993,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_0993.png",
+      "answer": "{2,6}",
+      "fact_expr": "G: Parabola;p: Real;A: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(A) = (m, 2*sqrt(3));PointOnCurve(A, G);Distance(A, Focus(G)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 994,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 995,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0995.png",
+      "answer": "(5/3,2*sqrt(3)/3)",
+      "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};Quadrant(A)=1;Abs(LineSegmentOf(F, A)) = 3*Abs(LineSegmentOf(F, B))",
+      "coords": {}
+    },
+    {
+      "index": 996,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_0996.png",
+      "answer": "2*sqrt(3)/3",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;O: Origin;F: Point;N: Point;M:Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;l1:Line;l2:Line;Expression(l1)=(y=(b/a)*x);Expression(l2)=(y=-(b/a)*x);L:Line;PointOnCurve(F,L);IsPerpendicular(l1,L);FootPoint(l1,L)=M;Intersection(L,l2)=N;Abs(LineSegmentOf(O, F)) = Abs(LineSegmentOf(F, N));Asymptote(C)={l1,l2}",
+      "coords": {}
+    },
+    {
+      "index": 997,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 998,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 999,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_0999.png",
+      "answer": "1/2",
+      "fact_expr": "C: Parabola;G: Line;M: Point;A: Point;B: Point;Expression(C) = (y = x^2/4);Coordinate(M) = (1, -1);PointOnCurve(Focus(C), G);Slope(G) = k;Intersection(G, C) = {A, B};DotProduct(VectorOf(A, M), VectorOf(B, M)) = 0;k:Number",
+      "coords": {
+        "M": [
+          1.0,
+          -1.0
+        ]
+      }
+    },
+    {
+      "index": 1000,
+      "success": true,
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+      "error": null,
+      "params": {
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+        "direction": "up"
+      },
+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1000.png",
+      "answer": "2*sqrt(3)",
+      "fact_expr": "G: Parabola;M: Point;x1:Number;y1:Number;Expression(G) = (x^2 = 4*y);Coordinate(M) = (x1, y1);PointOnCurve(M, G);Distance(M, Focus(G)) = 3;O:Origin",
+      "coords": {}
+    },
+    {
+      "index": 1001,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "left"
+      },
+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1001.png",
+      "answer": "x=2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = -8*x)",
+      "coords": {}
+    },
+    {
+      "index": 1002,
+      "success": true,
+      "conic_type": "hyperbola",
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1002.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2/6 - y^2/3 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 1003,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1003.png",
+      "answer": "3",
+      "fact_expr": "C: Parabola;A: Point;B: Point;D: Point;Expression(C) = (y^2 = 4*x);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);Coordinate(D) = (m, 0);x1:Number;x2:Number;y1:Number;y2:Number;m:Number;PointOnCurve(A, C);PointOnCurve(B, C);Intersection(PerpendicularBisector(LineSegmentOf(A,B)), xAxis) = D;x1 + x2 = 2",
+      "coords": {}
+    },
+    {
+      "index": 1004,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
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+          0.0
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+        "radius": 2.0
+      },
+      "optimization": {
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+        "converged": true
+      },
+      "output_path": "results/test/circle/problem_1004.png",
+      "answer": "(x-2)^2+y^2=1",
+      "fact_expr": "G: Circle;A: Point;P: Point;O: Origin;B: Point;x1:Number;y1:Number;Expression(G) = (x^2 + y^2 = 4);Coordinate(A) = (4, 0);Coordinate(P) = (x1, y1);PointOnCurve(B, G);VectorOf(O, A) + VectorOf(O, B) = 2*VectorOf(O, P)",
+      "coords": {}
+    },
+    {
+      "index": 1005,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1005.png",
+      "answer": "-9",
+      "fact_expr": "G: Hyperbola;k: Real;Expression(G) = (x^2/4 + y^2/k = 1);Distance(OneOf(Focus(G)),Asymptote(G))=3",
+      "coords": {}
+    },
+    {
+      "index": 1006,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 2.070901870727539,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "iterations": 106
+      },
+      "output_path": "results/test/hyperbola/problem_1006.png",
+      "answer": "pm*1",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;m: Number;A: Point;O: Origin;B: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x = m);l1: Line;l2: Line;Asymptote(C) = {l1, l2} ;Intersection(G, l1) = A;Intersection(G, l2) = B;Area(TriangleOf(A, O, B)) = sqrt(2);Eccentricity(C) = sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 1007,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1008,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1008.png",
+      "answer": "25",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/9 + x^2/a = 1);a: Number;FocalLength(C) = 8",
+      "coords": {}
+    },
+    {
+      "index": 1009,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1009.png",
+      "answer": "x^2/6-y^2/2=1",
+      "fact_expr": "G: Hyperbola;C: Hyperbola;H: Point;Expression(G) = (x^2/3 - y^2 = 1);Coordinate(H) = (3, -1);PointOnCurve(H, C);Asymptote(G) = Asymptote(C)",
+      "coords": {
+        "H": [
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+          -1.0
+        ]
+      }
+    },
+    {
+      "index": 1010,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1011,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 2.8284271247461903,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 8.0
+      },
+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1011.png",
+      "answer": "18+2*sqrt(17)",
+      "fact_expr": "P: Point;Coordinate(P) = (0, 2);G: Ellipse;Expression(G) = (x^2/16 + y^2/8 = 1);A: Point;B: Point;x1:Number;y1:Number;x2:Number;y2:Number;PointOnCurve(A, G);PointOnCurve(B, G);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);lambda: Real;VectorOf(A, P) = lambda*VectorOf(P, B)",
+      "coords": {
+        "P": [
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+          2.0
+        ]
+      }
+    },
+    {
+      "index": 1012,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1012.png",
+      "answer": "16",
+      "fact_expr": "G: Ellipse;A: Point;B: Point;C: Point;Expression(G) = (x^2/25 + y^2/16 = 1);Coordinate(A) = (-3, 0);Coordinate(B) = (3, 0);PointOnCurve(C, G)",
+      "coords": {
+        "A": [
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+          0.0
+        ],
+        "B": [
+          3.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 1013,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1013.png",
+      "answer": "3",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;M: Point;PointOnCurve(M, G);Negation(Vertex(G) = M);O: Origin;VectorOf(O, M) = 2*VectorOf(O, N);N: Point;P: Point;H: Line;PointOnCurve(N,H);IsPerpendicular(LineOf(O,M),H);Intersection(H,xAxis) = P",
+      "coords": {}
+    },
+    {
+      "index": 1014,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1014.png",
+      "answer": "1",
+      "fact_expr": "C: Ellipse;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/4 + y^2 = 1);Focus(C) = {F1, F2};PointOnCurve(P, C)",
+      "coords": {}
+    },
+    {
+      "index": 1015,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1015.png",
+      "answer": "3/2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;H: Line;PointOnCurve(F, H);A: Point;B: Point;Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, F)) = 3",
+      "coords": {}
+    },
+    {
+      "index": 1016,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1016.png",
+      "answer": "(0,sqrt(5)/5]",
+      "fact_expr": "l: Line;C: Ellipse;b: Number;a: Number;G: Circle;A: Point;B: Point;a > b;b > 0;l1:Line;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(RightFocus(C),l1);IsPerpendicular(l1,xAxis);Intersection(l1,C)={A,B};PointOnCurve(LeftFocus(C),l);PointOnCurve(UpperVertex(C),l);IsDiameter(LineSegmentOf(A,B),G);IsIntersect(l,G)",
+      "coords": {}
+    },
+    {
+      "index": 1017,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1017.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "C: Hyperbola;a: Number;b: Number;P: Point;H: Point;F1: Point;F2:Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C)=F1;RightFocus(C)=F2;PointOnCurve(P,RightPart(C));L:Line;PointOnCurve(P,L);IsPerpendicular(L,OneOf(Asymptote(C)));FootPoint(L,OneOf(Asymptote(C)))=H;Min(Abs(LineSegmentOf(P, F1)) + Abs(LineSegmentOf(P, H))) = 4*a",
+      "coords": {}
+    },
+    {
+      "index": 1018,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1019,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1019.png",
+      "answer": "-2",
+      "fact_expr": "Gamma: Ellipse;Expression(Gamma) = (x^2/a^2 + y^2/b^2 = 1);a: Number;b: Number;a > b;b > 0;Eccentricity(Gamma) = sqrt(2)/2;A: Point;B: Point;C: Point;PointOnCurve(Vertex(TriangleOf(A, B, C)), Gamma) = True;MidPoint(LineSegmentOf(A, B)) = D;MidPoint(LineSegmentOf(B, C)) = E;MidPoint(LineSegmentOf(A, C)) = F;D: Point;E: Point;F: Point;k1: Number;k2: Number;k3: Number;Slope(OverlappingLine(LineSegmentOf(A, B))) = k1;Slope(OverlappingLine(LineSegmentOf(B, C))) = k2;Slope(OverlappingLine(LineSegmentOf(A, C))) = k3;Negation(k1 = 0);Negation(k2 = 0);Negation(k3 = 0);O: Origin;Slope(LineOf(O, D)) + Slope(LineOf(O, E)) + Slope(LineOf(O, F)) = 1",
+      "coords": {}
+    },
+    {
+      "index": 1020,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1021,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1021.png",
+      "answer": "sqrt(10)/2",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;H: Line;A: Point;B: Point;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F2, H);Intersection(H, RightPart(C)) = {A,B};IsPerpendicular(LineSegmentOf(A,F1),LineSegmentOf(A,B));4*Abs(LineSegmentOf(A,F1))=3*Abs(LineSegmentOf(A,B))",
+      "coords": {}
+    },
+    {
+      "index": 1022,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1022.png",
+      "answer": "1/2",
+      "fact_expr": "M: Ellipse;b: Number;a: Number;c: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(M) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(M) = F1;RightFocus(M) = F2;PointOnCurve(P,M);Max(DotProduct(VectorOf(P, F1), VectorOf(P,F2))) = 3*c^2;c^2 = a^2-b^2",
+      "coords": {}
+    },
+    {
+      "index": 1023,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1023.png",
+      "answer": "2*sqrt(5)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2 + 4*y^2 = 16);A: Point;B: Point;P: Point;Coordinate(P) = (2, -1);PointOnCurve(P, LineOf(A,B));Intersection(LineOf(A,B),G)={A,B};Slope(LineOf(A,B)) = 1/2",
+      "coords": {
+        "P": [
+          2.0,
+          -1.0
+        ]
+      }
+    },
+    {
+      "index": 1024,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1024.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "G: Ellipse;P: Point;O: Origin;B:Point;A:Point;Expression(G) = (x^2/4 + y^2 = 1);Coordinate(P) = (x0,y0);PointOnCurve(P,G);WhenMin(Abs(LineSegmentOf(A,B)));x0:Number;y0:Number;l1:Line;TangentOfPoint(P,G)=l1;Expression(l1)=(x0*x/4+y0=1);Intersection(l2,yAxis)=A;Intersection(l1,yAxis)=B",
+      "coords": {}
+    },
+    {
+      "index": 1025,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1025.png",
+      "answer": "5/2",
+      "fact_expr": "G: Parabola;F: Point;P: Point;y1: Number;Expression(G) = (y^2 = 2*x);Coordinate(P) = (2, y1);Focus(G) = F;PointOnCurve(P, G)",
+      "coords": {}
+    },
+    {
+      "index": 1026,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1026.png",
+      "answer": "y=pm*sqrt(6)*x",
+      "fact_expr": "F2: Point;F1: Point;C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, l);l: Line;Intersection(l, LeftPart(C)) = P;Intersection(l, RightPart(C)) = Q;Q: Point;P: Point;AngleOf(Q, P, F2) = ApplyUnit(60, degree);Abs(LineSegmentOf(P, Q)) = Abs(LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 1027,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1027.png",
+      "answer": "x^2/(4*a^2)+y^2/(4*b^2)=1",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;O: Origin;Q: Point;P: Point;F1: Point;F2: Point;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, G);Focus(G)={F1,F2};VectorOf(O, Q) = VectorOf(P, F1) + VectorOf(P, F2)",
+      "coords": {}
+    },
+    {
+      "index": 1028,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          3.0,
+          0.0
+        ],
+        "radius": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/circle/problem_1028.png",
+      "answer": "y^2=-12*x",
+      "fact_expr": "F: Circle;Expression(F) = (y^2 + (x + 3)^2 = 1);l: Line;Expression(l) = (x = 2);M: Circle;IsTangent(l, M);IsOutTangent(F, M);M1: Point;Center(M) = M1",
+      "coords": {}
+    },
+    {
+      "index": 1029,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1029.png",
+      "answer": "y=pm*(3/2)*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/9 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 1030,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1030.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "C: Parabola;G: Line;A: Point;B: Point;M: Point;P: Point;l1: Line;l2: Line;Expression(C) = (y^2 = 4*x);Expression(G) = (x=-2);Coordinate(M) = (0, 1);PointOnCurve(P,G);TangentOfPoint(P, C) = {l1, l2};TangentPoint(l1,C)=A;TangentPoint(l2,C)=B",
+      "coords": {
+        "M": [
+          0.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 1031,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.23606797749979,
+        "major_axis": "y",
+        "x_coef": 5.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1031.png",
+      "answer": "sqrt(15)/5",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/5 + y^2/9 = 1);F1: Point;F2: Point;Focus(C) = {F1, F2};P: Point;PointOnCurve(P, C);Abs(LineSegmentOf(P, F1)) = Abs(LineSegmentOf(F1, F2));r: Number;Radius(InscribedCircle(TriangleOf(P, F1, F2))) = r",
+      "coords": {}
+    },
+    {
+      "index": 1032,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1032.png",
+      "answer": "y^2=8*x",
+      "fact_expr": "G: Parabola;O: Origin;Vertex(G) = O;PointOnCurve(Focus(G), xAxis);H: Line;Expression(H) = (2*x - y = 0);A: Point;B: Point;Intersection(H, G) = {A, B};P: Point;Coordinate(P) = (1, 2);MidPoint(LineSegmentOf(A, B)) = P",
+      "coords": {
+        "P": [
+          1.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 1033,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1033.png",
+      "answer": "2",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(C) = F;A: Point;LeftVertex(C) = A;O: Origin;G: Circle;IsDiameter(LineSegmentOf(O,F),G);Intersection(G,OneOf(Asymptote(C))) = P;P: Point;e: Number;Abs(LineSegmentOf(P, A)) = Abs(LineSegmentOf(P, F));Eccentricity(C) = e",
+      "coords": {}
+    },
+    {
+      "index": 1034,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1034.png",
+      "answer": "y=pm*2*x",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;B: Point;F: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);OneOf(Focus(C))=F;OneOf(Endpoint(ImageinaryAxis(C)))=B;PointOnCurve(MidPoint(LineSegmentOf(F,B)), C)",
+      "coords": {}
+    },
+    {
+      "index": 1035,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1035.png",
+      "answer": "-8",
+      "fact_expr": "H: Parabola;Expression(H) = (x^2 = a*y);a: Number;G: Hyperbola;Expression(G) = (-x^2 + y^2/3 = 1);Focus(H) = LowerFocus(G)",
+      "coords": {}
+    },
+    {
+      "index": 1036,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1037,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1038,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1038.png",
+      "answer": "x^2/4-(3*y^2)/4=1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Expression(Asymptote(G)) = (y = pm*sqrt(3)/3*x);Distance(Vertex(G), Asymptote(G)) = 1",
+      "coords": {}
+    },
+    {
+      "index": 1039,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1040,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1040.png",
+      "answer": "pm*3",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2 - y^2 = 1);A: Point;RightVertex(C) = A;PointOnCurve(A, l);l: Line;Intersection(l, Asymptote(C)) = {P, Q};P: Point;Q: Point;VectorOf(P, A) = 2*VectorOf(A, Q)",
+      "coords": {}
+    },
+    {
+      "index": 1041,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1041.png",
+      "answer": "sqrt(13)/2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;F1: Point;P: Point;F2: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) =F1;RightFocus(G)=F2;PointOnCurve(P,G);AngleOf(F1,P,F2)=2*pi/3;Abs(LineSegmentOf(F1, P)) = 3*Abs(LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 1042,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.23606797749979,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 5.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1042.png",
+      "answer": "-1",
+      "fact_expr": "Gamma: Ellipse;Expression(Gamma) = (x^2/5 + y^2 = 1);F1: Point;F2: Point;LeftFocus(Gamma) = F1;RightFocus(Gamma) = F2;P: Point;PointOnCurve(P, Gamma);Abs(LineSegmentOf(P, F1))*Abs(LineSegmentOf(P, F2)) = 3",
+      "coords": {}
+    },
+    {
+      "index": 1043,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1043.png",
+      "answer": "6 +sqrt(10)",
+      "fact_expr": "C: Ellipse;M: Point;N: Point;F1: Point;F2: Point;Expression(C) = (x^2/2 + y^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;UpperVertex(C)=N;Abs(VectorOf(M, N))^2 = 2*DotProduct(VectorOf(M, F1), VectorOf(M, F2))",
+      "coords": {}
+    },
+    {
+      "index": 1044,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1044.png",
+      "answer": "{25, -7}",
+      "fact_expr": "C: Curve;a: Number;Expression(C) = (y^2/9 + x^2/a = 1);FocalLength(C) = 8",
+      "coords": {}
+    },
+    {
+      "index": 1045,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.6457513110645907,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 7.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1045.png",
+      "answer": "[7,9)",
+      "fact_expr": "H: Curve;Expression(H) = (x = sqrt(a - y^2));a: Number;G: Ellipse;Expression(G) = (x^2/9 + y^2/7 = 1);NumIntersection(H, G) = 2",
+      "coords": {}
+    },
+    {
+      "index": 1046,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 6.708203932499369,
+        "b": 4.47213595499958,
+        "major_axis": "x",
+        "x_coef": 45.0,
+        "y_coef": 20.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1046.png",
+      "answer": "(-3,4)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/45 + y^2/20 = 1);P:Point;PointOnCurve(P,G);F1:Point;F2:Point;Focus(G)={F1,F2};IsPerpendicular(LineSegmentOf(P,F1),LineSegmentOf(P,F2));Quadrant(P)=2",
+      "coords": {}
+    },
+    {
+      "index": 1047,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1047.png",
+      "answer": "9",
+      "fact_expr": "G: Ellipse;Expression(G)=(x^2/m+y^2/9=1);m:Number;m>9;C:Hyperbola;Expression(C)=(x^2/n-y^2/9=1);n>0;n:Number;F1:Point;F2:Point;P:Point;Focus(G)={F1,F2};Focus(C)={F1,F2};OneOf(Intersection(G,C))=P",
+      "coords": {}
+    },
+    {
+      "index": 1048,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1048.png",
+      "answer": "8/3",
+      "fact_expr": "C: Parabola;G: Line;F: Point;A: Point;B: Point;l: Line;Expression(C) = (y^2 = 4*x);Focus(C) = F;Directrix(C) = l;PointOnCurve(F, G);Intersection(G, l) = A;OneOf(Intersection(G,C))=B;VectorOf(F, A) = 3*VectorOf(F, B)",
+      "coords": {}
+    },
+    {
+      "index": 1049,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1049.png",
+      "answer": "sqrt(3)-1",
+      "fact_expr": "l: Line;C: Ellipse;a: Number;b: Number;A: Point;F: Point;B: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;Expression(l)=(y = sqrt(3)*x);Intersection(l, C) = {A, B};IsPerpendicular(LineSegmentOf(A, F), LineSegmentOf(B, F))",
+      "coords": {}
+    },
+    {
+      "index": 1050,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1051,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_focus_mismatch",
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1051.png",
+      "answer": "x^2/24+y^2/8=1",
+      "fact_expr": "E: Ellipse;b: Number;a: Number;G: Line;A: Point;B: Point;F: Point;M: Point;a > b;b > 0;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (4, 0);Coordinate(M) = (1, -1);RightFocus(E) = F;PointOnCurve(F, G);Intersection(G, E) = {A, B};MidPoint(LineSegmentOf(A, B)) = M",
+      "coords": {
+        "F": [
+          4.0,
+          0.0
+        ],
+        "M": [
+          1.0,
+          -1.0
+        ]
+      },
+      "validation_reasons": [
+        "ellipse_focus_mismatch"
+      ]
+    },
+    {
+      "index": 1052,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1052.png",
+      "answer": "4",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G)",
+      "coords": {}
+    },
+    {
+      "index": 1053,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1053.png",
+      "answer": "y^2=8*x",
+      "fact_expr": "C: Parabola;P: Point;O: Origin;m:Number;Coordinate(P) = (4, m);Vertex(C) = O;PointOnCurve(Focus(C), xAxis);PointOnCurve(P, C);Distance(P, Focus(C)) = 6",
+      "coords": {}
+    },
+    {
+      "index": 1054,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1055,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1055.png",
+      "answer": "10",
+      "fact_expr": "G: Parabola;Focus(G) = F;Coordinate(F) = (0, 1);F: Point;P: Point;Coordinate(P) = (0, 2);l: Line;PointOnCurve(P,l) = True;Intersection(l, G) = {A, B};A: Point;B: Point;Abs(LineSegmentOf(A, B)) = 4*sqrt(6)",
+      "coords": {
+        "F": [
+          0.0,
+          1.0
+        ],
+        "P": [
+          0.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 1056,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1056.png",
+      "answer": "[1,9)",
+      "fact_expr": "G: Ellipse;m: Real;H: Line;k: Number;Expression(G) = (x^2/9 + y^2/m = 1);Expression(H) = (y = k*x + 1);PointOnCurve(Focus(G),xAxis);IsIntersect(H,G)",
+      "coords": {}
+    },
+    {
+      "index": 1057,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 10.0,
+        "b": 6.0,
+        "major_axis": "x",
+        "x_coef": 100.0,
+        "y_coef": 36.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1057.png",
+      "answer": "14",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/100 + y^2/36 = 1);P: Point;PointOnCurve(P, G);F1: Point;F2: Point;OneOf(Focus(G)) = F1;Distance(P, F1) = 6;OneOf(Focus(G)) = F2;Negation(F1= F2)",
+      "coords": {}
+    },
+    {
+      "index": 1058,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1058.png",
+      "answer": "1/3",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;c:Number;Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);Focus(G)={F1,F2};P: Point;PointOnCurve(P, G);Area(TriangleOf(P, F1, F2)) =(sqrt(2)/2)*b^2",
+      "coords": {}
+    },
+    {
+      "index": 1059,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1059.png",
+      "answer": "5/2",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;LeftFocus(C) = F;B1: Point;B2: Point;ImageinaryAxis(C) = LineSegmentOf(B1, B2);M: Point;O: Origin;MidPoint(LineSegmentOf(O, B1)) = M;G: Line;PointOnCurve(F, G);PointOnCurve(M, G);A: Point;Intersection(G, C) = A;LineSegmentOf(F, M) = 2*LineSegmentOf(M, A)",
+      "coords": {}
+    },
+    {
+      "index": 1060,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1060.png",
+      "answer": "x^2-y^2/2=1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Slope(OneOf(Asymptote(G))) = sqrt(2);H: Parabola;Expression(H) = (y^2 = 4*(sqrt(3)*x));RightFocus(G) = Focus(H)",
+      "coords": {}
+    },
+    {
+      "index": 1061,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1062,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1063,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 3.0,
+        "direction": "left"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1063.png",
+      "answer": "(-3, 0)",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = -12*x)",
+      "coords": {}
+    },
+    {
+      "index": 1064,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1064.png",
+      "answer": "sqrt(10)/2",
+      "fact_expr": "H: Line;Expression(H) = (y = 1 - x);G: Ellipse;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Intersection(H, G) = {A, B};A: Point;B: Point;IsPerpendicular(LineSegmentOf(O, A), LineSegmentOf(O, B)) = True;O: Origin;e: Number;Eccentricity(G) = e;In(e, [1/2, sqrt(3)/2])",
+      "coords": {}
+    },
+    {
+      "index": 1065,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.9960020780563354,
+        "b": 1.4970015287399292,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "iterations": 2
+      },
+      "output_path": "results/test/hyperbola/problem_1065.png",
+      "answer": "y = pm*2*x",
+      "fact_expr": "G: Hyperbola;a: Number;H: Point;a>0;Expression(G) = (-y^2/16 + x^2/a^2 = 1);Coordinate(H) = (2, 0);PointOnCurve(H, G)",
+      "coords": {
+        "H": [
+          2.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 1066,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1066.png",
+      "answer": "5",
+      "fact_expr": "G: Parabola;H: Line;a: Real;O: Origin;A: Point;B: Point;F:Point;Expression(H) = (x = a);Expression(G) = (y^2 = 4*x);Intersection(H, G) = {A, B};Focus(G)=F;Orthocenter(TriangleOf(O,A,B))=F",
+      "coords": {}
+    },
+    {
+      "index": 1067,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 4.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1067.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 16*x);F: Point;Focus(C) = F;A: Point;Coordinate(A) = (-4, 0);P: Point;PointOnCurve(P, C)",
+      "coords": {
+        "A": [
+          -4.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 1068,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1068.png",
+      "answer": "3",
+      "fact_expr": "G: Parabola;M: Point;F: Point;Expression(G) = (y = x^2/4);PointOnCurve(M, G);Focus(G) = F;Distance(M, F) = 4",
+      "coords": {}
+    },
+    {
+      "index": 1069,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1070,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1070.png",
+      "answer": "1",
+      "fact_expr": "A: Point;B: Point;C: Curve;Expression(C) = (y = x^2/4);PointOnCurve(A, C);PointOnCurve(B, C);XCoordinate(A) + XCoordinate(B) = 4",
+      "coords": {}
+    },
+    {
+      "index": 1071,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.6457513110645907,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1071.png",
+      "answer": "{1,13}",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2/9 - y^2/7 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C) = True;P: Point;Abs(LineSegmentOf(P, F1)) = 7",
+      "coords": {}
+    },
+    {
+      "index": 1072,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1072.png",
+      "answer": "3",
+      "fact_expr": "l: Line;C: Parabola;A: Point;F: Point;B: Point;Expression(C)=(y^2 = 2*(p*x));p:Number;p>0;Focus(C) = F;PointOnCurve(F, l);Inclination(l) = ApplyUnit(60, degree);Intersection(l, C) = {A, B};Quadrant(A)=1;Quadrant(B)=4",
+      "coords": {}
+    },
+    {
+      "index": 1073,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1073.png",
+      "answer": "7/2",
+      "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;Expression(G) = (y^2 = 2*x);Coordinate(A) = (3, 2);Focus(G) = F;PointOnCurve(P, G)",
+      "coords": {
+        "A": [
+          3.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 1074,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1075,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1075.png",
+      "answer": "8",
+      "fact_expr": "G: Ellipse;A: Point;B: Point;F2: Point;F1: Point;LeftFocus(G) = F1;RightFocus(G) = F2;l: Line;PointOnCurve(F1,l) = True;Intersection(l,G) = {A,B};Expression(G) = (x^2/4 + y^2/3 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 1076,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 6.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 36.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1076.png",
+      "answer": "12",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/36 + y^2/16 = 1);F1: Point;F2: Point;Focus(C) = {F1, F2};P: Point;PointOnCurve(P, C)",
+      "coords": {}
+    },
+    {
+      "index": 1077,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1077.png",
+      "answer": "2",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;F1: Point;A: Point;B: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, G);L1:Line;L2:Line;Asymptote(C)={L1,L2};Intersection(G, L1) = A;Intersection(G,L2) =B;VectorOf(F1, A) = VectorOf(A, B);DotProduct(VectorOf(F1,B),VectorOf(F2,B))=0",
+      "coords": {}
+    },
+    {
+      "index": 1078,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1079,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1079.png",
+      "answer": "13",
+      "fact_expr": "G: Parabola;F: Point;A: Point;P: Point;Expression(G) = (y^2 = 8*x);Focus(G) = F;Coordinate(A)=(6,3);PointOnCurve(P,G);Negation(PointOnCurve(P,LineOf(A,F)))",
+      "coords": {
+        "A": [
+          6.0,
+          3.0
+        ]
+      }
+    },
+    {
+      "index": 1080,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1080.png",
+      "answer": "3",
+      "fact_expr": "H: Line;G: Parabola;Expression(G) = (y^2 = 4*x);A: Point;B: Point;Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = 8",
+      "coords": {}
+    },
+    {
+      "index": 1081,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.0,
+        "major_axis": "y",
+        "x_coef": 9.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1081.png",
+      "answer": "8+2*sqrt(7)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/16 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);Negation(P=Endpoint(MajorAxis(G)))",
+      "coords": {}
+    },
+    {
+      "index": 1082,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "down"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1082.png",
+      "answer": "y=1",
+      "fact_expr": "G: Parabola;Expression(G) = (y = -x^2/4)",
+      "coords": {}
+    },
+    {
+      "index": 1083,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1083.png",
+      "answer": "x^2/9-y^2/3=1",
+      "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H)=(x^2 + 4*y^2 = 16);Focus(H) = Focus(G);Expression(OneOf(Asymptote(G))) = (x + sqrt(3)*y = 0)",
+      "coords": {}
+    },
+    {
+      "index": 1084,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1084.png",
+      "answer": "2*sqrt(3)/3",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;M: Point;F: Point;N: Point;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;L:Line;L1:Line;L2:Line;PointOnCurve(F, L);IsPerpendicular(L,L1);FootPoint(L,L1)=M;Intersection(L,L2)=N;2*VectorOf(M, F) = VectorOf(F, N);OneOf(Asymptote(C))=L1;OneOf(Asymptote(C))=L2;Negation(L1=L2)",
+      "coords": {}
+    },
+    {
+      "index": 1085,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1085.png",
+      "answer": "3*x+2*y-4=0",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2/3 = 1);A: Point;B: Point;PointOnCurve(A,C) = True;PointOnCurve(B,C) = True;Negation(A=B);P: Point;Coordinate(P) = (1, 1/2);VectorOf(A, P) = VectorOf(P, B)",
+      "coords": {
+        "P": [
+          1.0,
+          0.5
+        ]
+      }
+    },
+    {
+      "index": 1086,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 4.898979485566356,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1086.png",
+      "answer": "8",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2 - y^2/24 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;A: Point;PointOnCurve(A, RightPart(C))",
+      "coords": {}
+    },
+    {
+      "index": 1087,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.4142135623730951,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1087.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/2 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 1088,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1088.png",
+      "answer": "x+2*y-3=0",
+      "fact_expr": "P: Point;Coordinate(P) = (1, 1);L: Line;PointOnCurve(P, L) = True;G: Ellipse;Expression(G) = (x^2/4 + y^2/2 = 1);Intersection(L, G) = {M, N};M: Point;N: Point;MidPoint(LineSegmentOf(M, N)) = P",
+      "coords": {
+        "P": [
+          1.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 1089,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1089.png",
+      "answer": "(3+sqrt(10))/4",
+      "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;F: Point;Q: Point;A: Point;B: Point;e: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightVertex(C) = A;RightFocus(C) = F;PointOnCurve(A,l);Intersection(l,OneOf(Asymptote(C)))=Q;OneOf(Intersection(LineOf(Q,F), C)) = B;DotProduct(VectorOf(A, Q), VectorOf(A, B)) = DotProduct(VectorOf(A, Q), VectorOf(F, B));VectorOf(B, Q) = 4*VectorOf(F, Q);Eccentricity(C) = e",
+      "coords": {}
+    },
+    {
+      "index": 1090,
+      "success": true,
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+      "optimization": {
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+        "converged": true,
+        "iterations": 102
+      },
+      "output_path": "results/test/parabola/problem_1090.png",
+      "answer": "5/2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;M: Point;Coordinate(M) = (2, 2);PointOnCurve(M, G)",
+      "coords": {
+        "M": [
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+          2.0
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+      }
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+    {
+      "index": 1091,
+      "success": true,
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1091.png",
+      "answer": "1",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;H: Line;O: Origin;M: Point;a: Number;Negation(a=0);Coordinate(M) = (a, 0);PointOnCurve(M, H);A: Point;B: Point;Intersection(H, G) = {A, B};k1: Number;k2: Number;Slope(LineSegmentOf(O, A)) = k1;Slope(LineSegmentOf(O, B)) = k2;k1*k2 = -2*p",
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+    {
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+      "success": true,
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+        "direction": "up"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1092.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;Expression(G) = (y = x^2/4);M: Point;PointOnCurve(M, G);Distance(M, Focus(G)) = 3",
+      "coords": {}
+    },
+    {
+      "index": 1093,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 1094,
+      "success": true,
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+        "y_coef": 1.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1094.png",
+      "answer": "4*sqrt(2)",
+      "fact_expr": "C: Ellipse;G: Line;A: Point;B: Point;F2: Point;F1: Point;Expression(C) = (x^2/2 + y^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, G);Intersection(G, C) = {A, B}",
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+    },
+    {
+      "index": 1095,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1095.png",
+      "answer": "7/2",
+      "fact_expr": "F: Point;Focus(G) = F;G: Parabola;Expression(G) = (y^2 = 2*x);A: Point;B: Point;PointOnCurve(A, G);PointOnCurve(B, G);Abs(LineSegmentOf(A, F)) + Abs(LineSegmentOf(B, F)) = 8",
+      "coords": {}
+    },
+    {
+      "index": 1096,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.0,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1096.png",
+      "answer": "4",
+      "fact_expr": "C: Hyperbola;F1: Point;P: Point;F2: Point;Expression(C) = (x^2 - y^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);AngleOf(F1, P, F2) = ApplyUnit(60, degree)",
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+    },
+    {
+      "index": 1097,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "a": 3.0599746704101562,
+        "b": 2.576634645462036,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "iterations": 116
+      },
+      "output_path": "results/test/hyperbola/problem_1097.png",
+      "answer": "4*sqrt(3)",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/4 + x^2/m = 1);m: Number;m>0;F1: Point;F2: Point;Coordinate(F1) = (-4, 0);Coordinate(F2) = (4, 0);Focus(C) = {F1, F2};M: Point;PointOnCurve(M, C);AngleOf(F1, M, F2) = ApplyUnit(60, degree)",
+      "coords": {
+        "F1": [
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+          0.0
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+        "F2": [
+          4.0,
+          0.0
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+      }
+    },
+    {
+      "index": 1098,
+      "success": true,
+      "conic_type": "parabola",
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1098.png",
+      "answer": "12",
+      "fact_expr": "C: Parabola;B: Point;A: Point;D: Point;E: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F,LineOf(A,B));PointOnCurve(F,LineOf(D,E));Intersection(LineOf(A,B),C)={A,B};Intersection(LineOf(D,E),C)={D,E};Slope(LineOf(A,B))=k1;Slope(LineOf(D,E))=k2;k1:Number;k2:Number;k1^2+k2^2=4",
+      "coords": {}
+    },
+    {
+      "index": 1099,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 40.0,
+        "direction": "up"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1099.png",
+      "answer": "y=-40",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 160*y)",
+      "coords": {}
+    },
+    {
+      "index": 1100,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1101,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1102,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1102.png",
+      "answer": "(6,8)",
+      "fact_expr": "l: Line;C: Ellipse;A: Point;F: Point;B: Point;k:Number;Negation(k=0);Expression(C) = (x^2/4 + y^2 = 1);LeftFocus(C) = F;Expression(l)=(y = k*x) ;Intersection(l, C) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 1103,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1103.png",
+      "answer": "7",
+      "fact_expr": "G: Parabola;H: Line;F: Point;A: Point;B: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (2*x + y - 4 = 0);Intersection(G, H) = {A, B};Focus(G) = F",
+      "coords": {}
+    },
+    {
+      "index": 1104,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1105,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1105.png",
+      "answer": "(1,sqrt(3))",
+      "fact_expr": "A1: Point;A2: Point;C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > 0;b > 0;LeftVertex(C) = A1;RightVertex(C) = A2;M: Point;PointOnCurve(M, C);k1: Number;k2: Number;Slope(LineOf(M, A1)) = k1;Slope(LineOf(M, A2)) = k2;k1 * k2 < 2",
+      "coords": {}
+    },
+    {
+      "index": 1106,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1106.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;e: Number;Eccentricity(G) = e;Expression(Asymptote(G)) = (y = pm*x*sqrt(6))",
+      "coords": {}
+    },
+    {
+      "index": 1107,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1108,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1108.png",
+      "answer": "40",
+      "fact_expr": "l: Line;G: Hyperbola;A: Point;B: Point;F2: Point;F1: Point;Expression(G) = (x^2 - 16*y^2 = 16);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, l);Intersection(l, LeftPart(G)) = {A, B};Abs(LineSegmentOf(A, B)) = 12",
+      "coords": {}
+    },
+    {
+      "index": 1109,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1109.png",
+      "answer": "2*sqrt(3)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;H: Parabola;Expression(H) = (y^2 = 8*x);OneOf(Focus(G))=Focus(H);Distance(Focus(G),Asymptote(G))=1",
+      "coords": {}
+    },
+    {
+      "index": 1110,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1111,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1112,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 4.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1112.png",
+      "answer": "10",
+      "fact_expr": "G: Parabola;M: Point;Expression(G) = (y^2 = 16*x);PointOnCurve(M, G);XCoordinate(M) = 6;F: Point;Focus(G) = F",
+      "coords": {}
+    },
+    {
+      "index": 1113,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 3.872983346207417,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1113.png",
+      "answer": "13",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/15 = 1);P: Point;PointOnCurve(P, RightPart(G));H: Circle;Expression(H) = (y^2 + (x + 4)^2 = 4);K: Circle;Expression(K) = (y^2 + (x - 4)^2 = 1);Z1: Line;Z2: Line;TangentOfPoint(P, H) = Z1;TangentOfPoint(P, K) = Z2;M: Point;TangentPoint(Z1, H) = M;N: Point;TangentPoint(Z2, K) = N",
+      "coords": {}
+    },
+    {
+      "index": 1114,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1114.png",
+      "answer": "sqrt(7)",
+      "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;A: Point;F2: Point;B: Point;O: Origin;F1: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, l);Intersection(l, C) = {A, B};IsPerpendicular(LineSegmentOf(B, F1),LineSegmentOf(O, A));Abs(LineSegmentOf(A, F2)) = Abs(LineSegmentOf(B, F2))",
+      "coords": {}
+    },
+    {
+      "index": 1115,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1115.png",
+      "answer": "4*sqrt(2)",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;G: Line;PointOnCurve(F, G);Slope(G) = 1;Intersection(G, C) = {P, Q};P: Point;Q: Point;Projection(P, Directrix(C)) = M;Projection(Q, Directrix(C)) = N;M: Point;N: Point",
+      "coords": {}
+    },
+    {
+      "index": 1116,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.366584062576294,
+        "b": 2.364173412322998,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.00017838670464698225,
+        "converged": true,
+        "iterations": 89
+      },
+      "output_path": "results/test/hyperbola/problem_1116.png",
+      "answer": "27",
+      "fact_expr": "G: Hyperbola;m: Real;Negation(m=0);Expression(G) = (x^2/9 - y^2/m = 1);Eccentricity(G) = 2",
+      "coords": {}
+    },
+    {
+      "index": 1117,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1117.png",
+      "answer": "3*sqrt(3)",
+      "fact_expr": "G: Parabola;M: Point;x1: Number;y1: Number;Expression(G) = (y^2 = 6*x);Coordinate(M) = (x1, y1);PointOnCurve(M, G);Distance(M, Focus(G)) = 9/2;O: Origin",
+      "coords": {}
+    },
+    {
+      "index": 1118,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1119,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1120,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1120.png",
+      "answer": "3",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/4 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 1121,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1121.png",
+      "answer": "1/2",
+      "fact_expr": "l: Line;G: Ellipse;a: Number;b: Number;A: Point;F: Point;P: Point;O: Origin;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftVertex(G) = A;RightFocus(G) = F;PointOnCurve(F, l);IsPerpendicular(xAxis, l);PointOnCurve(P, l);AngleOf(A, P, O) = ApplyUnit(30, degree)",
+      "coords": {}
+    },
+    {
+      "index": 1122,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 0.5,
+        "b": 0.8660254037844386,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1122.png",
+      "answer": "",
+      "fact_expr": "l: Line;G: Hyperbola;H: Ellipse;Expression(G) = (12*x^2 - 4*y^2 = 3);Expression(l) = (y = x + 3);PointOnCurve(P,l);PointOnCurve(P,H);Focus(G)=Focus(H);WhenMin(MajorAxis(H));P:Point",
+      "coords": {}
+    },
+    {
+      "index": 1123,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1123.png",
+      "answer": "3*x+pm*2*y=0",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/9 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 1124,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
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+        "y_coef": 9.0
+      },
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1124.png",
+      "answer": "pi",
+      "fact_expr": "F1: Point;F2: Point;Focus(G) = {F1, F2};G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);P: Point;PointOnCurve(P, G) = True;DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0",
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+        "direction": "up"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1125.png",
+      "answer": "3",
+      "fact_expr": "G: Parabola;l1: Line;l2: Line;P: Point;Expression(G) = (y = x^2/4);Expression(l1)=(4*x - 3*y - 12 = 0);Expression(l2)=(y+1=0);PointOnCurve(P, G)",
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+    },
+    {
+      "index": 1126,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1127,
+      "success": true,
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+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1127.png",
+      "answer": "[sqrt(2)/2,1)",
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+    },
+    {
+      "index": 1128,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1129,
+      "success": true,
+      "conic_type": "parabola",
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+        "direction": "up"
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+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1129.png",
+      "answer": "y=-a/4",
+      "fact_expr": "G: Parabola;a: Number;Expression(G) = (x^2 = a*y);Negation(a=0)",
+      "coords": {}
+    },
+    {
+      "index": 1130,
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+      "conic_type": "parabola",
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+        "direction": "up"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1130.png",
+      "answer": "(1, 1)",
+      "fact_expr": "G: Parabola;H: Line;Expression(G) = (y = x^2);Expression(H) = (2*x - y = 4);Z: Point;PointOnCurve(Z, G);WhenMin(Distance(Z, H))",
+      "coords": {}
+    },
+    {
+      "index": 1131,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1132,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1132.png",
+      "answer": "[-sqrt(3)/3,sqrt(3)/3]",
+      "fact_expr": "G: Hyperbola;H: Line;F: Point;Expression(G) = (x^2/12 - y^2/4 = 1);RightFocus(G) = F;PointOnCurve(F,H);NumIntersection(H,RightPart(G))=1",
+      "coords": {}
+    },
+    {
+      "index": 1133,
+      "success": true,
+      "conic_type": "ellipse",
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+      "params": {
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1133.png",
+      "answer": "16",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/m = 1);m: Real;PointOnCurve(Focus(G), yAxis);Eccentricity(G) = sqrt(3)/2",
+      "coords": {}
+    },
+    {
+      "index": 1134,
+      "success": true,
+      "conic_type": "parabola",
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1134.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);P: Point;PointOnCurve(P, G);Distance(P, Focus(G)) = 3",
+      "coords": {}
+    },
+    {
+      "index": 1135,
+      "success": true,
+      "conic_type": "ellipse",
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1135.png",
+      "answer": "sqrt(10)/4",
+      "fact_expr": "E: Ellipse;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;LeftFocus(E) = F;l: Line;O: Origin;PointOnCurve(O, l);P: Point;Q: Point;Intersection(l, E) = {P, Q};Abs(LineSegmentOf(P, F)) = 3*Abs(LineSegmentOf(Q, F));AngleOf(P, F, Q) = ApplyUnit(90, degree)",
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+    },
+    {
+      "index": 1136,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1136.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Distance(Focus(G),Asymptote(G)) = a",
+      "coords": {}
+    },
+    {
+      "index": 1137,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "vertical"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1137.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/4 + y^2/5 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 1138,
+      "success": true,
+      "conic_type": "ellipse",
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1138.png",
+      "answer": "(2,0)",
+      "fact_expr": "C: Ellipse;P: Point;F: Point;Expression(C) = (x^2/4 + y^2/3 = 1);RightFocus(C) = F;PointOnCurve(P, C);Abs(LineSegmentOf(P, F)) = 1",
+      "coords": {}
+    },
+    {
+      "index": 1139,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1139.png",
+      "answer": "[1,3]",
+      "fact_expr": "G: Ellipse;P: Point;F: Point;Expression(G) = (x^2/4 + y^2/3 = 1);LeftFocus(G) = F;PointOnCurve(P, G)",
+      "coords": {}
+    },
+    {
+      "index": 1140,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1140.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "G: Parabola;p: Number;O: Origin;M: Point;F: Point;P: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G)=F;PointOnCurve(P, G);PointOnCurve(M, LineSegmentOf(P,F));Abs(LineSegmentOf(P, M)) = 2*Abs(LineSegmentOf(M, F))",
+      "coords": {}
+    },
+    {
+      "index": 1141,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1142,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1142.png",
+      "answer": "sqrt(5)/2",
+      "fact_expr": "G: Hyperbola;a: Number;b: Number;H: Ellipse;l: Line;A: Point;B: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a > b;b > 0;Expression(H) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(Focus(H), l);IsPerpendicular(l,xAxis);Intersection(l, H) = {A, B};LineSegmentOf(A, B) = a/2",
+      "coords": {}
+    },
+    {
+      "index": 1143,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1143.png",
+      "answer": "4",
+      "fact_expr": "G: Ellipse;m: Real;Expression(G) = (x^2/25 + y^2/m^2 = 1);FocalLength(G) = 6;m>0;m<5",
+      "coords": {}
+    },
+    {
+      "index": 1144,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1144.png",
+      "answer": "1",
+      "fact_expr": "G: Hyperbola;n: Number;H: Ellipse;m: Number;F1: Point;P: Point;F2: Point;n>0;Expression(G) = (-y^2 + x^2/n = 1);m>1;Expression(H) = (y^2 + x^2/m = 1);Focus(G) = {F1,F2};Focus(H) = {F1,F2};OneOf(Intersection(G,H))=P",
+      "coords": {}
+    },
+    {
+      "index": 1145,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1145.png",
+      "answer": "13",
+      "fact_expr": "G: Hyperbola;H: Ellipse;m: Number;n: Number;P: Point;F1: Point;F2: Point;m>0;n>0;Expression(H) = (y^2/9 + x^2/m^2 = 1);Expression(G) = (-y^2/4 + x^2/n^2 = 1);Focus(H) = {F1, F2};Focus(G) = {F1, F2};OneOf(Intersection(H,G))=P",
+      "coords": {}
+    },
+    {
+      "index": 1146,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1146.png",
+      "answer": "3",
+      "fact_expr": "G: Hyperbola;A: Point;F1: Point;F2: Point;Expression(G) = (x^2 - y^2/3 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(A,LeftPart(G));IsPerpendicular(LineSegmentOf(A,F1),LineSegmentOf(A,F2))",
+      "coords": {}
+    },
+    {
+      "index": 1147,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1147.png",
+      "answer": "7*x+8*y-22=0",
+      "fact_expr": "l: Line;G: Ellipse;a: Number;b: Number;A: Point;B: Point;M: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(M) = (2, 1);FocalLength(G)=6;MinorAxis(G)=(sqrt(7)/4)*MajorAxis(G);Intersection(l,G)={A,B};IsChordOf(LineSegmentOf(A,B),G);MidPoint(LineSegmentOf(A,B))=M",
+      "coords": {
+        "M": [
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+          1.0
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+    },
+    {
+      "index": 1148,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1148.png",
+      "answer": "10",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);P: Point;PointOnCurve(P, G) = True;L: Line;PointOnCurve(P, L) = True;IsPerpendicular(L, Directrix(G)) = True;FootPoint(L, Directrix(G)) = M;M: Point;Abs(LineSegmentOf(P, M)) = 5;F: Point;Focus(G) = F",
+      "coords": {}
+    },
+    {
+      "index": 1149,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 9.0,
+        "b": 5.196152422706632,
+        "major_axis": "x",
+        "x_coef": 81.0,
+        "y_coef": 27.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1149.png",
+      "answer": "36",
+      "fact_expr": "G: Ellipse;H:Line;P: Point;Q: Point;F2: Point;F1: Point;Expression(G) = (x^2/81 + y^2/27 = 1);Focus(G) = {F1, F2};PointOnCurve(F1,H);Intersection(H, G) = {P, Q}",
+      "coords": {}
+    },
+    {
+      "index": 1150,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1150.png",
+      "answer": "x^2/16+y^2/4=1",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = sqrt(3)/2;Length(MinorAxis(G))=4",
+      "coords": {}
+    },
+    {
+      "index": 1151,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1151.png",
+      "answer": "(5,pm*2*sqrt(5))",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);P:Point;Distance(P,Focus(G))=6;PointOnCurve(P,G)",
+      "coords": {}
+    },
+    {
+      "index": 1152,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1152.png",
+      "answer": "(2*sqrt(3))/3",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;LeftFocus(C) = F;A: Point;Coordinate(A) = (0, 2*b);Inclination(LineOf(A, F)) = ApplyUnit(45, degree)",
+      "coords": {}
+    },
+    {
+      "index": 1153,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1153.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "C: Parabola;A: Point;P: Point;Expression(C) = (y^2 = 4*x);Coordinate(A) = (0, 2);PointOnCurve(P, C)",
+      "coords": {
+        "A": [
+          0.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 1154,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1155,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1155.png",
+      "answer": "x^2/20-y^2/5=1",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(P) = (2, 1);FocalLength(C) = 10;PointOnCurve(P, Asymptote(C))",
+      "coords": {
+        "P": [
+          2.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 1156,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1157,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1157.png",
+      "answer": "4*sqrt(3)",
+      "fact_expr": "G: Hyperbola;m: Number;H: Parabola;m>0;Expression(G) = (x^2/4 - y^2/m^2 = 1);Expression(H) = (y^2 = 16*x);Focus(H) = RightFocus(G)",
+      "coords": {}
+    },
+    {
+      "index": 1158,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1159,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1160,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1160.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;H: Line;PointOnCurve(F, H) = True;Intersection(H, G) = {A, B};A: Point;B: Point;l: Line;Directrix(G) = l;Intersection(H, l) = C;C: Point;Abs(LineSegmentOf(A, F)) > Abs(LineSegmentOf(B, F));Abs(LineSegmentOf(B, C)) = 2*Abs(LineSegmentOf(B, F))",
+      "coords": {}
+    },
+    {
+      "index": 1161,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1161.png",
+      "answer": "sqrt(3)+1",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;O: Origin;P: Point;F2: Point;F1: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P,RightPart(G));DotProduct((VectorOf(O, F2) + VectorOf(O, P)),VectorOf(F2, P)) = 0;Abs(VectorOf(P, F1)) = sqrt(3)*Abs(VectorOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 1162,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1162.png",
+      "answer": "x^2/4+y^2/3=1",
+      "fact_expr": "C: Ellipse;a: Number;O: Origin;F: Point;P: Point;Expression(C) = (y^2/3 + x^2/a^2 = 1);RightFocus(C)=F;PointOnCurve(P,C);Abs(LineSegmentOf(O,F))=Abs(LineSegmentOf(F,P))",
+      "coords": {}
+    },
+    {
+      "index": 1163,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1163.png",
+      "answer": "28/3",
+      "fact_expr": "l: Line;C: Parabola;A: Point;F: Point;B: Point;Expression(C) = (y^2 = 4*x);Expression(l) = (y = sqrt(3)*x + b);Focus(C) = F;Distance(F, l) = sqrt(3);Intersection(l, C) = {A, B};b:Number",
+      "coords": {}
+    },
+    {
+      "index": 1164,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1164.png",
+      "answer": "2*sqrt(5)",
+      "fact_expr": "C: Parabola;Expression(C) = (x^2 = 8*y);F: Point;Focus(C) = F;O: Origin;P: Point;PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(P, F), LineSegmentOf(O, F))",
+      "coords": {}
+    },
+    {
+      "index": 1165,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1165.png",
+      "answer": "10",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 1166,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 2.8284271247461903,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1166.png",
+      "answer": "32",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2 - y^2/8 = 1);F: Point;RightFocus(C) = F;P: Point;PointOnCurve(P, LeftPart(C));A: Point;Coordinate(A) = (0, 6*sqrt(6));PointOnCurve(A, yAxis)",
+      "coords": {
+        "A": [
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+          14.696938456699067
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+      }
+    },
+    {
+      "index": 1167,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.4142135623730951,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1167.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/2 = 1);P: Point;PointOnCurve(P, RightPart(G));F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;I: Point;Incenter(TriangleOf(P,F1,F2)) = I;lambda: Real;Area(TriangleOf(P,F1,F2)) = 2*Area(TriangleOf(I,P,F2)) + (1+1/lambda)*Area(TriangleOf(I,F1,F2))",
+      "coords": {}
+    },
+    {
+      "index": 1168,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 2.298015331604694,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 1.93011531933813e-06,
+        "converged": true,
+        "iterations": 91
+      },
+      "output_path": "results/test/hyperbola/problem_1168.png",
+      "answer": "{sqrt(6)/2,sqrt(3)}",
+      "fact_expr": "G: Hyperbola;Expression(OneOf(Asymptote(G))) = (y=sqrt(2)*x)",
+      "coords": {}
+    },
+    {
+      "index": 1169,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.0,
+        "major_axis": "y",
+        "x_coef": 1.0,
+        "y_coef": 4.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1169.png",
+      "answer": "3*sqrt(6)/8",
+      "fact_expr": "l: Line;PointOnCurve(Focus(G), l) = True;Intersection(l,G) = {P,Q};G: Ellipse;Expression(G) = (x^2 + y^2/2 = 1);P: Point;Q: Point;M: Point;Intersection(PerpendicularBisector(LineSegmentOf(P,Q)),xAxis) = M",
+      "coords": {}
+    },
+    {
+      "index": 1170,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 2.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1170.png",
+      "answer": "y=pm*(1/2)*x\n12",
+      "fact_expr": "C: Hyperbola;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/16 - y^2/4 = 1);LeftFocus(C) =F1;RightFocus(C)=F2;PointOnCurve(P, C);Abs(LineSegmentOf(P, F1)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 1171,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1172,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1172.png",
+      "answer": "[sqrt(2)/2,1]",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);F: Point;Focus(G) = F;Intersection(Directrix(G),yAxis) = M;M: Point;PointOnCurve(N,G) = True;N: Point;lambda: Number;Abs(LineSegmentOf(N, F)) = lambda*Abs(LineSegmentOf(M, N))",
+      "coords": {}
+    },
+    {
+      "index": 1173,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1173.png",
+      "answer": "1/4",
+      "fact_expr": "G: Ellipse;a: Number;M: Point;A: Point;B: Point;N: Point;F1: Point;F2: Point;a>1;Expression(G) = (y^2 + x^2/a^2 = 1);RightVertex(G) = A;LowerVertex(G) = B;LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(M, G);PointOnCurve(N, G);Max(Area(TriangleOf(M, A, B))) = 1 + sqrt(2)",
+      "coords": {}
+    },
+    {
+      "index": 1174,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1174.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "C2: Hyperbola;C1: Parabola;A: Point;p: Number;a: Number;b: Number;a > 0;b > 0;p > 0;Expression(C1) = (y^2 = 2*p*x);Expression(C2) = (x^2/a^2 - y^2/b^2 = 1);Intersection(C1, OneOf(Asymptote(C2))) = A;Distance(A, Directrix(C1)) = p",
+      "coords": {}
+    },
+    {
+      "index": 1175,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1175.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;a: Number;a > 0;Expression(G) = (-y^2/9 + x^2/a^2 = 1);Expression(OneOf(Asymptote(G))) = (3*x - 2*y = 0)",
+      "coords": {}
+    },
+    {
+      "index": 1176,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1176.png",
+      "answer": "1",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);M: Point;Coordinate(M) = (-1, 2);PointOnCurve(Focus(C), G);Slope(G) = k;k: Number;G: Line;Intersection(G, C) = {A, B};A: Point;B: Point;DotProduct(VectorOf(M, A), VectorOf(M, B)) = 0",
+      "coords": {
+        "M": [
+          -1.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 1177,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1178,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1178.png",
+      "answer": "[2*a, (4*sqrt(3)*a)/3]",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;e: Number;Eccentricity(C) = e;e = 2;G: Line;PointOnCurve(F2, G);A: Point;B: Point;Intersection(G, RightPart(C)) = {A, B};Quadrant(A) = 1;M: Point;N: Point;Incenter(TriangleOf(A, F1, F2)) = M;Incenter(TriangleOf(B, F1, F2)) = N;a>0;b>0",
+      "coords": {}
+    },
+    {
+      "index": 1179,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 6.555423624376999e-06,
+        "converged": true,
+        "iterations": 116
+      },
+      "output_path": "results/test/parabola/problem_1179.png",
+      "answer": "1/2",
+      "fact_expr": "C: Parabola;p: Number;G: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Coordinate(G) = (1, 1);PointOnCurve(G, C)",
+      "coords": {
+        "G": [
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+          1.0
+        ]
+      }
+    },
+    {
+      "index": 1180,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1180.png",
+      "answer": "sqrt(10)/2",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;P: Point;Q: Point;F2: Point;F1: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, G);Intersection(G, LeftPart(C)) = {P, Q};IsPerpendicular(VectorOf(P,F2),VectorOf(P,Q));Perimeter(TriangleOf(P,Q,F2))=12*a",
+      "coords": {}
+    },
+    {
+      "index": 1181,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1182,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_focus_mismatch",
+      "params": {
+        "a": 1.681220143106833,
+        "b": 2.2412689930799665,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 4.985250663519028e-06,
+        "converged": true,
+        "iterations": 95
+      },
+      "output_path": "results/test/hyperbola/problem_1182.png",
+      "answer": "x^2/36-y^2/64=1",
+      "fact_expr": "G: Hyperbola;F: Point;O:Origin;Center(G)=O;Coordinate(F) = (10, 0);OneOf(Focus(G))=F;Expression(Asymptote(G)) = (y = pm*(4/3)*x)",
+      "coords": {
+        "F": [
+          10.0,
+          0.0
+        ]
+      },
+      "validation_reasons": [
+        "hyperbola_focus_mismatch"
+      ]
+    },
+    {
+      "index": 1183,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "down"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1183.png",
+      "answer": "2",
+      "fact_expr": "Q: Point;Coordinate(Q) = (-2*sqrt(2), 0);G: Parabola;Expression(G) = (x^2 = -4*y);P: Point;Coordinate(P) = (x1, y1);x1: Number;y1: Number;PointOnCurve(P, G) = True",
+      "coords": {
+        "Q": [
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+          0.0
+        ]
+      }
+    },
+    {
+      "index": 1184,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1184.png",
+      "answer": "3",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;G: Parabola;Expression(G) = (x^2 = 4*y);Distance(Focus(G), Asymptote(C)) = 1/3",
+      "coords": {}
+    },
+    {
+      "index": 1185,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.4142135623730951,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1185.png",
+      "answer": "2*sqrt(3)+3*sqrt(2)",
+      "fact_expr": "P: Point;m: Number;Coordinate(P) = (m, 2);C: Hyperbola;Expression(C) = (x^2/2 - y^2/2 = 1);PointOnCurve(P, RightPart(C));F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;Q: Point;Intersection(LineSegmentOf(P, F1), LeftPart(C)) = Q",
+      "coords": {}
+    },
+    {
+      "index": 1186,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1186.png",
+      "answer": "sqrt(10)/ 2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;A: Point;PointOnCurve(A, G) = True;AngleOf(F1, A, F2) = ApplyUnit(90, degree);Abs(LineSegmentOf(A, F1)) = 3*Abs(LineSegmentOf(A, F2))",
+      "coords": {}
+    },
+    {
+      "index": 1187,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1187.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;a: Number;b: Number;H: Line;P: Point;Q: Point;F2: Point;A: Point;B: Point;F1: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) =F1;RightFocus(G)=F2;PointOnCurve(F1,H);IsPerpendicular(xAxis, H);Intersection(H, LeftPart(G)) = {A, B};Intersection(LineSegmentOf(A, F2), yAxis) = P ;Intersection(LineSegmentOf(B, F2), yAxis) = Q;Perimeter(TriangleOf(P, Q, F2)) = 16",
+      "coords": {}
+    },
+    {
+      "index": 1188,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1188.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;a: Number;a>0;Expression(G) = (-y^2/9 + x^2/a^2 = 1);Expression(Asymptote(G)) = ((2*y)+(3*(pm*x)) = 0)",
+      "coords": {}
+    },
+    {
+      "index": 1189,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 6.928203230275509,
+        "major_axis": "x",
+        "x_coef": 100.0,
+        "y_coef": 48.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1189.png",
+      "answer": "48",
+      "fact_expr": "C: Ellipse;Q: Point;F1: Point;F2: Point;Expression(C) = (x^2/100 + y^2/48 = 1);LeftFocus(C) =F1;RightFocus(C)=F2;PointOnCurve(Q, C);Abs(LineSegmentOf(Q, F1)) = (2/3)*Abs(LineSegmentOf(Q, F2))",
+      "coords": {}
+    },
+    {
+      "index": 1190,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1190.png",
+      "answer": "9",
+      "fact_expr": "C: Parabola;Expression(C) = (x^2 = 2*p*y);p: Number;p > 0;F: Point;Focus(C) = F;Distance(F, Directrix(C)) = 4;PointOnCurve(F, l) = True;PointOnCurve(R, l) = True;R: Point;Coordinate(R) = (m, 0);m: Number;l: Line;Intersection(C, l) = {P, Q};P: Point;Q: Point;VectorOf(R, P) = VectorOf(P, F)",
+      "coords": {}
+    },
+    {
+      "index": 1191,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1192,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1192.png",
+      "answer": "8",
+      "fact_expr": "G: Ellipse;H: Line;F2: Point;A: Point;B: Point;F1: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Focus(G)={F1,F2};PointOnCurve(F1, H);Intersection(H, G) = {A,B};Abs(LineSegmentOf(F2, A)) + Abs(LineSegmentOf(F2, B)) = 12",
+      "coords": {}
+    },
+    {
+      "index": 1193,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_ecc_mismatch",
+      "params": {
+        "a": 1.9257268905639648,
+        "b": 1.7993911322594118,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 2.5576271215184017e-15,
+        "converged": true,
+        "iterations": 2
+      },
+      "output_path": "results/test/ellipse/problem_1193.png",
+      "answer": "sqrt(3)/3",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = 1/2",
+      "coords": {},
+      "validation_reasons": [
+        "ellipse_ecc_mismatch"
+      ]
+    },
+    {
+      "index": 1194,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1194.png",
+      "answer": "{sqrt(7)/2, sqrt(21)/3}",
+      "fact_expr": "H: Ellipse;Expression(H) = (y^2/b^2 + x^2/a^2 = 1);Eccentricity(H) = 1/2;b: Number;a: Number;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 1195,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1195.png",
+      "answer": "3+2*sqrt(2)",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);Focus(G) = F;F: Point;PointOnCurve(F,H) = True;H: Line;Intersection(H,G) = {A,B};A: Point;B: Point",
+      "coords": {}
+    },
+    {
+      "index": 1196,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1197,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1198,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1198.png",
+      "answer": "a/2-p/2",
+      "fact_expr": "G: Parabola;p: Number;a:Number;A: Point;B: Point;M: Point;p>0;a>=2*p;Expression(G) = (y^2 = 2*(p*x));IsChordOf(LineSegmentOf(A,B),G);Length(LineSegmentOf(A,B))=a;MidPoint(LineSegmentOf(A, B)) = M",
+      "coords": {}
+    },
+    {
+      "index": 1199,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1199.png",
+      "answer": "4",
+      "fact_expr": "C: Parabola;p: Number;A: Point;Expression(C) = (y^2 = 2*(p*x));Coordinate(A) = (-2, 0);PointOnCurve(A, Directrix(C))",
+      "coords": {
+        "A": [
+          -2.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 1200,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1200.png",
+      "answer": "2*sqrt(15)",
+      "fact_expr": "H: Line;Expression(H) = (y = k*x - 2);k: Number;G: Parabola;Expression(G) = (y^2 = 8*x);A: Point;B: Point;Intersection(H, G) = {A, B};XCoordinate(MidPoint(LineSegmentOf(A, B))) = 2",
+      "coords": {}
+    },
+    {
+      "index": 1201,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 3.0,
+        "direction": "left"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1201.png",
+      "answer": "x=8/3",
+      "fact_expr": "G: Hyperbola;a: Number;H: Parabola;Expression(G) = (-y^2 + x^2/a^2 = 1);Expression(H) = (y^2 = -12*x);LeftFocus(G) = Focus(H)",
+      "coords": {}
+    },
+    {
+      "index": 1202,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
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+          0.0
+        ],
+        "radius": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/circle/problem_1202.png",
+      "answer": "{sqrt(6),sqrt(2)}",
+      "fact_expr": "l: Line;C: Circle;A: Point;O: Origin;B: Point;b: Number;Expression(C) = (x^2 + y^2 = 4);Slope(l)=1;b>0;Intercept(l,yAxis)=b;Intersection(l, C) = {A, B};Area(TriangleOf(A, O, B)) = sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 1203,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.2135674953460693,
+        "b": 2.4270472526550293,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 4.453341261978494e-06,
+        "converged": true,
+        "iterations": 125
+      },
+      "output_path": "results/test/hyperbola/problem_1203.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;F1: Point;P: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C)=F1;RightFocus(C)=F2;PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(F1, P), LineSegmentOf(F2, P));Area(TriangleOf(P, F1, F2)) = 8;Eccentricity(C)=sqrt(5)",
+      "coords": {}
+    },
+    {
+      "index": 1204,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1204.png",
+      "answer": "(1,(2/3)*sqrt(3))",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;F: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F;PointOnCurve(F, H);Inclination(H) = ApplyUnit(30, degree);NumIntersection(H, RightPart(G)) = 2",
+      "coords": {}
+    },
+    {
+      "index": 1205,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 3.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1205.png",
+      "answer": "4",
+      "fact_expr": "P: Point;G: Parabola;Expression(G) = (y^2 = 12*x);PointOnCurve(P, G) = True;Q: Point;H: Circle;Expression(H) = (x^2 + (y - 4)^2 = 1);PointOnCurve(Q, H) = True;I: Line;Expression(I) = (x = -3)",
+      "coords": {}
+    },
+    {
+      "index": 1206,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1207,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1207.png",
+      "answer": "3/2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x);A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G);PointOnCurve(Focus(G), LineSegmentOf(A, B));Abs(LineSegmentOf(A, B)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 1208,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1209,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1210,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1210.png",
+      "answer": "3",
+      "fact_expr": "G: Parabola;A: Point;Expression(G) = (x^2 = 4*y);Coordinate(A) = (2*sqrt(2), 2);PointOnCurve(A,G)",
+      "coords": {
+        "A": [
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+          2.0
+        ]
+      }
+    },
+    {
+      "index": 1211,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1211.png",
+      "answer": "",
+      "fact_expr": "C: Parabola;Expression(C) = (x^2 = 2*y);F: Point;Focus(C) = F;l: Line;Directrix(C) = l",
+      "coords": {}
+    },
+    {
+      "index": 1212,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1213,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
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+          -0.0
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+        "radius": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/circle/problem_1213.png",
+      "answer": "(y^2 = 80 - 20*x)&(x <= 4)",
+      "fact_expr": "G: Circle;H: Line;Z: Circle;Expression(G) = (2*x + x^2 + y^2 - 15 = 0);Expression(H) = (x = 5);IsTangent(Z, H);IsOutTangent(Z, G)",
+      "coords": {}
+    },
+    {
+      "index": 1214,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1214.png",
+      "answer": "x^2/4-y^2/3=1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;H: Ellipse;Expression(H) = (x^2/16 + y^2/9 = 1);Focus(G) = Focus(H);Eccentricity(G) = Eccentricity(H)*2",
+      "coords": {}
+    },
+    {
+      "index": 1215,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1216,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1216.png",
+      "answer": "2*x+y-17=0",
+      "fact_expr": "G: Ellipse;H: LineSegment;P: Point;Expression(G) = (x^2 + 4*y^2 = 4);Coordinate(P) = (8, 1);MidPoint(H)=P;IsChordOf(H,G)",
+      "coords": {
+        "P": [
+          8.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 1217,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1217.png",
+      "answer": "4",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 + y^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G)",
+      "coords": {}
+    },
+    {
+      "index": 1218,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1218.png",
+      "answer": "-5",
+      "fact_expr": "P: Point;Coordinate(P) = (x1, y1);x1: Number;y1: Number;G: Curve;PointOnCurve(P, G);Expression(G) = (x^2/4 + y^2/3 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 1219,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1220,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1220.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;F1: Point;P: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G) = F2;AngleOf(F1, P, F2) = ApplyUnit(90, degree);Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 1221,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1221.png",
+      "answer": "3",
+      "fact_expr": "G: Parabola;A: Point;P: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (2, sqrt(15));PointOnCurve(P, G);d1:Number;d2:Number;Distance(P, yAxis) = d1;Distance(P, A) = d2",
+      "coords": {
+        "A": [
+          2.0,
+          3.872983346207417
+        ]
+      }
+    },
+    {
+      "index": 1222,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1222.png",
+      "answer": "25/3, 3",
+      "fact_expr": "G: Ellipse;m: Number;m>0;e: Number;Expression(G) = (x^2/5 + y^2/m = 1);Eccentricity(G) = e;e = sqrt(10)/5",
+      "coords": {}
+    },
+    {
+      "index": 1223,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.535473169108841,
+        "b": 3.535473169108841,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 4.3822798825688845e-07,
+        "converged": true,
+        "iterations": 93
+      },
+      "output_path": "results/test/hyperbola/problem_1223.png",
+      "answer": "(x^2/9-y^2/16=1)&(x>0)",
+      "fact_expr": "G: Hyperbola;F1: Point;F2: Point;Coordinate(F1) = (-5, 0);Coordinate(F2) = (5, 0);PointOnCurve(P,H);Distance(P,F1)-Distance(P,F2)=6;P:Point",
+      "coords": {
+        "F1": [
+          -5.0,
+          0.0
+        ],
+        "F2": [
+          5.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 1224,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1224.png",
+      "answer": "sqrt(3)/3",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;O: Origin;Center(C) = O;C: Ellipse;OneOf(Focus(C)) = F;Q: Point;OneOf(Focus(C)) = Q;Negation(F = Q);Eccentricity(C) = sqrt(2)/2;P: Point;PointOnCurve(P, C) = True;AngleOf(F, P, Q) = ApplyUnit(60, degree)",
+      "coords": {}
+    },
+    {
+      "index": 1225,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1225.png",
+      "answer": "2/3",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;P: Point;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Focus(G)={F1,F2};PointOnCurve(P,G) ;AngleOf(F1, P, F2) = pi/3;R:Number;r:Number;Radius(CircumCircle(TriangleOf(F1,P,F2)))=R;Radius(InscribedCircle(TriangleOf(F1,P,F2)))=r;R=4*r",
+      "coords": {}
+    },
+    {
+      "index": 1226,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1226.png",
+      "answer": "1",
+      "fact_expr": "G: Parabola;p: Number;l: Line;B: Point;A: Point;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;PointOnCurve(F,l);Inclination(l)=ApplyUnit(30,degree);Intersection(l, G) = {A, B};Length(LineSegmentOf(A,B))=8",
+      "coords": {}
+    },
+    {
+      "index": 1227,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1227.png",
+      "answer": "6",
+      "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(Focus(G), l);Intersection(l,G)={A,B};XCoordinate(MidPoint(LineSegmentOf(A,B))) = 2",
+      "coords": {}
+    },
+    {
+      "index": 1228,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1228.png",
+      "answer": "(4,2)",
+      "fact_expr": "A: Point;B: Point;G: Parabola;H: Line;Expression(G) = (y^2 = 4*x);Expression(H) = (x - y = 2);Intersection(H, G) = {A,B}",
+      "coords": {}
+    },
+    {
+      "index": 1229,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1229.png",
+      "answer": "pm*sqrt(7)",
+      "fact_expr": "C: Ellipse;l: Line;m: Real;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/4 + y^2/3 = 1);Expression(l) = (y = x + m);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, l);Min(Abs(LineSegmentOf(P, F1)) + Abs(LineSegmentOf(P, F2))) = 4",
+      "coords": {}
+    },
+    {
+      "index": 1230,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1230.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;H: Line;Expression(H) = (2*x - y + 1 = 0);IsParallel(OneOf(Asymptote(G)), H) = True",
+      "coords": {}
+    },
+    {
+      "index": 1231,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.23606797749979,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 5.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1231.png",
+      "answer": "8*sqrt(5)/5",
+      "fact_expr": "G: Ellipse;H: Line;t: Number;F: Point;M: Point;N: Point;Expression(G) = (x^2/5 + y^2/4 = 1);Expression(H) = (x = t);LeftFocus(G)=F;Intersection(H, G) = {M, N};WhenMax(Perimeter(TriangleOf(F,M,N)))",
+      "coords": {}
+    },
+    {
+      "index": 1232,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1232.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 + y^2/2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) - Abs(LineSegmentOf(P, F2)) = 2",
+      "coords": {}
+    },
+    {
+      "index": 1233,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1233.png",
+      "answer": "4",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);Focus(G) = {F1, F2};F1: Point;F2: Point;A: Point;PointOnCurve(A, G)",
+      "coords": {}
+    },
+    {
+      "index": 1234,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1234.png",
+      "answer": "(sqrt(5),+oo)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = 2*x);IsIntersect(G, H)",
+      "coords": {}
+    },
+    {
+      "index": 1235,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1236,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1236.png",
+      "answer": "1",
+      "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 + y^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0",
+      "coords": {}
+    },
+    {
+      "index": 1237,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1237.png",
+      "answer": "5*sqrt(2)+5",
+      "fact_expr": "G: Hyperbola;H: Circle;M: Point;F1: Point;F2:Point;A: Point;Expression(H) = (x^2 + (y - 5)^2 = 1);Expression(G) = (x^2/9 - y^2/16 = 1);PointOnCurve(A, H);LeftFocus(G) =F1;RightFocus(G)=F2;PointOnCurve(M, RightPart(G))",
+      "coords": {}
+    },
+    {
+      "index": 1238,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 3.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1238.png",
+      "answer": "25/3",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 12*x);F: Point;Focus(G)=F;M: Point;PointOnCurve(M,G);L:Line;PointOnCurve(M,L);IsPerpendicular(Directrix(G),L);N: Point;FootPoint(Directrix(G),L)=N;Abs(LineSegmentOf(N, F)) = 10",
+      "coords": {}
+    },
+    {
+      "index": 1239,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1239.png",
+      "answer": "pi/2",
+      "fact_expr": "G: Parabola;p: Number;H: Line;A1: Point;F: Point;B1: Point;A: Point;B: Point;p>0;Expression(G) = (y^2 = 2*p*x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};Projection(A,Directrix(G))=A1;Projection(B,Directrix(G))=B1",
+      "coords": {}
+    },
+    {
+      "index": 1240,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1240.png",
+      "answer": "{1,16}",
+      "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2/4 + y^2/m = 1);Eccentricity(G) = sqrt(3)/2",
+      "coords": {}
+    },
+    {
+      "index": 1241,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.8284271247461903,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 8.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1241.png",
+      "answer": "(0,4)",
+      "fact_expr": "l: Line;C: Ellipse;O: Origin;M: Point;N: Point;Q: Point;lambda:Real;Expression(C) = (x^2/8 + y^2/4 = 1);Expression(l) = (y = 2*m + x/m);m:Number;Negation(m=0);Intersection(l, C) = {M, N};PointOnCurve(Q, C);VectorOf(O, M) + VectorOf(O, N) = lambda*VectorOf(O, Q);Negation(lambda=0)",
+      "coords": {}
+    },
+    {
+      "index": 1242,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1242.png",
+      "answer": "pm*4/3",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, C) = {A, B};Abs(LineSegmentOf(A, F)) = 4*Abs(LineSegmentOf(B, F))",
+      "coords": {}
+    },
+    {
+      "index": 1243,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_focus_mismatch",
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1243.png",
+      "answer": "-4",
+      "fact_expr": "G: Hyperbola;m: Real;F1: Point;Expression(G) = (y^2/12 + x^2/m = 1);Coordinate(F1) = (0, 4);OneOf(Focus(G)) = F1",
+      "coords": {
+        "F1": [
+          0.0,
+          4.0
+        ]
+      },
+      "validation_reasons": [
+        "ellipse_focus_mismatch"
+      ]
+    },
+    {
+      "index": 1244,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1244.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;M: Point;F1: Point;F2: Point;e: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(M, LeftPart(G));Abs(LineSegmentOf(M, F1)) = 2*Abs(LineSegmentOf(F1, F2));Cos(AngleOf(M, F1, F2)) = -5/16;Eccentricity(G) = e",
+      "coords": {}
+    },
+    {
+      "index": 1245,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 6.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 36.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1245.png",
+      "answer": "sqrt(10)",
+      "fact_expr": "G: Ellipse;l: Line;M: Point;Expression(G) = (x^2/36 + y^2/9 = 1);Coordinate(M) = (4, 2);MidPoint(InterceptChord(l, G)) = M",
+      "coords": {
+        "M": [
+          4.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 1246,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1246.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "H: Line;Expression(H) = (x - 2*y + 3 = 0);G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Intersection(H, G) = {A, B};A: Point;B: Point;P: Point;Coordinate(P) = (-1, 1);MidPoint(LineSegmentOf(A, B)) = P",
+      "coords": {
+        "P": [
+          -1.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 1247,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1247.png",
+      "answer": "1",
+      "fact_expr": "G: Parabola;p: Number;P: Point;M: Point;F: Point;l: Line;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;Directrix(G) = l;PointOnCurve(P, G);PointOnCurve(P,LineSegmentOf(P,M));IsPerpendicular(LineSegmentOf(P,M),l);FootPoint(LineSegmentOf(P,M),l)=M;Abs(LineSegmentOf(M, P)) = Abs(LineSegmentOf(M, F));Area(TriangleOf(P, M, F)) = sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 1248,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1248.png",
+      "answer": "2*sqrt(5)",
+      "fact_expr": "C: Parabola;Expression(C) = (x^2 = 4*y);F: Point;Focus(C) = F;A: Point;PointOnCurve(A, C) = True;PointOnCurve(A, Z) = True;l: Line;Directrix(C) = l;IsPerpendicular(Z, l) = True;Z: Line;FootPoint(Z, l) = B;B: Point;Intersection(LineSegmentOf(F, B), xAxis) = D;D: Point;Abs(LineSegmentOf(A, F)) = 5",
+      "coords": {}
+    },
+    {
+      "index": 1249,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1249.png",
+      "answer": "2",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Inclination(OneOf(Asymptote(C))) = ApplyUnit(120, degree)",
+      "coords": {}
+    },
+    {
+      "index": 1250,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1250.png",
+      "answer": "6",
+      "fact_expr": "C: Parabola;G: Line;A: Point;B: Point;M: Point;E: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;Intersection(Directrix(C), xAxis) = E;PointOnCurve(F, G);Intersection(G, C) = {A, B};MidPoint(LineSegmentOf(A,B)) = M;Abs(LineSegmentOf(M, E)) = sqrt(11)",
+      "coords": {}
+    },
+    {
+      "index": 1251,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1251.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "C: Hyperbola;F1: Point;P: Point;F2: Point;O: Origin;Expression(C) = (x^2 - y^2/3 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);Tan(AngleOf(F1, P, F2)) = 4*sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 1252,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1252.png",
+      "answer": "9",
+      "fact_expr": "C: Ellipse;M: Point;F1: Point;F2: Point;Expression(C) = (x^2/9 + y^2/4 = 1);Focus(C) = {F1, F2};PointOnCurve(M, C)",
+      "coords": {}
+    },
+    {
+      "index": 1253,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "y",
+        "x_coef": 1.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1253.png",
+      "answer": "x^2+2*y^2=1",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2 + y^2/2 = 1);P: Point;L: Line;PointOnCurve(P,G);PointOnCurve(P, L);IsPerpendicular(L, xAxis);Q: Point;FootPoint(L, xAxis) = Q;M: Point;MidPoint(LineSegmentOf(P, Q)) = M",
+      "coords": {}
+    },
+    {
+      "index": 1254,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_focus_mismatch",
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "vertical"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1254.png",
+      "answer": "16",
+      "fact_expr": "G: Hyperbola;m: Number;F: Point;Expression(G) = (-x^2/9 + y^2/m = 1);Coordinate(F) = (0, 5);OneOf(Focus(G)) = F",
+      "coords": {
+        "F": [
+          0.0,
+          5.0
+        ]
+      },
+      "validation_reasons": [
+        "hyperbola_focus_mismatch"
+      ]
+    },
+    {
+      "index": 1255,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1255.png",
+      "answer": "4*sqrt(3)/3",
+      "fact_expr": "G: Ellipse;H: Line;k: Number;Expression(G) = (x^2/4 + y^2 = 1);Expression(H) = (y = k*x + 1)",
+      "coords": {}
+    },
+    {
+      "index": 1256,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1257,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1257.png",
+      "answer": "x=5*p/2",
+      "fact_expr": "G: Parabola;p: Number;B: Point;A: Point;O: Origin;p>0;Expression(G) = (y^2 = 2*p*x);PointOnCurve(A,G);PointOnCurve(B,G);Abs(LineSegmentOf(O, A)) = Abs(LineSegmentOf(O, B));Orthocenter(TriangleOf(A,O,B))=Focus(G)",
+      "coords": {}
+    },
+    {
+      "index": 1258,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1258.png",
+      "answer": "(2,pm*2*sqrt(2))",
+      "fact_expr": "C: Parabola;P: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(P, C);Abs(LineSegmentOf(P, F)) = 3",
+      "coords": {}
+    },
+    {
+      "index": 1259,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1259.png",
+      "answer": "2*sqrt(3)",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, C) = {A, B};Abs(LineSegmentOf(A, F)) = Abs(LineSegmentOf(B, F));G: Circle;IsDiameter(LineSegmentOf(A, B), G)",
+      "coords": {}
+    },
+    {
+      "index": 1260,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1260.png",
+      "answer": "2*sqrt(2)/3",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 1261,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1261.png",
+      "answer": "sqrt(3)-1",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b + x^2/a = 1);b: Number;a: Number;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);AngleOf(P, F1, F2) = pi/6;AngleOf(P, F2, F1) = pi/3;e: Number;Eccentricity(G) = e",
+      "coords": {}
+    },
+    {
+      "index": 1262,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.366584062576294,
+        "b": 2.364173412322998,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.00017838670464698225,
+        "converged": true,
+        "iterations": 89
+      },
+      "output_path": "results/test/hyperbola/problem_1262.png",
+      "answer": "1/2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;A: Point;F: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (-a*y + b*x = 0);Eccentricity(G) = 2;RightVertex(G) = A;RightFocus(G) = F;d1: Number;d2: Number;Distance(A, H) = d1;Distance(F, H) = d2",
+      "coords": {}
+    },
+    {
+      "index": 1263,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1263.png",
+      "answer": "125/16",
+      "fact_expr": "k: Number;Slope(l) = k;k = 3/4;l: Line;G: Parabola;Expression(G) = (x^2 = 4*y);PointOnCurve(Focus(G), l);A: Point;B: Point;Intersection(l, G) = {A, B};M: Point;Intersection(TangentOfPoint(A, G), TangentOfPoint(B, G)) = M",
+      "coords": {}
+    },
+    {
+      "index": 1264,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1264.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;H: Point;XCoordinate(H) = 6;PointOnCurve(H, G);Distance(H, Focus(G)) = 8",
+      "coords": {}
+    },
+    {
+      "index": 1265,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1265.png",
+      "answer": "y=pm*(sqrt(5)/2)*x",
+      "fact_expr": "E: Ellipse;Expression(E) = (x^2/11 + y^2/2 = 1);C: Hyperbola;Expression(C) = (-y^2/5 + x^2/a^2 = 1);a: Number;a>0;Focus(E) = Focus(C)",
+      "coords": {}
+    },
+    {
+      "index": 1266,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1266.png",
+      "answer": "",
+      "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2/5 + y^2/m = 1)",
+      "coords": {}
+    },
+    {
+      "index": 1267,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "down"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1267.png",
+      "answer": "(0, -1/2)",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = -2*y)",
+      "coords": {}
+    },
+    {
+      "index": 1268,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1268.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;k: Real;Expression(G) = (-y^2/4 + x^2/k = 1);Coordinate(OneOf(Focus(G)))=(sqrt(6),0)",
+      "coords": {}
+    },
+    {
+      "index": 1269,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1270,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1270.png",
+      "answer": "pm*4",
+      "fact_expr": "G: Parabola;a: Number;H: Ellipse;Expression(G) = (y^2 = a*x);Expression(H) = (x^2/4 + y^2/3 = 1);OneOf(Focus(H)) = Focus(G)",
+      "coords": {}
+    },
+    {
+      "index": 1271,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1271.png",
+      "answer": "2",
+      "fact_expr": "l: Line;Slope(l) = 1;C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;B: Point;D: Point;Intersection(l, C) = {B, D};M: Point;Coordinate(M)=(1,3);MidPoint(LineSegmentOf(B, D)) = M",
+      "coords": {
+        "M": [
+          1.0,
+          3.0
+        ]
+      }
+    },
+    {
+      "index": 1272,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1272.png",
+      "answer": "1/3",
+      "fact_expr": "Gamma: Parabola;l: Line;O: Origin;F: Point;A: Point;B: Point;Expression(Gamma) = (y^2 = 4*x);Expression(l) = (sqrt(3)*x - y - sqrt(3) = 0);Intersection(l,Gamma)={A,B};Intersection(l,xAxis)=F;lambda:Number;mu:Number;lambda<=mu;VectorOf(O, F) = lambda*VectorOf(O, A) + mu*VectorOf(O, B)",
+      "coords": {}
+    },
+    {
+      "index": 1273,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1274,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1274.png",
+      "answer": "sqrt(5)/3",
+      "fact_expr": "C: Ellipse;G: Circle;a:Number;b: Number;H: Line;P: Point;F1: Point;F2: Point;A: Point;Q: Point;Expression(C)=(y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2 + y^2 = b^2);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F2, H);TangentPoint(H,G)=A;Intersection(H,C) = {P,Q};IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2));a>b;b>0",
+      "coords": {}
+    },
+    {
+      "index": 1275,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1275.png",
+      "answer": "2",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;A: Point;PointOnCurve(A, C);Abs(LineSegmentOf(A, F)) = 5;O: Origin",
+      "coords": {}
+    },
+    {
+      "index": 1276,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.23606797749979,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 5.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1276.png",
+      "answer": "12",
+      "fact_expr": "G: Ellipse;l1: Line;l2:Line;A: Point;F: Point;B: Point;C: Point;D: Point;Expression(G) = (x^2/9 + y^2/5 = 1);Expression(l1) = (x - y - 2 = 0);Expression(l2)=(x-y+2=0);LeftFocus(G) = F;Intersection(l1,G)={A,B};Intersection(l2,G)={C,D}",
+      "coords": {}
+    },
+    {
+      "index": 1277,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1278,
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+        "note": "using explicit params"
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+      "answer": "(0, 1/32)",
+      "fact_expr": "G: Parabola;Expression(G) = (y = 8*x^2)",
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+    {
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+        "note": "using explicit params"
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+      "output_path": "results/test/hyperbola/problem_1280.png",
+      "answer": "1/2",
+      "fact_expr": "H: Line;Expression(H) = (k*x - y + 1 = 0);k: Number;G: Hyperbola;Expression(G) = (x^2/2 - y^2 = 1);Intersection(H, G) = {A, B};A: Point;B: Point;Negation(A = B);M: Point;Coordinate(M) = (3, 0);PointOnCurve(M, xAxis);Distance(M, A) = Distance(M, B)",
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+        "note": "using explicit params"
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+      "answer": "sqrt(17)",
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+      "error": "Unsupported or unparseable expression"
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+        "note": "using explicit params"
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+      "output_path": "results/test/parabola/problem_1283.png",
+      "answer": "pi/3",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;M: Point;Coordinate(M) = (x1, y1);x1: Number;y1: Number;N: Point;Coordinate(N) = (x2, y2);x2: Number;y2: Number;PointOnCurve(M, C);PointOnCurve(N, C);x1 + x2 + 2 = 2*Abs(LineSegmentOf(M, N))",
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+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1284.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Hyperbola;a: Number;H: Ellipse;F1: Point;F2: Point;P:Point;Expression(G) = (-y^2 + x^2/a^2 = 1);Expression(H) = (x^2/6 + y^2/2 = 1);Focus(G)={F1,F2};Focus(H)={F1,F2};OneOf(Intersection(H,G))=P",
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+    {
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+        "iterations": 125
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+      "output_path": "results/test/hyperbola/problem_1285.png",
+      "answer": "y = pm*x/2",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-x^2/b^2 + y^2/a^2 = 1);Eccentricity(C) = sqrt(5)",
+      "coords": {}
+    },
+    {
+      "index": 1286,
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1286.png",
+      "answer": "3",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);H: Line;D: Point;Coordinate(D) = (2, 0);PointOnCurve(D, H);A: Point;B: Point;Intersection(H, G) = {A, B};Length(LineSegmentOf(A, B)) = 10;C: Point;MidPoint(LineSegmentOf(A, B)) = C",
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+          0.0
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+      }
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+    {
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+        "y_coef": 1.0
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+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1287.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/3 + y^2 = 1);P: Point;PointOnCurve(P, G);H: Line;Expression(H) = (x - y + 6 = 0)",
+      "coords": {}
+    },
+    {
+      "index": 1288,
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+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1288.png",
+      "answer": "sqrt(13)+2*sqrt(3)",
+      "fact_expr": "G: Hyperbola;M: Point;P: Point;F: Point;Expression(G) = (x^2/3 - y^2 = 1);Coordinate(M) = (4, 3);LeftFocus(G) = F;PointOnCurve(P, RightPart(G))",
+      "coords": {
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+          3.0
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+      }
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+    {
+      "index": 1289,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1289.png",
+      "answer": "sqrt(7)/2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;A: Point;B: Point;LeftVertex(G) = A;RightVertex(G) = B;C: Point;Coordinate(C) = (0, 2*b);PointOnCurve(B, PerpendicularBisector(LineSegmentOf(A, C)))",
+      "coords": {}
+    },
+    {
+      "index": 1290,
+      "success": true,
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+      "error": null,
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+        "b": 1.0,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1290.png",
+      "answer": "y^2/8-x^2/32=1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - 4*y^2 = 4);Z: Hyperbola;Asymptote(G) = Asymptote(Z);H: Point;Coordinate(H) = (2, 3);PointOnCurve(H, Z)",
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+        "H": [
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+          3.0
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+    },
+    {
+      "index": 1291,
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+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1291.png",
+      "answer": "y = pm*sqrt(2)*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;H: Line;PointOnCurve(F2, H) ;IsPerpendicular(H, xAxis) ;Intersection(H, G) = {A, B};A: Point;B: Point;Abs(LineSegmentOf(F1, F2)) = (sqrt(3)/2)*Abs(LineSegmentOf(A, B))",
+      "coords": {}
+    },
+    {
+      "index": 1292,
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+      "conic_type": "ellipse",
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+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1292.png",
+      "answer": "y=pm*sqrt(2)/2*x",
+      "fact_expr": "G: Hyperbola;H: Ellipse;P: Point;Expression(H) = (x^2/4 + y^2 = 1);Coordinate(P) = (2, 1);Focus(G) = Focus(H);PointOnCurve(P,G)",
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+          1.0
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+    },
+    {
+      "index": 1293,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1293.png",
+      "answer": "[sqrt(7)/2,sqrt(3)]",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;M: Point;F1: Point;F2: Point;a>0;b>0;lambda:Number;lambda>=1/3;lambda<=1/2;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(M, G);Abs(LineSegmentOf(M, F1)) = lambda*Abs(LineSegmentOf(M, F2));AngleOf(F1, M, F2) = ApplyUnit(60, degree)",
+      "coords": {}
+    },
+    {
+      "index": 1294,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1294.png",
+      "answer": "5*sqrt(2)/2",
+      "fact_expr": "G: Parabola;H: Line;M: Point;O: Origin;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 8*x);Coordinate(M) = (1, 0);PointOnCurve(M, H);Intersection(H, G) = {A, B};Focus(G) = F;Abs(LineSegmentOf(A, F)) = 6",
+      "coords": {
+        "M": [
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+          0.0
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+      }
+    },
+    {
+      "index": 1295,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 2.8284271247461903,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1295.png",
+      "answer": "8*sqrt(3)",
+      "fact_expr": "F1: Point;F2: Point;G: Hyperbola;Expression(G) = (x^2/4 - y^2/8 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);LineSegmentOf(P, F1) = 2*LineSegmentOf(P, F2)",
+      "coords": {}
+    },
+    {
+      "index": 1296,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1296.png",
+      "answer": "pm*2*sqrt(2)/3",
+      "fact_expr": "C: Parabola;G: Line;F: Point;A: Point;B: Point;k: Real;Expression(C) = (y^2 = 8*x);Expression(G) = (y = k*(x + 2));Intersection(G, C) = {A, B};Focus(C) = F;Abs(VectorOf(F, A)) = 2*Abs(VectorOf(F, B))",
+      "coords": {}
+    },
+    {
+      "index": 1297,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "down"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1297.png",
+      "answer": "(0,-3/2)",
+      "fact_expr": "G: Parabola;Expression(G) = (y = -x^2/6)",
+      "coords": {}
+    },
+    {
+      "index": 1298,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 1299,
+      "success": true,
+      "conic_type": "ellipse",
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+        "x_coef": 9.0,
+        "y_coef": 4.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1299.png",
+      "answer": "2*sqrt(5)",
+      "fact_expr": "G: Ellipse;Expression(G)=(x^2/9+y^2/4=1)",
+      "coords": {}
+    },
+    {
+      "index": 1300,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1300.png",
+      "answer": "(sqrt(2),+oo)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;A: Point;PointOnCurve(A, RightPart(G));Distance(F2, LineOf(A, F1)) = 2*a",
+      "coords": {}
+    },
+    {
+      "index": 1301,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1302,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1302.png",
+      "answer": "1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 1303,
+      "success": true,
+      "conic_type": "ellipse",
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+        "y_coef": 25.0
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+      "optimization": {
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+        "converged": true,
+        "iterations": 2
+      },
+      "output_path": "results/test/ellipse/problem_1303.png",
+      "answer": "y = pm*25/4",
+      "fact_expr": "G: Ellipse;O: Origin;Center(G) = O;Eccentricity(G) = 4/5;Coordinate(OneOf(Focus(G)))=(0,4)",
+      "coords": {}
+    },
+    {
+      "index": 1304,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1304.png",
+      "answer": "0",
+      "fact_expr": "G: Ellipse;H: Line;E: Point;N: Point;M: Point;P: Point;l:Line;Expression(G) = (x^2/2 + y^2 = 1);PointOnCurve(Center(G),H);Intersection(H,G)={M,N};Quadrant(M)=1;PointOnCurve(M,l);IsPerpendicular(l,xAxis);FootPoint(l,xAxis)=E;OneOf(Intersection(LineOf(N,E),G))=P",
+      "coords": {}
+    },
+    {
+      "index": 1305,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1305.png",
+      "answer": "sqrt(10)/2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;A: Point;B: Point;LeftVertex(G) = A;RightVertex(G) = B;C: Point;Coordinate(C) = (0, sqrt(2)*b);PointOnCurve(B, PerpendicularBisector(LineSegmentOf(A, C)))",
+      "coords": {}
+    },
+    {
+      "index": 1306,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1306.png",
+      "answer": "(9/4)*p",
+      "fact_expr": "l: Line;C: Parabola;p: Number;A: Point;B: Point;M: Point;F: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};Intersection(l,Directrix(C)) = M;VectorOf(A, B) + VectorOf(A, M) = 0",
+      "coords": {}
+    },
+    {
+      "index": 1307,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1307.png",
+      "answer": "pm*1",
+      "fact_expr": "l: Line;C: Parabola;B: Point;A: Point;P: Point;F: Point;Q: Point;Expression(C) = (y^2 = 4*x);Coordinate(P) = (-1, 0);Focus(C) = F;PointOnCurve(P, l);Intersection(l, C) = {A, B};MidPoint(LineSegmentOf(A,B)) = Q;Abs(LineSegmentOf(F, Q)) = 2",
+      "coords": {
+        "P": [
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+          0.0
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+      }
+    },
+    {
+      "index": 1308,
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "results/test/parabola/problem_1308.png",
+      "answer": "{pi/3,2*pi/3}",
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+    {
+      "index": 1309,
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true
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+      "output_path": "results/test/parabola/problem_1309.png",
+      "answer": "sqrt(2)/2",
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+      "error": "Unsupported or unparseable expression"
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 1312,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1313,
+      "success": true,
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+        "note": "using explicit params"
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+      "answer": "(1,5/3]",
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+    {
+      "index": 1314,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "results/test/hyperbola/problem_1314.png",
+      "answer": "sqrt(15)/3",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;A: Point;P: Point;B: Point;O:Origin;e: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(A, G);PointOnCurve(B, G);PointOnCurve(P, G);Negation(A=B);Negation(A=P);Negation(B=P);PointOnCurve(O,LineSegmentOf(A,B));Slope(LineOf(P, B)) = k2;Slope(LineOf(P,A))=k1;k1:Number;k2:Number;k1*k2=2/3;Eccentricity(G) = e",
+      "coords": {}
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+    {
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+        "x_coef": 25.0,
+        "y_coef": 16.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "results/test/ellipse/problem_1315.png",
+      "answer": "100/3",
+      "fact_expr": "G: Parabola;H: Ellipse;A: Point;B: Point;Expression(H) = (x^2/25 + y^2/16 = 1);Vertex(G)=Center(H);LeftDirectrix(H)=Directrix(G);Intersection(RightDirectrix(H),G)={A,B}",
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+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1316.png",
+      "answer": "y^2/20+x^2/4=1",
+      "fact_expr": "G: Ellipse;H: Point;Expression(G) = (x^2/9 + y^2/25 = 1);Coordinate(H) = (sqrt(3), sqrt(5));C:Ellipse;Focus(C)=Focus(G);PointOnCurve(H,C)",
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+        "note": "using explicit params"
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+      "answer": "1",
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+    {
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+      "output_path": "results/test/hyperbola/problem_1318.png",
+      "answer": "x^2/3 - y^2/6 = 1",
+      "fact_expr": "G: Hyperbola;F1: Point;F2: Point;Coordinate(F1) = (-3, 0);Coordinate(F2) = (3, 0);Focus(G) = {F1, F2};Length(ImageinaryAxis(G)) = sqrt(2)*Length(RealAxis(G))",
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+        "F2": [
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+      "index": 1319,
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1319.png",
+      "answer": "pm*(sqrt(3)/3)",
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+    {
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+        "direction": "up"
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+      "optimization": {
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+        "converged": true,
+        "iterations": 81
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+      "output_path": "results/test/parabola/problem_1320.png",
+      "answer": "x^2=4*y",
+      "fact_expr": "C: Parabola;O: Origin;Vertex(C) = O;PointOnCurve(Focus(C), yAxis);G: Point;Coordinate(G) = (2, 1);PointOnCurve(G, C)",
+      "coords": {
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+          1.0
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+      "index": 1321,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 1322,
+      "success": true,
+      "conic_type": "ellipse",
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+        "y_coef": 1.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "results/test/ellipse/problem_1322.png",
+      "answer": "(pm*sqrt(3),0)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2 + 4*y^2 = 4)",
+      "coords": {}
+    },
+    {
+      "index": 1323,
+      "success": true,
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+        "y_coef": 3.0
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+        "note": "using explicit params"
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+      "output_path": "results/test/ellipse/problem_1323.png",
+      "answer": "(0,sqrt(3)-1]",
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+      "coords": {}
+    },
+    {
+      "index": 1324,
+      "success": true,
+      "conic_type": "parabola",
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1324.png",
+      "answer": "(4, 2)",
+      "fact_expr": "G: Parabola;H: Line;B: Point;A: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (x - y = 2);Intersection(H, G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 1325,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_ecc_mismatch",
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+        "converged": true,
+        "iterations": 2
+      },
+      "output_path": "results/test/ellipse/problem_1325.png",
+      "answer": "{16/3,3}",
+      "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2/4 + y^2/m = 1);Eccentricity(G) = 1/2",
+      "coords": {},
+      "validation_reasons": [
+        "ellipse_ecc_mismatch"
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+    },
+    {
+      "index": 1326,
+      "success": true,
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+        "converged": true
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+      "output_path": "results/test/parabola/problem_1326.png",
+      "answer": "x=-1",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;Coordinate(Focus(G)) = (1,0)",
+      "coords": {}
+    },
+    {
+      "index": 1327,
+      "success": true,
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+        "direction": "up"
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+        "note": "using explicit params"
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+      "output_path": "results/test/parabola/problem_1327.png",
+      "answer": "(0,1/16)",
+      "fact_expr": "G: Parabola;F: Point;Expression(G) = (y = 4*x^2);Focus(G) = F",
+      "coords": {}
+    },
+    {
+      "index": 1328,
+      "success": true,
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "results/test/parabola/problem_1328.png",
+      "answer": "8",
+      "fact_expr": "G: Hyperbola;m: Number;H: Parabola;p: Real;Expression(G) = (x^2/(12 - m) - y^2/(m + 4) = 1);p>0;Expression(H) = (y^2 = 2*(p*x));Focus(H) = RightFocus(G)",
+      "coords": {}
+    },
+    {
+      "index": 1329,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1329.png",
+      "answer": "8*x - y - 15 = 0",
+      "fact_expr": "G: Parabola;H: Line;P: Point;A: Point;B: Point;Expression(G) = (y^2 = 16*x);Coordinate(P) = (2, 1);PointOnCurve(P, H);Intersection(H, G) = {A, B};VectorOf(P, A) + VectorOf(P, B) = 0",
+      "coords": {
+        "P": [
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+          1.0
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+    {
+      "index": 1330,
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+      "error": null,
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+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "results/test/ellipse/problem_1330.png",
+      "answer": "1/3",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;H: Circle;e: Number;a > b;b > 0;Expression(H) = (y^2 + (x - 2)^2 = 1);PointOnCurve(OneOf(Focus(G)),H);PointOnCurve(OneOf(Vertex(G)),H);Eccentricity(G) = e",
+      "coords": {}
+    },
+    {
+      "index": 1331,
+      "success": true,
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+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1331.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;P: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(P,G);Distance(P, Focus(G)) = Distance(P, yAxis)*2",
+      "coords": {}
+    },
+    {
+      "index": 1332,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1333,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 2.4270472526550293,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 4.453341261978494e-06,
+        "converged": true,
+        "iterations": 125
+      },
+      "output_path": "results/test/hyperbola/problem_1333.png",
+      "answer": "-21/5",
+      "fact_expr": "A: Point;B: Point;G: Hyperbola;b: Number;P: Point;F1: Point;F2: Point;b>0;Expression(G) = (x^2/4 - y^2/b^2 = 1);Coordinate(B) = (0, b);Eccentricity(G)=sqrt(5)/2;Focus(G)={F1,F2};LeftVertex(G)=A;PointOnCurve(P, LineSegmentOf(A, B))",
+      "coords": {}
+    },
+    {
+      "index": 1334,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 4.0,
+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1334.png",
+      "answer": "8",
+      "fact_expr": "G: Hyperbola;k: Number;H: Parabola;Expression(G) = (x^2 - y^2 = k);Expression(H) = (y^2 = 16*x);OneOf(Focus(G)) = Focus(H)",
+      "coords": {}
+    },
+    {
+      "index": 1335,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1336,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 2.5275303393962245,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "iterations": 95
+      },
+      "output_path": "results/test/hyperbola/problem_1336.png",
+      "answer": "x^2-y^2/4=1",
+      "fact_expr": "C: Hyperbola;Expression(Asymptote(C)) = (y = pm*2*x)",
+      "coords": {}
+    },
+    {
+      "index": 1337,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 2.364173412322998,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "iterations": 89
+      },
+      "output_path": "results/test/hyperbola/problem_1337.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = 2;Distance(Focus(G),Asymptote(G))=sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 1338,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1339,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1340,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1340.png",
+      "answer": "x^2/5-y^2/20=1",
+      "fact_expr": "l: Line;Expression(l) = (y = 2*x + 10);G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;PointOnCurve(OneOf(Focus(G)), l);IsParallel(l, OneOf(Asymptote(G)))",
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+      "answer": "[3/2, +oo)",
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+      "answer": "1/4",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+        "note": "using explicit params"
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+      "output_path": "results/test/ellipse/problem_1356.png",
+      "answer": "x^2/7-y^2=1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2 + x^2/m = 1);m: Number;H: Ellipse;Expression(H) = (x^2/12 + y^2/4 = 1);Focus(G) = Focus(H)",
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+      "output_path": "results/test/hyperbola/problem_1357.png",
+      "answer": "",
+      "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (x^2/36 - y^2/m = 1);FocalLength(G) = 18",
+      "coords": {}
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+    {
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 1359,
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+      "optimization": {
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+        "note": "using explicit params"
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+      "output_path": "results/test/hyperbola/problem_1359.png",
+      "answer": "sqrt(2)",
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+        "converged": true
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+      "output_path": "results/test/parabola/problem_1360.png",
+      "answer": "1/9",
+      "fact_expr": "A: Point;M: Point;G: Hyperbola;a: Real;H: Parabola;p: Number;m: Number;Expression(G) = (-y^2 + x^2/a = 1);p>0;m > 0;Expression(H) = (y^2 = 2*(p*x));Coordinate(M) = (1, m);PointOnCurve(M, H);Distance(M, Focus(H)) = 5;LeftVertex(G) = A;IsParallel(OneOf(Asymptote(G)), LineOf(A, M))",
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+    {
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 1362,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
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+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "results/test/hyperbola/problem_1363.png",
+      "answer": "2*sqrt(3)/3",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);c: Number;FocalLength(G) = 2*c;Distance(Focus(G),Asymptote(G))=c/2",
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+    {
+      "index": 1364,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 1365,
+      "success": true,
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+      "output_path": "results/test/parabola/problem_1365.png",
+      "answer": "2\nx=-1",
+      "fact_expr": "G: Parabola;p: Number;Expression(G) = (y^2 = 2*(p*x));Coordinate(Focus(G)) = (1, 0)",
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+    {
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+        "note": "using explicit params"
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+      "answer": "sqrt(3)",
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+    {
+      "index": 1367,
+      "success": true,
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+        "converged": true
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+      "output_path": "results/test/parabola/problem_1367.png",
+      "answer": "9*sqrt(3)",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*p*x);p: Number;p>0;F: Point;Focus(C) = F;G: Ellipse;Expression(G) = (4*x^2/9 + y^2/b^2 = 1);b: Number;RightVertex(G) = F;l: Line;Directrix(C) = l;A: Point;PointOnCurve(A, C) = True;PointOnCurve(A, LineSegmentOf(A, B)) = True;IsPerpendicular(LineSegmentOf(A, B), l) = True;FootPoint(LineSegmentOf(A, B), l) = B;B: Point;Slope(LineOf(B, F)) = k_BF;k_BF: Number;k_BF = -sqrt(3)",
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+    {
+      "index": 1368,
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+        "note": "using explicit params"
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+      "output_path": "results/test/ellipse/problem_1368.png",
+      "answer": "4",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);M: Point;MidPoint(LineSegmentOf(F1, P)) = M;O: Origin;Abs(LineSegmentOf(O, M)) = 3",
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+    {
+      "index": 1369,
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+        "converged": true,
+        "note": "using explicit params"
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+      "answer": "sqrt(5)/2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: LineSegment;L:LineSegment;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(F,H);PointOnCurve(F,L);OneOf(Focus(G)) = F ;F: Point;L1:Line;L2:Line;Asymptote(G)={L1,L2};IsPerpendicular(H,L1);IsPerpendicular(L,L2);H + L = a",
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+    {
+      "index": 1370,
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "results/test/hyperbola/problem_1370.png",
+      "answer": "y^2=16*x",
+      "fact_expr": "Vertex(H) = O;O: Origin;G: Hyperbola;Expression(G) = (x^2/12 - y^2/4 = 1);RightFocus(G) = Focus(H);H: Parabola",
+      "coords": {}
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+    {
+      "index": 1371,
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+      "optimization": {
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+        "iterations": 122
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+      "output_path": "results/test/hyperbola/problem_1371.png",
+      "answer": "3*sqrt(5)/5",
+      "fact_expr": "G: Hyperbola;a: Number;b: Number;F1: Point;F2: Point;P: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-sqrt(3)/2, 0);Coordinate(F2) = (sqrt(3)/2, 0);Focus(G) = {F1, F2};Quadrant(P) = 1;PointOnCurve(P, G);Tan(AngleOf(P, F1, F2)) = 1/2;Tan(AngleOf(P, F2, F1)) = -2",
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+        "F2": [
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+      "index": 1372,
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+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "results/test/hyperbola/problem_1372.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Hyperbola;b: Number;H: Ellipse;Expression(G) = (x^2/8 - y^2/b^2 = 1);Expression(H) = (x^2/2 + y^2 = 1);Directrix(G)=Directrix(H)",
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+    },
+    {
+      "index": 1373,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 1374,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "major_axis": "x",
+        "x_coef": 3.0,
+        "y_coef": 1.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "results/test/ellipse/problem_1374.png",
+      "answer": "{(-sqrt(2)/2,sqrt(30)/6),(-sqrt(2)/2,-sqrt(30)/6)}",
+      "fact_expr": "G: Ellipse;F1: Point;A: Point;F2: Point;B: Point;Expression(G) = (x^2/3 + y^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(A, G);PointOnCurve(B, G);VectorOf(F1, A) = 3*VectorOf(F2, B)",
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+    },
+    {
+      "index": 1375,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 1376,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1376.png",
+      "answer": "4*sqrt(5)/5",
+      "fact_expr": "G: Parabola;l1:Line;l2:Line;P:Point;Expression(G) = (y^2 = 4*x);Expression(l1)=(2*x-y+2=0);Expression(l2)=(x=-1);PointOnCurve(P,G)",
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+    },
+    {
+      "index": 1377,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1377.png",
+      "answer": "4/3",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/m = 1);m: Number;Radius(H) = Eccentricity(G);Center(H) = RightFocus(G);H: Circle;IsTangent(H, OneOf(Asymptote(G))) = True",
+      "coords": {}
+    },
+    {
+      "index": 1378,
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+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "iterations": 2
+      },
+      "output_path": "results/test/hyperbola/problem_1378.png",
+      "answer": "9",
+      "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (x^2/16 - y^2/m = 1);Eccentricity(G) = 5/4",
+      "coords": {}
+    },
+    {
+      "index": 1379,
+      "success": true,
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+        "x_coef": 16.0,
+        "y_coef": 8.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1379.png",
+      "answer": "2*sqrt(42)/7",
+      "fact_expr": "G: Ellipse;H: Circle;A: Point;B: Point;Q: Point;l1: Line;l2: Line;Expression(G) = (x^2/16 + y^2/8 = 1);Expression(H) = (y^2 + (x - 1)^2 = 1);PointOnCurve(Q, G);TangentOfPoint(Q, H) = {l1, l2};TangentPoint(l1,H)=A;TangentPoint(l2,H)=B",
+      "coords": {}
+    },
+    {
+      "index": 1380,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "up"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1380.png",
+      "answer": "2",
+      "fact_expr": "H: Line;Expression(H) = (y = b + x);b: Number;G: Parabola;Expression(G) = (x^2 = 2*y);Intersection(H, G) = {A, B};A: Point;B: Point;O: Origin;IsPerpendicular(LineSegmentOf(O, A), LineSegmentOf(O, B)) = True",
+      "coords": {}
+    },
+    {
+      "index": 1381,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1381.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;A: Point;B: Point;O: Origin;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = -8*x);l1:Line;l2:Line;Asymptote(G)={l1,l2};Intersection(l1, Directrix(G)) = A;Intersection(l2, Directrix(G)) = B;Area(TriangleOf(A, B, O)) = 4*sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 1382,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1382.png",
+      "answer": "25/3",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);P: Point;PointOnCurve(P, G);H: Circle;Expression(H) = (x^2 + (y + 2)^2 = 1);A: Point;B: Point;IsDiameter(LineSegmentOf(A, B), H)",
+      "coords": {}
+    },
+    {
+      "index": 1383,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 2.364173412322998,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "iterations": 89
+      },
+      "output_path": "results/test/hyperbola/problem_1383.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;a: Number;b: Number;H: Parabola;p: Number;A: Point;B:Point;O: Origin;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);p>0;Expression(H) = (y^2 = 2*(p*x));L1:Line;L2:Line;Asymptote(G)={L1,L2};Intersection(L1,Directrix(H))=A;Intersection(L2,Directrix(H))=B;Eccentricity(G)=2;Area(TriangleOf(A,O,B))=sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 1384,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "vertical"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1384.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/b^2 + y^2/a^2 = 1);a: Number;b: Number;a>0;b>0;Expression(OneOf(Asymptote(G))) = (x - 2*y = 0)",
+      "coords": {}
+    },
+    {
+      "index": 1385,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1385.png",
+      "answer": "(1/2,2/3)",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;H: Line;A:Point;B: Point;e: Number;k:Number;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftVertex(C)=A;PointOnCurve(A,H);Slope(H)=k;Intersection(H,C)={A,B};RightFocus(C)=F;IsPerpendicular(LineSegmentOf(B,F),xAxis);1/3<k;k<1/2;Eccentricity(C) = e;F:Point",
+      "coords": {}
+    },
+    {
+      "index": 1386,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1386.png",
+      "answer": "[sqrt(3)/3,sqrt(2)/2]",
+      "fact_expr": "M: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;e: Number;c:Number;a > b;b > 0;Expression(M) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(M) =F1;RightFocus(M)=F2;PointOnCurve(P,M);Range(Max(Abs(LineSegmentOf(P,F1))*Abs(LineSegmentOf(P,F2))))=[2*c^2,3*c^2];Eccentricity(M) = e;c=sqrt(a^2-b^2)",
+      "coords": {}
+    },
+    {
+      "index": 1387,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "y",
+        "x_coef": 3.0,
+        "y_coef": 12.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1387.png",
+      "answer": "6",
+      "fact_expr": "G: Parabola;p: Number;H: Ellipse;p>0;Expression(G) = (x^2 = 2*(p*y));Expression(H) = (x^2/3 + y^2/12 = 1);Focus(G) = OneOf(Focus(H))",
+      "coords": {}
+    },
+    {
+      "index": 1388,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1388.png",
+      "answer": "sqrt(13)/2",
+      "fact_expr": "F: Point;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;LeftFocus(G) = F;PointOnCurve(F, l) = True;l: Line;H: Circle;Expression(H) = (x^2 + y^2 = a^2);IsTangent(l, H) = True;TangentPoint(l, H) = A;A: Point;Intersection(l, RightPart(G)) = B;B: Point;VectorOf(F, A) = VectorOf(F, B)/3",
+      "coords": {}
+    },
+    {
+      "index": 1389,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1389.png",
+      "answer": "(5/4,-3/4)",
+      "fact_expr": "G: Hyperbola;H: Line;P: Point;Expression(G) = (x^2 - y^2 = 1);Expression(H) = (y = x);PointOnCurve(P, RightPart(G));Distance(P, H) = sqrt(2)",
+      "coords": {}
+    },
+    {
+      "index": 1390,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1390.png",
+      "answer": "4/3",
+      "fact_expr": "l: Line;G: Parabola;E: Point;F: Point;P: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;Directrix(G) = l;PointOnCurve(P, G);PointOnCurve(P, LineSegmentOf(P, E));IsPerpendicular(LineSegmentOf(P, E), l);FootPoint(LineSegmentOf(P, E), l) = E;Inclination(LineOf(E, F)) = ApplyUnit(150, degree)",
+      "coords": {}
+    },
+    {
+      "index": 1391,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1391.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;F: Point;LeftFocus(G) = F;A: Point;RightVertex(G) = A;B: Point;Coordinate(B) = (0, b);P: Point;MidPoint(LineSegmentOf(A, B)) = P;Abs(LineSegmentOf(P, F)) = sqrt(6)*Abs(LineSegmentOf(O, P));O: Origin",
+      "coords": {}
+    },
+    {
+      "index": 1392,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1392.png",
+      "answer": "y=pm*(1/2)*x\nsqrt(5)/2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 1393,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1393.png",
+      "answer": "2",
+      "fact_expr": "G: Ellipse;M: Point;F1: Point;O: Origin;N: Point;Expression(G) = (x^2/25 + y^2/9 = 1);PointOnCurve(M,G);LeftFocus(G)=F1;Distance(M,F1)=6;MidPoint(LineSegmentOf(M,F1))=N",
+      "coords": {}
+    },
+    {
+      "index": 1394,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.3300262689590454,
+        "b": 2.30381441116333,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 6.658357051492203e-06,
+        "converged": true,
+        "iterations": 122
+      },
+      "output_path": "results/test/hyperbola/problem_1394.png",
+      "answer": "x^2/9 - y^2/27 = 1",
+      "fact_expr": "G: Hyperbola;a: Number;b: Number;H: Parabola;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 24*x);Expression(OneOf(Asymptote(G))) = (y = sqrt(3)*x);PointOnCurve(OneOf(Focus(G)), Directrix(H))",
+      "coords": {}
+    },
+    {
+      "index": 1395,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1396,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1396.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;H: Line;P: Point;Q: Point;Expression(G) = (y^2 = 2*x);Coordinate(P)=(x1,y1);Coordinate(Q)=(x2,y2);PointOnCurve(Focus(G), H);Intersection(H, G)={P,Q};x1:Number;x2:Number;y1:Number;y2:Number;x1+x2=3",
+      "coords": {}
+    },
+    {
+      "index": 1397,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1397.png",
+      "answer": "x^2/4+y^2/3=1",
+      "fact_expr": "l: Line;G: Ellipse;b: Number;F2: Point;B: Point;F1: Point;A1: Point;Expression(G) = (x^2/4 + y^2/b^2 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(F1, l);Intersection(l, G) = {A1, B};Max(LineSegmentOf(A1, F2) + LineSegmentOf(B, F2)) = 5",
+      "coords": {}
+    },
+    {
+      "index": 1398,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1398.png",
+      "answer": "4*sqrt(3)/3",
+      "fact_expr": "G: Ellipse;P: Point;Q: Point;Expression(G) = (x^2/4 + y^2/1 = 1);Coordinate(P) = (0, 1);PointOnCurve(Q, G)",
+      "coords": {
+        "P": [
+          0.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 1399,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1399.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;F:Point;OneOf(Focus(G))=F;l:Line;IsParallel(l,Asymptote(G));Intersection(l, yAxis) = P;P:Point;O:Origin;MidPoint(LineSegmentOf(O,P))=Endpoint(ImageinaryAxis(G));PointOnCurve(F,l)",
+      "coords": {}
+    },
+    {
+      "index": 1400,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          -5.0,
+          0.0
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+        "radius": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/circle/problem_1400.png",
+      "answer": "10",
+      "fact_expr": "C1: Curve;C2: Curve;C3: Curve;P: Point;Q: Point;R: Point;Expression(C1) = (x^2/16 - y^2/9 = 1);Expression(C2) = (y^2 + (x - 5)^2 = 1);Expression(C3) = (y^2 + (x + 5)^2 = 1);PointOnCurve(P, C1);PointOnCurve(Q, C2);PointOnCurve(R, C3)",
+      "coords": {}
+    },
+    {
+      "index": 1401,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1401.png",
+      "answer": "3*x+4*y-7=0",
+      "fact_expr": "Gamma: Ellipse;Expression(Gamma) = (x^2/4 + y^2/3 = 1);P: Point;Coordinate(P) = (1, 1);l: Line;PointOnCurve(P, l);A: Point;B: Point;Intersection(l, Gamma) = {A, B};IsChordOf(LineSegmentOf(A, B), Gamma);MidPoint(LineSegmentOf(A, B)) = P",
+      "coords": {
+        "P": [
+          1.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 1402,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1403,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1403.png",
+      "answer": "2*sqrt(7)/7",
+      "fact_expr": "l: Line;G: Ellipse;b: Number;a: Number;H: Circle;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(LeftFocus(G), l);Inclination(l) = pi/3;IsDiameter(MajorAxis(G), H);Length(InterceptChord(l, H)) = FocalLength(G)",
+      "coords": {}
+    },
+    {
+      "index": 1404,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1404.png",
+      "answer": "[2, 3]",
+      "fact_expr": "H: Circle;Expression(H) = (x^2 + y^2 = a^2);a: Real;a>0;G: Ellipse;Expression(G) = (x^2/9 + y^2/4 = 1);IsIntersect(H, G)",
+      "coords": {}
+    },
+    {
+      "index": 1405,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 7.0,
+        "b": 4.898979485566356,
+        "major_axis": "x",
+        "x_coef": 49.0,
+        "y_coef": 24.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1405.png",
+      "answer": "24",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;F2:Point;Expression(G) = (x^2/49 + y^2/24 = 1);Focus(G)={F1,F2};PointOnCurve(P,G);IsPerpendicular(LineSegmentOf(P,F1),LineSegmentOf(P,F2))",
+      "coords": {}
+    },
+    {
+      "index": 1406,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1406.png",
+      "answer": "(1, 0)",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;H: Point;Coordinate(H) = (-1, 1);PointOnCurve(H, Directrix(G))",
+      "coords": {
+        "H": [
+          -1.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 1407,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1407.png",
+      "answer": "(a^2+b^2)/4",
+      "fact_expr": "P: Point;PointOnCurve(P, E);E: Hyperbola;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;l1: Line;Expression(l1) = (y = (b/a)*x);l2: Line;Expression(l2) = (y = (-b/a)*x);Asymptote(E) = {l1, l2};Z1: Line;Z2: Line;PointOnCurve(P, Z1);PointOnCurve(P, Z2);IsParallel(Z1, l1);IsParallel(Z2, l2);R: Point;Q: Point;Intersection(Z1, l2) = R;Intersection(Z2, l1) = Q",
+      "coords": {}
+    },
+    {
+      "index": 1408,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1409,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1409.png",
+      "answer": "x-y-2=0",
+      "fact_expr": "G: Parabola;M: Point;N: Point;F:Point;Expression(G) = (y^2 = 4*x);PointOnCurve(M,G);PointOnCurve(N,G);Negation(M=N);Focus(G)=F;YCoordinate(MidPoint(LineSegmentOf(M,N)))=2;Abs(LineSegmentOf(M,F))+Abs(LineSegmentOf(N,F))=10",
+      "coords": {}
+    },
+    {
+      "index": 1410,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.7320508075688772,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 3.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1410.png",
+      "answer": "sqrt(3)/3",
+      "fact_expr": "G: Ellipse;A: Point;B: Point;Expression(G) = (x^2/3 + y^2 = 1);UpperVertex(G) = A;LeftVertex(G) = B",
+      "coords": {}
+    },
+    {
+      "index": 1411,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1412,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1412.png",
+      "answer": "4",
+      "fact_expr": "A: Point;B: Point;C:Curve;Expression(C)=(y^2=4*x);IsChordOf(LineSegmentOf(A,B),C);PointOnCurve(Focus(C),LineSegmentOf(A,B));Abs(LineSegmentOf(A, B)) = 10",
+      "coords": {}
+    },
+    {
+      "index": 1413,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1413.png",
+      "answer": "(1+sqrt(5))/2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;RightFocus(G) = F;A: Point;B: Point;LeftVertex(G) = A;RightVertex(G) = B;P: Point;PointOnCurve(P, G);O: Origin;Abs(LineSegmentOf(O, P)) = Abs(LineSegmentOf(O, F));DotProduct(VectorOf(O, F), VectorOf(O, A)) + DotProduct(VectorOf(P, A), VectorOf(P, B)) = 0",
+      "coords": {}
+    },
+    {
+      "index": 1414,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1414.png",
+      "answer": "1/3",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;B: Point;A: Point;F: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F;LeftVertex(G) = A;UpperVertex(G) = B;Distance(F, LineOf(A,B)) = (2*b)/sqrt(17)",
+      "coords": {}
+    },
+    {
+      "index": 1415,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1415.png",
+      "answer": "2*sqrt(6)",
+      "fact_expr": "G:Parabola;P: Point;O: Origin;F: Point;Expression(G) = (y^2 = 8*x);Focus(G) = F;PointOnCurve(P,G);Abs(LineSegmentOf(P, F)) = 5",
+      "coords": {}
+    },
+    {
+      "index": 1416,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1416.png",
+      "answer": "2*sqrt(2)/3",
+      "fact_expr": "E: Ellipse;b: Number;a: Number;A: Point;P: Point;B: Point;a > b;b > 0;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);LeftVertex(E) = A;RightVertex(E) = B;PointOnCurve(P, E);Tan(AngleOf(P, A, B)) = 1/2;Tan(AngleOf(P, B, A)) = 2/9",
+      "coords": {}
+    },
+    {
+      "index": 1417,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1417.png",
+      "answer": "sqrt(10)/4",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;A: Point;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(A, G);AngleOf(F1, A, F2) = ApplyUnit(90, degree);Abs(LineSegmentOf(A, F1)) = 3*Abs(LineSegmentOf(A, F2))",
+      "coords": {}
+    },
+    {
+      "index": 1418,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1419,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.366584062576294,
+        "b": 2.364173412322998,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.00017838670464698225,
+        "converged": true,
+        "iterations": 89
+      },
+      "output_path": "results/test/hyperbola/problem_1419.png",
+      "answer": "(-oo,-32]+(0,oo)",
+      "fact_expr": "l: Line;G: Hyperbola;b: Number;a: Number;M: Point;O: Origin;A: Point;B: Point;a > b;b > 0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(M) = (4, 0);PointOnCurve(M,l);L1:Line;L2:Line;Asymptote(G)={L1,L2};Intersection(l,L1)=A;Intersection(l,L2)=B;Negation(A=B);Eccentricity(G) = 2",
+      "coords": {
+        "M": [
+          4.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 1420,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1421,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1422,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1422.png",
+      "answer": "4/5\nx=pm*(25/4)",
+      "fact_expr": "G:Curve;Expression(G)=(x^2/25+y^2/9=1)",
+      "coords": {}
+    },
+    {
+      "index": 1423,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "vertical"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1423.png",
+      "answer": "y=pm*(sqrt(2)/2)*x",
+      "fact_expr": "F1: Point;F2: Point;Focus(E) = {F1, F2};E: Hyperbola;Expression(E) = (-x^2/b^2 + y^2/a^2 = 1);b: Number;a: Number;a>0;b>0;P: Point;PointOnCurve(P, E);O: Origin;Distance(P, O) = b;Sin(AngleOf(P, F2, F1)) = 3*Sin(AngleOf(P, F1, F2))",
+      "coords": {}
+    },
+    {
+      "index": 1424,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1425,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1425.png",
+      "answer": "sqrt(2) - 1",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;P: Point;F2: Point;e: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;Z: Line;PointOnCurve(F2, Z);IsPerpendicular(Z, xAxis);OneOf(Intersection(Z, G)) = P;AngleOf(F1, P, F2) = ApplyUnit(45, degree);Eccentricity(G) = e",
+      "coords": {}
+    },
+    {
+      "index": 1426,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1427,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1428,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1428.png",
+      "answer": "4",
+      "fact_expr": "G: Ellipse;m: Real;m>1;Expression(G) = (y^2 + x^2/m = 1);Length(MajorAxis(G)) = 2*Length(MinorAxis(G))",
+      "coords": {}
+    },
+    {
+      "index": 1429,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.23606797749979,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 5.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1429.png",
+      "answer": "y^2=4*x",
+      "fact_expr": "H: Ellipse;Expression(H) = (x^2/5 + y^2/4 = 1);F: Point;RightFocus(H) = F;G: Parabola;Focus(G) = F",
+      "coords": {}
+    },
+    {
+      "index": 1430,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.449489742783178,
+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 6.0,
+        "y_coef": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1430.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;p: Real;H: Ellipse;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (x^2/6 + y^2/2 = 1);Focus(G) = RightFocus(H)",
+      "coords": {}
+    },
+    {
+      "index": 1431,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1432,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1432.png",
+      "answer": "(x - 2*sqrt(5)) + y^2 = 25",
+      "fact_expr": "G: Hyperbola;H: Circle;Expression(G) = (x^2/4 - y^2/16 = 1);RightFocus(G) = Center(H);Length(InterceptChord(Asymptote(G),H))=6",
+      "coords": {}
+    },
+    {
+      "index": 1433,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1433.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);M: Point;PointOnCurve(M, G) = True;Distance(M, Focus(G)) = 5",
+      "coords": {}
+    },
+    {
+      "index": 1434,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1434.png",
+      "answer": "10",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/6 + y^2/m = 1);m: Number;Coordinate(OneOf(Focus(G))) = (0,2)",
+      "coords": {}
+    },
+    {
+      "index": 1435,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.03125,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1435.png",
+      "answer": "65/16",
+      "fact_expr": "G: Parabola;H: Line;B: Point;A: Point;M:Point;Expression(G) = (x^2 = y/8);PointOnCurve(Focus(G), H);Intersection(H, G) = {A, B};MidPoint(LineSegmentOf(A,B))=M;YCoordinate(M)=2",
+      "coords": {}
+    },
+    {
+      "index": 1436,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1436.png",
+      "answer": "5/2",
+      "fact_expr": "G: Ellipse;H: Line;P: Point;F2: Point;F1: Point;Expression(G) = (x^2/4 + y^2/3 = 1);Focus(G) = {F1,F2};PointOnCurve(F1,H);IsPerpendicular(H,xAxis);OneOf(Intersection(H,G)) = P;IsIntersect(G, H)",
+      "coords": {}
+    },
+    {
+      "index": 1437,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1437.png",
+      "answer": "(x-3/8)^2 + y^2 = 9/64",
+      "fact_expr": "l: Line;G: Ellipse;A: Point;B: Point;M: Point;O: Origin;D: Point;Expression(G) = (x^2/4 + y^2 = 1);t: Real;Coordinate(M) = (1, t);Intersection(l, G) = {A, B};MidPoint(LineSegmentOf(A, B)) = M;PerpendicularBisector(LineSegmentOf(A, B)) = l;IsPerpendicular(LineSegmentOf(O, D), l);FootPoint(LineSegmentOf(O, D), l) = D",
+      "coords": {}
+    },
+    {
+      "index": 1438,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1438.png",
+      "answer": "3*sqrt(2)/2",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;PointOnCurve(N, Directrix(C));N: Point;L: Line;PointOnCurve(N, L) ;IsPerpendicular(LineSegmentOf(N, F), L);Intersection(L, yAxis) = M;M: Point;PointOnCurve(E, C);E: Point;2*VectorOf(N, E) = VectorOf(N, F) + VectorOf(N, M)",
+      "coords": {}
+    },
+    {
+      "index": 1439,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1439.png",
+      "answer": "sqrt(3)/2",
+      "fact_expr": "E: Ellipse;b: Number;a: Number;M: Point;A: Point;B: Point;C: Point;D: Point;a > b;b > 0;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);Endpoint(MajorAxis(E)) = {A, B};Endpoint(MinorAxis(E)) = {C, D};Abs(LineSegmentOf(M, A))/Abs(LineSegmentOf(M, B)) = 2;Max(TriangleOf(M, A, B)) = 8;Min(TriangleOf(M, C, D)) = 1",
+      "coords": {}
+    },
+    {
+      "index": 1440,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1440.png",
+      "answer": "7*sqrt(2)/8",
+      "fact_expr": "G: Parabola;Expression(G) = (y = x^2);P: Point;PointOnCurve(P, G);H: Line;Expression(H) = (x - y - 2 = 0)",
+      "coords": {}
+    },
+    {
+      "index": 1441,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1441.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;M: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(M, G);Distance(M, Focus(G)) = 3",
+      "coords": {}
+    },
+    {
+      "index": 1442,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1442.png",
+      "answer": "sqrt(2)\n(3/4)*sqrt(2)",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;l: Line;Directrix(G) = l;A: Point;Coordinate(A) = (0, 2);B: Point;MidPoint(LineSegmentOf(F, A)) = B;PointOnCurve(B, G)",
+      "coords": {
+        "A": [
+          0.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 1443,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1443.png",
+      "answer": "(0,(sqrt(6)-sqrt(2))/4)",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;G: Line;A: Point;F: Point;B: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F;PointOnCurve(O,G);Intersection(G, C) = {A, B};AngleOf(A, F, B) >= ApplyUnit(150, degree);O:Origin",
+      "coords": {}
+    },
+    {
+      "index": 1444,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 2.23606797749979,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1444.png",
+      "answer": "sqrt(13) - 2",
+      "fact_expr": "G: Hyperbola;A: Point;B: Point;P: Point;F: Point;Expression(G) = (x^2/4 - y^2/5 = 1);Coordinate(A) = (0, 2);PointOnCurve(P, RightPart(G));Abs(LineSegmentOf(A, B)) = 2;RightFocus(G) = F",
+      "coords": {
+        "A": [
+          0.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 1445,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1445.png",
+      "answer": "8",
+      "fact_expr": "G: Hyperbola;H: Line;M: Point;F2: Point;N: Point;Expression(G) = (x^2/4 - y^2/3 = 1);RightFocus(G) = F2;F1: Point;LeftFocus(G) = F1;PointOnCurve(F1, H);Intersection(H, LeftPart(G)) = {M, N}",
+      "coords": {}
+    },
+    {
+      "index": 1446,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1446.png",
+      "answer": "sqrt(10)/2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;F: Point;E: Point;O: Origin;P: Point;l1: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2 + y^2 = a^2/4);Coordinate(F) = (-c, 0);LeftFocus(G) = F;TangentOfPoint(F, H) = l1;TangentPoint(l1,H)=E;Intersection(OverlappingLine(LineSegmentOf(F,E)),RightPart(G))=P;VectorOf(O,E)=(VectorOf(O,F)+VectorOf(O,P))/2;c:Number;c>0",
+      "coords": {}
+    },
+    {
+      "index": 1447,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.23606797749979,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 5.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1447.png",
+      "answer": "5/3",
+      "fact_expr": "G: Ellipse;H: Line;O: Origin;A: Point;B: Point;Expression(G) = (x^2/5 + y^2/4 = 1);PointOnCurve(RightFocus(G), H);Slope(H) = 2;Intersection(H, G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 1448,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1448.png",
+      "answer": "-2",
+      "fact_expr": "C: Parabola;G: Circle;M: Point;A: Point;B: Point;x0:Number;y0:Number;Expression(C) = (x^2 = 8*y);Coordinate(M) = (x0, y0);PointOnCurve(M,LineOf(M,A));PointOnCurve(M,LineOf(M,B));TangentPoint(LineOf(M,A),C)=A;TangentPoint(LineOf(M,B),C)=B;IsDiameter(LineSegmentOf(A,B),G);PointOnCurve(M,G)",
+      "coords": {}
+    },
+    {
+      "index": 1449,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1449.png",
+      "answer": "-4",
+      "fact_expr": "G: Parabola;P: Point;Q: Point;A: Point;l1:Line;l2:Line;Expression(G) = (x^2 = 2*y);PointOnCurve(P,G);PointOnCurve(Q,G);XCoordinate(P)=4;XCoordinate(Q)=-2;TangentOnPoint(P,G)=l1;TangentOnPoint(Q,G)=l2;Intersection(l1,l2)=A",
+      "coords": {}
+    },
+    {
+      "index": 1450,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1450.png",
+      "answer": "8",
+      "fact_expr": "H: Parabola;Expression(H) = (x^2 = p*y);p: Number;G: Hyperbola;Expression(G) = (-x^2 + y^2/3 = 1);Focus(H) = UpperFocus(G)",
+      "coords": {}
+    },
+    {
+      "index": 1451,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.7320508075688772,
+        "b": 2.0,
+        "orientation": "vertical"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1451.png",
+      "answer": "(-oo, -sqrt(3)/2) + (sqrt(3)/2, +oo)",
+      "fact_expr": "l: Line;G: Hyperbola;Expression(G) = (x^2/4 - y^2/3 = -1);O: Origin;PointOnCurve(O, l);NumIntersection(l, G) = 2",
+      "coords": {}
+    },
+    {
+      "index": 1452,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1452.png",
+      "answer": "(0,pm*3)",
+      "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/25 + y^2/9 = 1);PointOnCurve(P, G);F1:Point;F2:Point;Distance(P,F1)*Distance(P,F2) = m;WhenMax(m);m:Number;Focus(G)={F1,F2}",
+      "coords": {}
+    },
+    {
+      "index": 1453,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "y",
+        "x_coef": 3.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1453.png",
+      "answer": "1/2",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/3 + y^2/4 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 1454,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1454.png",
+      "answer": "x^2-y^2/3=1",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;l1: Line;l2: Line;F: Point;O: Origin;P: Point;Q: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F;PointOnCurve(Q, l1);Expression(l1)=(y = -x*b/a);PointOnCurve(P, l2);Expression(l2)=(y = x*(b/a));2*VectorOf(F, Q) = VectorOf(F, P);DotProduct(VectorOf(F, P), VectorOf(O, Q)) = 0;Area(TriangleOf(F, O, P)) = sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 1455,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1456,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1456.png",
+      "answer": "2",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;A: Point;B: Point;PointOnCurve(A, C);PointOnCurve(B, C);M: Point;Coordinate(M) = (2, 2);MidPoint(LineSegmentOf(A, B)) = M",
+      "coords": {
+        "M": [
+          2.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 1457,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1457.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;M: Point;m: Number;Coordinate(M) = (1, m);PointOnCurve(M, G);Distance(M, Focus(G)) = 5;C: Hyperbola;Expression(C) = (x^2 - y^2/b^2 = 1);b: Number;b>0;A: Point;LeftVertex(C) = A;IsPerpendicular(OneOf(Asymptote(C)), LineOf(A, M))",
+      "coords": {}
+    },
+    {
+      "index": 1458,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1458.png",
+      "answer": "[sqrt(3)/2,1)",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;P: Point;F2: Point;LeftFocus(C)=F1;RightFocus(C)=F2;PointOnCurve(P,C);AngleOf(F1,P,F2)=ApplyUnit(120,degree)",
+      "coords": {}
+    },
+    {
+      "index": 1459,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1460,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_point_off_curve",
+      "params": {
+        "a": 1.9902384281158447,
+        "b": 1.827926735623088,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 1.2790036214234405e-06,
+        "converged": true,
+        "iterations": 106
+      },
+      "output_path": "results/test/ellipse/problem_1460.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;A: Point;RightVertex(C) = A;B: Point;UpperVertex(C) = B;P: Point;Coordinate(P) = (2, 1);PointOnCurve(P, C);WhenMin(Length(LineSegmentOf(A, B)))",
+      "coords": {
+        "P": [
+          2.0,
+          1.0
+        ]
+      },
+      "validation_reasons": [
+        "ellipse_point_off_curve"
+      ]
+    },
+    {
+      "index": 1461,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1462,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1462.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);P: Point;PointOnCurve(P, G);F: Point;Focus(G) = F;A: Point;Coordinate(A) = (2, 3)",
+      "coords": {
+        "A": [
+          2.0,
+          3.0
+        ]
+      }
+    },
+    {
+      "index": 1463,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1463.png",
+      "answer": "sqrt(3)-1",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;F: Point;P: Point;O: Origin;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F;PointOnCurve(P, C);Quadrant(P)=1;AngleOf(F, P, O) = pi/6;Abs(LineSegmentOf(P, F)) = sqrt(3)*Abs(LineSegmentOf(O, F))",
+      "coords": {}
+    },
+    {
+      "index": 1464,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1464.png",
+      "answer": "20",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/9 + x^2/a^2 = 1);a: Number;a>3;F1: Point;F2: Point;Focus(G) = {F1, F2};Abs(LineSegmentOf(F1, F2)) = 8;A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G) ;PointOnCurve(F1, LineSegmentOf(A, B))",
+      "coords": {}
+    },
+    {
+      "index": 1465,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1466,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1466.png",
+      "answer": "sqrt(3)/3",
+      "fact_expr": "C: Parabola;p: Number;A: Point;B: Point;F: Point;D: Point;p>0;Expression(C) = (y^2 = 2*(p*x));PointOnCurve(A, C);PointOnCurve(B, C);Focus(C) = F;AngleOf(A, F, B) = ApplyUnit(120, degree);MidPoint(LineSegmentOf(A, B))=D;Distance(D, Directrix(C)) = d;d:Number",
+      "coords": {}
+    },
+    {
+      "index": 1467,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "left"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1467.png",
+      "answer": "x = 2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = -8*x)",
+      "coords": {}
+    },
+    {
+      "index": 1468,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1468.png",
+      "answer": "4*sqrt(7)",
+      "fact_expr": "G: Ellipse;H: Line;F1: Point;A:Point;C:Circle;Expression(G) = (x^2/25 + y^2/16 = 1);Expression(H) = (3*x + 4*y - 21 = 0);LeftFocus(G)=F1;Center(C)=F1;RightVertex(G)=A;PointOnCurve(A,C)",
+      "coords": {}
+    },
+    {
+      "index": 1469,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.7320508075688772,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1469.png",
+      "answer": "x^2/16+y^2/12=1",
+      "fact_expr": "e: Number;Eccentricity(H) = e;e = 1/2;G: Hyperbola;Expression(G) = (x^2/3 - y^2 = 1);Focus(H) = Focus(G);H: Ellipse",
+      "coords": {}
+    },
+    {
+      "index": 1470,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "y",
+        "x_coef": 3.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1470.png",
+      "answer": "1/2",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/3 + y^2/4 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 1471,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1471.png",
+      "answer": "16",
+      "fact_expr": "l: Line;G: Parabola;M: Point;N: Point;A: Point;B: Point;Expression(G) = (y^2 = 8*x);Coordinate(M) = (2, 0);Coordinate(N) = (3, 1);PointOnCurve(M, l);PointOnCurve(N, l);Intersection(l,G)={A,B}",
+      "coords": {
+        "M": [
+          2.0,
+          0.0
+        ],
+        "N": [
+          3.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 1472,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          -3.0,
+          0.0
+        ],
+        "radius": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/circle/problem_1472.png",
+      "answer": "y^2=12*x",
+      "fact_expr": "l: Line;F: Circle;G:Circle;Expression(F) = (y^2 + (x - 3)^2 = 4);Expression(l)=(x=-1);IsOutTangent(F,G);IsTangent(l,G)",
+      "coords": {}
+    },
+    {
+      "index": 1473,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1474,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1474.png",
+      "answer": "sqrt(3)/2",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;A: Point;P: Point;B: Point;D: Point;C: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);RightVertex(G) = A;UpperVertex(G) = B;PointOnCurve(P, G);Intersection(LineSegmentOf(A, P), yAxis) = C;Intersection(LineSegmentOf(B, P), xAxis) = D;Abs(LineSegmentOf(A, D))*Abs(LineSegmentOf(B, C)) = a^2",
+      "coords": {}
+    },
+    {
+      "index": 1475,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_focus_mismatch",
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1475.png",
+      "answer": "x^2/4-y^2/3=1",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Point;F: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(H) = (2, sqrt(3));Coordinate(F) = (-sqrt(7), 0);PointOnCurve(H, OneOf(Asymptote(G)));OneOf(Focus(G)) = F",
+      "coords": {
+        "H": [
+          2.0,
+          1.7320508075688772
+        ],
+        "F": [
+          -2.6457513110645907,
+          0.0
+        ]
+      },
+      "validation_reasons": [
+        "hyperbola_focus_mismatch"
+      ]
+    },
+    {
+      "index": 1476,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 9.0,
+        "b": 5.196152422706632,
+        "major_axis": "x",
+        "x_coef": 81.0,
+        "y_coef": 27.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1476.png",
+      "answer": "36",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/81 + y^2/27 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};H: Line;PointOnCurve(F1, H);P: Point;Q: Point;Intersection(H, G) = {P, Q}",
+      "coords": {}
+    },
+    {
+      "index": 1477,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1477.png",
+      "answer": "4*sqrt(3)/3",
+      "fact_expr": "l: Line;G: Parabola;A: Point;O: Origin;B: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(Focus(G), l);Inclination(l) = pi/3;Intersection(l, G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 1478,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1478.png",
+      "answer": "sqrt(15)",
+      "fact_expr": "G: Parabola;A: Point;B: Point;M: Point;Expression(G) = (y^2 = 4*x);Coordinate(M) = (1, 1);IsChordOf(LineSegmentOf(A, B), G);MidPoint(LineSegmentOf(A, B)) = M",
+      "coords": {
+        "M": [
+          1.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 1479,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1479.png",
+      "answer": "1/(Cos(alpha)-Sin(alpha))",
+      "fact_expr": "F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;IsDiameter(LineSegmentOf(F1,F2),H);H: Circle;Intersection(H,RightPart(G)) = P;P: Point;AngleOf(P, F1, F2) = alpha;alpha: Number",
+      "coords": {}
+    },
+    {
+      "index": 1480,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1480.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x)",
+      "coords": {}
+    },
+    {
+      "index": 1481,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1481.png",
+      "answer": "4",
+      "fact_expr": "G: Ellipse;H: Line;M: Point;N: Point;F2: Point;F1: Point;Expression(G) = (x^2/4 + y^2/3 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, H);Intersection(H, G) = {M, N};Area(InscribedCircle(TriangleOf(M, N, F2))) = pi",
+      "coords": {}
+    },
+    {
+      "index": 1482,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1482.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "C: Parabola;P: Point;F: Point;O: Origin;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(P, C);Abs(LineSegmentOf(P, F)) = 3",
+      "coords": {}
+    },
+    {
+      "index": 1483,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1483.png",
+      "answer": "-4/3",
+      "fact_expr": "G: Ellipse;a: Number;b: Number;H: Line;O: Origin;E: Point;P: Point;Q: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (b + x - 2*y = 0);Eccentricity(G)= sqrt(3)/3;Intersection(H, G) = {P, Q};MidPoint(LineSegmentOf(P,Q))=E",
+      "coords": {}
+    },
+    {
+      "index": 1484,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1485,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1485.png",
+      "answer": "ApplyUnit(90,degree)",
+      "fact_expr": "G: Parabola;H: Line;A: Point;O: Origin;B: Point;Expression(G) = (y^2 = 2*x);Expression(H) = (y = x - 2);Intersection(H, G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 1486,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1486.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1);P: Point;PointOnCurve(P, G);L1: Line;L2: Line;Asymptote(G) = {L1, L2};l1: Line;l2: Line;PointOnCurve(P, l1);PointOnCurve(P, l2);IsParallel(L1, l1);IsParallel(L2, l2);M: Point;N: Point;Intersection(l1, xAxis) = M;Intersection(l2, xAxis) = N;O: Origin",
+      "coords": {}
+    },
+    {
+      "index": 1487,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1487.png",
+      "answer": "sqrt(2)*27/16",
+      "fact_expr": "G: Parabola;P: Point;A: Point;B: Point;C: Point;l1: Line;l2: Line;Expression(G) = (y = x^2);Coordinate(P) = (0, 1);TangentOfPoint(C, G) = {l1, l2};TangentPoint(l1, G) = A;TangentPoint(l2, G) = B;VectorOf(C, P) = VectorOf(C, A)/3 + (2/3)*VectorOf(C, B)",
+      "coords": {
+        "P": [
+          0.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 1488,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1488.png",
+      "answer": "x^2/18+y^2/8=1",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;G: Circle;M: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(M) = (0, -1);Eccentricity(C) = sqrt(3)/3;Center(G)=M;IsIntersect(G,C);Max(Radius(G))=sqrt(15)",
+      "coords": {
+        "M": [
+          0.0,
+          -1.0
+        ]
+      }
+    },
+    {
+      "index": 1489,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1489.png",
+      "answer": "x=-1/2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x)",
+      "coords": {}
+    },
+    {
+      "index": 1490,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1490.png",
+      "answer": "12",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/25 + y^2/9 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G)",
+      "coords": {}
+    },
+    {
+      "index": 1491,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1491.png",
+      "answer": "[-5,5]",
+      "fact_expr": "G: Ellipse;P: Point;x1:Number;y1:Number;Expression(G) = (x^2/4 + y^2 = 1);Coordinate(P) = (x1, y1);PointOnCurve(P, G)",
+      "coords": {}
+    },
+    {
+      "index": 1492,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1493,
+      "success": false,
+      "error": "Circle from diameter: missing point coordinates"
+    },
+    {
+      "index": 1494,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1494.png",
+      "answer": "8",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);l: Line;PointOnCurve(Focus(G), l);P: Point;Q: Point;x1: Number;x2: Number;y1: Number;y2: Number;Coordinate(P) = (x1, y1);Coordinate(Q) = (x2, y2);Intersection(l, G) = {P, Q};XCoordinate(MidPoint(LineSegmentOf(P, Q))) = 3",
+      "coords": {}
+    },
+    {
+      "index": 1495,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1496,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1496.png",
+      "answer": "y = pm*x",
+      "fact_expr": "G: Hyperbola;a: Number;Expression(G) = (-y^2/8 + x^2/a^2 = 1);Coordinate(OneOf(Focus(G))) = (4, 0)",
+      "coords": {}
+    },
+    {
+      "index": 1497,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "left"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1497.png",
+      "answer": "(-1/2,0)",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = -2*x)",
+      "coords": {}
+    },
+    {
+      "index": 1498,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1498.png",
+      "answer": "sqrt(10)/2",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;A: Point;F2: Point;B: Point;F1: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, G);Intersection(G, LeftPart(C)) = {A, B};Abs(LineSegmentOf(A, F2)) = 6;Abs(LineSegmentOf(B, F2)) = 10;Abs(LineSegmentOf(A, B)) = 8",
+      "coords": {}
+    },
+    {
+      "index": 1499,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1500,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1500.png",
+      "answer": "sqrt(3)+1",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;c:Number;H: Circle;F1: Point;M: Point;F2: Point;a>0;b>0;c>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F2) = (c, 0);LeftFocus(G) = F1;RightFocus(G) = F2;Center(H)=F2;Radius(H)=c;OneOf(Intersection(H,RightPart(G)))=M;IsTangent(LineOf(F1,M),H)",
+      "coords": {}
+    },
+    {
+      "index": 1501,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1501.png",
+      "answer": "a^2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;P: Point;R: Point;Q: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, G);PointOnCurve(P, H);IsParallel(H, RealAxis(G));L1: Line;L2: Line;Asymptote(G) = {L1, L2};Intersection(H, L1) = R;Intersection(H, L2) = Q",
+      "coords": {}
+    },
+    {
+      "index": 1502,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1502.png",
+      "answer": "sqrt(3) - 1",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;H: Circle;F2: Point;M: Point;F1: Point;N: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F2;Center(H) = F2;PointOnCurve(Center(G), H);Intersection(H, G) = {M, N};LeftFocus(G) = F1;PointOnCurve(F1, LineOf(M, F1));IsTangent(LineOf(M, F1), H)",
+      "coords": {}
+    },
+    {
+      "index": 1503,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1503.png",
+      "answer": "4*sqrt(6)",
+      "fact_expr": "l: Line;G: Parabola;M: Point;A: Point;B: Point;Expression(G) = (y^2 = 4*x);Coordinate(M) = (2, 0);PointOnCurve(M, l);Slope(l) = 1;Intersection(l, G) = {A, B}",
+      "coords": {
+        "M": [
+          2.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 1504,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          0.0,
+          0.0
+        ],
+        "radius": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/circle/problem_1504.png",
+      "answer": "(x - 2)^2 + (y + 1)^2 = 1",
+      "fact_expr": "P: Point;Coordinate(P) = (4, -2);G: Circle;Expression(G) = (x^2 + y^2 = 4);Z: Point;PointOnCurve(Z, G)",
+      "coords": {}
+    },
+    {
+      "index": 1505,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1505.png",
+      "answer": "4*sqrt(2)",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;O: Origin;D: Point;E: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x = a);L1:Line;L2:Line;Asymptote(C)={L1,L2};Intersection(G,L1)=D;Intersection(G,L2)=E;Area(TriangleOf(O, D, E)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 1506,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1507,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 10.0,
+        "b": 6.0,
+        "major_axis": "x",
+        "x_coef": 100.0,
+        "y_coef": 36.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1507.png",
+      "answer": "66/5",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/100 + y^2/36 = 1);P: Point;PointOnCurve(P, G);Distance(P, RightDirectrix(G)) = 17/2",
+      "coords": {}
+    },
+    {
+      "index": 1508,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "y",
+        "x_coef": 1.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1508.png",
+      "answer": "5/2",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2 + y^2/5 = 1);A: Point;RightVertex(C) = A;P: Point;PointOnCurve(P, C) = True",
+      "coords": {}
+    },
+    {
+      "index": 1509,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_point_off_curve",
+      "params": {
+        "a": 1.6259966743174588,
+        "b": 2.298015331604694,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 1.93011531933813e-06,
+        "converged": true,
+        "iterations": 91
+      },
+      "output_path": "results/test/hyperbola/problem_1509.png",
+      "answer": "x^2/2-y^2/4=1",
+      "fact_expr": "G: Hyperbola;P: Point;O: Origin;Center(G)=O;Coordinate(P) = (1, -2);PointOnCurve(Focus(G), xAxis);PointOnCurve(P, G);Expression(OneOf(Asymptote(G)))=(y=sqrt(2)*x)",
+      "coords": {
+        "P": [
+          1.0,
+          -2.0
+        ]
+      },
+      "validation_reasons": [
+        "hyperbola_point_off_curve"
+      ]
+    },
+    {
+      "index": 1510,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1510.png",
+      "answer": "(1, 1)",
+      "fact_expr": "G: Parabola;H: Line;Expression(G) = (y = x^2);Expression(H) = (2*x - y = 4);Z: Point;PointOnCurve(Z, G);WhenMin(Distance(Z, H))",
+      "coords": {}
+    },
+    {
+      "index": 1511,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1512,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_focus_mismatch",
+      "params": {
+        "a": 1.3300262689590454,
+        "b": 2.30381441116333,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 6.658357051492203e-06,
+        "converged": true,
+        "iterations": 122
+      },
+      "output_path": "results/test/hyperbola/problem_1512.png",
+      "answer": "x^2 - y^2/3 = 1",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;F: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (2, 0);OneOf(Focus(G)) = F;Expression(Asymptote(G)) = (y = pm*sqrt(3)*x)",
+      "coords": {
+        "F": [
+          2.0,
+          0.0
+        ]
+      },
+      "validation_reasons": [
+        "hyperbola_focus_mismatch"
+      ]
+    },
+    {
+      "index": 1513,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1514,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1514.png",
+      "answer": "{0,1/8}",
+      "fact_expr": "H: Line;Expression(H) = (y = k*x + 2);G: Parabola;Expression(G) = (y^2 = x);NumIntersection(H, G) = 1;k: Real",
+      "coords": {}
+    },
+    {
+      "index": 1515,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1515.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/12 + x^2/a^2 = 1);a: Number;a>0;Inclination(OneOf(Asymptote(G))) = ApplyUnit(60, degree)",
+      "coords": {}
+    },
+    {
+      "index": 1516,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1516.png",
+      "answer": "2*sqrt(3)\ny=pm*(sqrt(2)/2)*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/2 - y^2 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 1517,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1518,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1518.png",
+      "answer": "5/3",
+      "fact_expr": "E: Hyperbola;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(E) = F1;RightFocus(E) = F2;P: Point;PointOnCurve(P, LeftPart(E));Abs(LineSegmentOf(P, F1)) = Abs(LineSegmentOf(F1, F2));G: Circle;Expression(G) = (x^2 + y^2 = a^2);IsTangent(LineOf(P, F2), G)",
+      "coords": {}
+    },
+    {
+      "index": 1519,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1519.png",
+      "answer": "10",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);F: Point;Focus(G) = F;P: Point;Coordinate(P) = (1, 4);l: Line;PointOnCurve(P,l) = True;Intersection(l,G) = {A,B};A: Point;B: Point;MidPoint(LineSegmentOf(A,B)) = P",
+      "coords": {
+        "P": [
+          1.0,
+          4.0
+        ]
+      }
+    },
+    {
+      "index": 1520,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1520.png",
+      "answer": "3/2",
+      "fact_expr": "l: Line;C: Ellipse;A: Point;B: Point;P: Point;Expression(C) = (x^2/4 + y^2/3 = 1);Coordinate(P) = (-1, 1/2);PointOnCurve(P, l);Intersection(l, C) = {A, B};IsChordOf(LineSegmentOf(A, B), C);MidPoint(LineSegmentOf(A, B)) = P",
+      "coords": {
+        "P": [
+          -1.0,
+          0.5
+        ]
+      }
+    },
+    {
+      "index": 1521,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 4.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1521.png",
+      "answer": "(x-2)^2 + (y - 4*sqrt(2))^2 = 36",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 16*x);H: Circle;PointOnCurve(Center(H), G);PointOnCurve(Vertex(G), H);PointOnCurve(Focus(G), H);Quadrant(Center(H)) = 1",
+      "coords": {}
+    },
+    {
+      "index": 1522,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1522.png",
+      "answer": "1",
+      "fact_expr": "l: Line;Expression(l) = (x - y + 1 = 0);C: Parabola;Expression(C) = (x^2 = 2*y);A: Point;B: Point;Intersection(l, C) = {A, B};P: Point;PointOnCurve(P, l);M: Point;N: Point;PointOnCurve(M, C);PointOnCurve(N, C);IsParallel(VectorOf(M, N), VectorOf(A, B));Abs(VectorOf(M, N)) < Abs(VectorOf(A, B))",
+      "coords": {}
+    },
+    {
+      "index": 1523,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1523.png",
+      "answer": "y=pm*(sqrt(2)/2)*x",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Length(ImageinaryAxis(G)) = 2;FocalLength(G) = 2*sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 1524,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 3.4641016151377544,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1524.png",
+      "answer": "2*sqrt(3)-2",
+      "fact_expr": "G: Hyperbola;H: Circle;P: Point;F: Point;N: Point;O: Origin;T: Point;Expression(G) = (x^2/4 - y^2/12 = 1);Expression(H) = (x^2 + y^2 = 4);LeftFocus(G)=F;TangentOfPoint(F,H)=LineOf(F,P);Intersection(LineOf(F,P),RightPart(G))=P;TangentPoint(LineOf(F,P),H)=T;MidPoint(LineSegmentOf(F,P)) = N",
+      "coords": {}
+    },
+    {
+      "index": 1525,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1525.png",
+      "answer": "-1/2",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/2 + y^2 = 1);F: Point;RightFocus(C) = F;G: Line;Expression(G) = (y = k*x + 1);k: Real;A: Point;B: Point;Intersection(G, C) = {A, B};IsPerpendicular(LineSegmentOf(A, F), LineSegmentOf(B, F))",
+      "coords": {}
+    },
+    {
+      "index": 1526,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1526.png",
+      "answer": "9",
+      "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 - y^2/9 = 1);PointOnCurve(P, G);LeftFocus(G)=F1;RightFocus(G)=F2;Abs(LineSegmentOf(P, F1)) = 5",
+      "coords": {}
+    },
+    {
+      "index": 1527,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1528,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1528.png",
+      "answer": "7",
+      "fact_expr": "l1: Line;b: Number;b>3;Expression(l1) = (x - sqrt(3)*y + b = 0);C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l2: Line;Directrix(C) = l2;A: Point;PointOnCurve(A, C);d1: Number;d2: Number;Distance(A, l1) = d1;Distance(A, l2) = d2;WhenMin(d1+d2);d1 = 2*d2",
+      "coords": {}
+    },
+    {
+      "index": 1529,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1529.png",
+      "answer": "2*x+4*y-3=0",
+      "fact_expr": "l: Line;C: Ellipse;A: Point;B: Point;Expression(C) = (x^2/2 + y^2 = 1);Coordinate(MidPoint(LineSegmentOf(A,B))) = (1/2, 1/2);Intersection(l, C) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 1530,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1531,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.23606797749979,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 5.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1531.png",
+      "answer": "(3/2, sqrt(15)/2)",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/9 + y^2/5 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;M: Point;PointOnCurve(M, C);Quadrant(M) = 1;Abs(LineSegmentOf(M, F1)) = Abs(LineSegmentOf(F1, F2))",
+      "coords": {}
+    },
+    {
+      "index": 1532,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1533,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1534,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1534.png",
+      "answer": "1/2",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;A: Point;B: Point;C: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(Vertex(TriangleOf(A,B,C)),G);IsPerpendicular(LineSegmentOf(A,B),xAxis);IsParallel(LineSegmentOf(A,C),xAxis)",
+      "coords": {}
+    },
+    {
+      "index": 1535,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1535.png",
+      "answer": "1/7",
+      "fact_expr": "P: Point;A: Point;G: Ellipse;b: Number;a: Number;B: Point;a > b;b > 0;e: Number;alpha: Number;beta: Number;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = e ;e = 1/2;LeftVertex(G) = A;RightVertex(G) = B;PointOnCurve(P, G);Negation(P = A);Negation(P = B);Inclination(LineOf(P, A)) = alpha;Inclination(LineOf(P, B)) = beta",
+      "coords": {}
+    },
+    {
+      "index": 1536,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1536.png",
+      "answer": "25/2",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;F1: Point;N: Point;F2: Point;M: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(M, LeftPart(C));L:Line;PointOnCurve(M, L);IsPerpendicular(L,OneOf(Asymptote(C)));FootPoint(L,OneOf(Asymptote(C)))=N;Min(Abs(LineSegmentOf(M,F2))+Abs(LineSegmentOf(M,N)))=10",
+      "coords": {}
+    },
+    {
+      "index": 1537,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1537.png",
+      "answer": "2*sqrt(2)/3",
+      "fact_expr": "l: Line;C: Parabola;B: Point;P: Point;A: Point;Expression(C) = (y^2 = 8*x);Coordinate(P) = (-2, 0);PointOnCurve(P, l);Intersection(l, C) = {A, B};Quadrant(A) = 1;Quadrant(B) = 1;MidPoint(LineSegmentOf(P,B)) = A",
+      "coords": {
+        "P": [
+          -2.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 1538,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.366584062576294,
+        "b": 2.364173412322998,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.00017838670464698225,
+        "converged": true,
+        "iterations": 89
+      },
+      "output_path": "results/test/hyperbola/problem_1538.png",
+      "answer": "{3,1/3}",
+      "fact_expr": "G: Hyperbola;m: Number;n: Number;Expression(G) = (-y^2/n + x^2/m = 1);Eccentricity(G) = 2;Negation(m*n=0)",
+      "coords": {}
+    },
+    {
+      "index": 1539,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1540,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1540.png",
+      "answer": "-b^2/a^2",
+      "fact_expr": "P: Point;A: Point;G: Ellipse;a: Number;b: Number;B: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Endpoint(MinorAxis(G)) = {A, B};PointOnCurve(P, G);Negation(P = A);Negation(P = B)",
+      "coords": {}
+    },
+    {
+      "index": 1541,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1542,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1542.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;H: Line;Expression(H) = (m - m*y + x = 0);PointOnCurve(Focus(G), H);m: Number;Intersection(H, G) = {A, B};O: Origin;A: Point;B: Point;Area(TriangleOf(O, A, B)) = 2*sqrt(2)",
+      "coords": {}
+    },
+    {
+      "index": 1543,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1543.png",
+      "answer": "3",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1);F1: Point;LeftFocus(G) = F1;PointOnCurve(F1, LineOf(A, B)) = True;Inclination(LineOf(A, B)) = pi/6;A: Point;B: Point;Intersection(LineOf(A, B), LeftPart(G)) = A;Intersection(LineOf(A, B), RightPart(G)) = B;IsChordOf(LineSegmentOf(A, B), G) = True",
+      "coords": {}
+    },
+    {
+      "index": 1544,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1545,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "left"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1545.png",
+      "answer": "6",
+      "fact_expr": "G: Parabola;P: Point;Expression(G) = (y^2 = -8*x);PointOnCurve(P, G);Distance(P, yAxis) = 4",
+      "coords": {}
+    },
+    {
+      "index": 1546,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1547,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1548,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1548.png",
+      "answer": "12",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;A: Point;B: Point;y1: Number;y2: Number;Coordinate(A) = (2, y1);Coordinate(B) = (1/2, y2);PointOnCurve(A, G);PointOnCurve(B, G);Quadrant(A) = 1;Quadrant(B) = 4;Abs(LineSegmentOf(A, F)) = 10",
+      "coords": {}
+    },
+    {
+      "index": 1549,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1549.png",
+      "answer": "y=pm*((4/3)*x)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, RightPart(G));Abs(LineSegmentOf(P, F2)) = Abs(LineSegmentOf(F1, F2));Distance(F2, LineOf(P, F1)) = Length(RealAxis(G))",
+      "coords": {}
+    },
+    {
+      "index": 1550,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1550.png",
+      "answer": "2/3",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2 + 2*y^2 = 2);H: Line;PointOnCurve(Focus(G), H) = True;Inclination(H) = ApplyUnit(45, degree);A: Point;B: Point;Intersection(H,G) = {A,B};O: Origin;Center(G) = O",
+      "coords": {}
+    },
+    {
+      "index": 1551,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.7320508075688772,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 3.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1551.png",
+      "answer": "",
+      "fact_expr": "G: Ellipse;F2: Point;B: Point;F1: Point;A: Point;Expression(G) = (x^2/3 + y^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(A, G);PointOnCurve(B, G);VectorOf(F1, A) = 5*VectorOf(F2, B)",
+      "coords": {}
+    },
+    {
+      "index": 1552,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1552.png",
+      "answer": "1",
+      "fact_expr": "l: Line;E: Parabola;F: Point;A: Point;B: Point;Expression(E) = (y^2 = 4*x);Focus(E) = F;PointOnCurve(F, l);Intersection(l, E) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 1553,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1553.png",
+      "answer": "{(2, 3), sqrt(13)}",
+      "fact_expr": "G: Ellipse;C: Curve;m: Real;Expression(G) = (x^2/9 + y^2/4 = 1);m>0;Expression(C) = (Abs(x) + Abs(y) = m);NumIntersection(C, G) = 4",
+      "coords": {}
+    },
+    {
+      "index": 1554,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1554.png",
+      "answer": "(1,2)",
+      "fact_expr": "G: Hyperbola;a: Number;b: Number;A: Point;B: Point;P: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (1, 2);Coordinate(B) = (-1, 2);IsPerpendicular(VectorOf(A,P),VectorOf(B,P));NumIntersection(Asymptote(G),Locus(P))=0",
+      "coords": {
+        "A": [
+          1.0,
+          2.0
+        ],
+        "B": [
+          -1.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 1555,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1555.png",
+      "answer": "1",
+      "fact_expr": "G: Hyperbola;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 - y^2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(F, P1), LineSegmentOf(F, P2))",
+      "coords": {}
+    },
+    {
+      "index": 1556,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.47213595499958,
+        "b": 2.23606797749979,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1556.png",
+      "answer": "10",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/20 - y^2/5 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 1557,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1558,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1558.png",
+      "answer": "3*sqrt(5)",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);N: Point;Coordinate(N) = (2, 0);H: Line;PointOnCurve(N, H);A: Point;B: Point;Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, N))/Abs(LineSegmentOf(B, N)) = 2",
+      "coords": {
+        "N": [
+          2.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 1559,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1559.png",
+      "answer": "[sqrt(5)/5,sqrt(3)/3]",
+      "fact_expr": "Gamma: Ellipse;a: Number;b: Number;A: Point;F1: Point;F2: Point;B: Point;a > b;b > 0;lambda:Number;In(lambda,[2,3]);Expression(Gamma) = (x^2/a^2 + y^2/b^2 = 1);LeftFocus(Gamma) = F1;RightFocus(Gamma) = F2;PointOnCurve(A, Gamma);PointOnCurve(B, Gamma);DotProduct(VectorOf(A,F1),VectorOf(F1,F2))=0;VectorOf(A,F2)=lambda*VectorOf(F2,B)",
+      "coords": {}
+    },
+    {
+      "index": 1560,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1560.png",
+      "answer": "4*sqrt(3)/3",
+      "fact_expr": "G: Ellipse;H: Line;A: Point;B: Point;Expression(G) = (x^2/2 + y^2 = 1);Slope(H)=1;Intersection(H, G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 1561,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1561.png",
+      "answer": "x = -1/4",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = x)",
+      "coords": {}
+    },
+    {
+      "index": 1562,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "down"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1562.png",
+      "answer": "y=2",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = -8*y)",
+      "coords": {}
+    },
+    {
+      "index": 1563,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1563.png",
+      "answer": "1",
+      "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;P: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (x + y + 2 = 0);Intersection(H,xAxis) = A;Intersection(H,yAxis) = B;PointOnCurve(P, G)",
+      "coords": {}
+    },
+    {
+      "index": 1564,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1565,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "vertical"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1565.png",
+      "answer": "sqrt(11)/2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/7 + y^2/4 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 1566,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1566.png",
+      "answer": "pm*(4/3)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;P: Point;PointOnCurve(P, RightPart(G));F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;O: Origin;H: Circle;A: Point;TangentPoint(LineOf(P, F1), H) = A;Center(H) = O;Radius(H) = a;PointOnCurve(F2, PerpendicularBisector(LineSegmentOf(P, F1)))",
+      "coords": {}
+    },
+    {
+      "index": 1567,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1567.png",
+      "answer": "2*sqrt(6)",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 + y^2/3 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(P,G);LineSegmentOf(P, F1) = 2*LineSegmentOf(P, F2)",
+      "coords": {}
+    },
+    {
+      "index": 1568,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1569,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1570,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 25.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1570.png",
+      "answer": "(25, 0)",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 100*x)",
+      "coords": {}
+    },
+    {
+      "index": 1571,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.8284271247461903,
+        "b": 2.0,
+        "major_axis": "y",
+        "x_coef": 4.0,
+        "y_coef": 8.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1571.png",
+      "answer": "x^2+y^2/2=1",
+      "fact_expr": "G: Ellipse;O: Origin;P: Point;Q:Point;Expression(G) = (x^2/4 + y^2/8 = 1);PointOnCurve(P, G);MidPoint(LineSegmentOf(O,P))=Q",
+      "coords": {}
+    },
+    {
+      "index": 1572,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1572.png",
+      "answer": "2",
+      "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;Expression(G) = (y^2 = 4*x);Intersection(l, G) = {A, B};Abs(LineSegmentOf(A, B)) = 6",
+      "coords": {}
+    },
+    {
+      "index": 1573,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1573.png",
+      "answer": "1+sqrt(2)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;M: Point;F2: Point;F1: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, H);Inclination(H)=ApplyUnit(45,degree);Intersection(H, RightPart(G)) = M;IsPerpendicular(LineSegmentOf(M,F2),xAxis)",
+      "coords": {}
+    },
+    {
+      "index": 1574,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1574.png",
+      "answer": "3",
+      "fact_expr": "C: Parabola;F: Point;N: Point;M: Point;l: Line;lambda:Number;lambda>0;Expression(C) = (y^2 = 4*x);Focus(C) = F;Directrix(C) = l;PointOnCurve(M, C);PointOnCurve(N, l);VectorOf(F, N) = lambda*VectorOf(F, M);Abs(LineSegmentOf(M, F)) = 4/3",
+      "coords": {}
+    },
+    {
+      "index": 1575,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1576,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1576.png",
+      "answer": "8",
+      "fact_expr": "C: Parabola;G: Line;A: Point;B: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F, G);Slope(G) = 1;Intersection(G, C) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 1577,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1577.png",
+      "answer": "1",
+      "fact_expr": "G: Hyperbola;H: Parabola;Expression(G) = (-x^2 + y^2/3 = 1);Expression(H) = (x^2 = 8*y)",
+      "coords": {}
+    },
+    {
+      "index": 1578,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 0.5,
+        "b": 0.8660254037844386,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1578.png",
+      "answer": "x^2/5+y^2/4=1",
+      "fact_expr": "l: Line;G: Hyperbola;H: Ellipse;P: Point;Expression(G) = (12*x^2 - 4*y^2 = 3);Expression(l) = (y = x + 3);PointOnCurve(P, l);PointOnCurve(P, H);Focus(G) = Focus(H);WhenMin(MajorAxis(H))",
+      "coords": {}
+    },
+    {
+      "index": 1579,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1579.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "A: Point;B: Point;E: Ellipse;Expression(E) = (y^2 + x^2/m = 1);m: Number;m > 1;Vertex(MinorAxis(E)) = {A, B};P: Point;PointOnCurve(P, E) = True;Negation(P = A);Negation(P = B);Slope(LineOf(A, P))*Slope(LineOf(B, P)) = -m/4",
+      "coords": {}
+    },
+    {
+      "index": 1580,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1580.png",
+      "answer": "3",
+      "fact_expr": "G: Parabola;P: Point;F: Point;Expression(G) = (x^2 = 4*y);Focus(G) = F;PointOnCurve(P, G);Distance(P, xAxis) = 2",
+      "coords": {}
+    },
+    {
+      "index": 1581,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1581.png",
+      "answer": "(-sqrt(10)/2,-sqrt(2)/2)+(-sqrt(2)/2,sqrt(2)/2)+(sqrt(2)/2,sqrt(10)/2)",
+      "fact_expr": "l: Line;C: Hyperbola;Expression(C) = (x^2/2 - y^2 = 1);Expression(l)=(y=k*x+2);NumIntersection(l,C)=2;k:Number",
+      "coords": {}
+    },
+    {
+      "index": 1582,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1582.png",
+      "answer": "pm*3*y+4*x=0",
+      "fact_expr": "C: Hyperbola;G: Ellipse;Expression(G)=(x^2/25+y^2/16=1);Focus(C)=Endpoint(MajorAxis(G));Endpoint(RealAxis(C))=Focus(G)",
+      "coords": {}
+    },
+    {
+      "index": 1583,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.23606797749979,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 5.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1583.png",
+      "answer": "x=-2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;H: Ellipse;Expression(H) = (x^2/9 + y^2/5 = 1);Focus(G) = RightFocus(H)",
+      "coords": {}
+    },
+    {
+      "index": 1584,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1584.png",
+      "answer": "(1,sqrt(2)]",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (-4*x + x^2 + y^2 + 2 = 0);IsIntersect(Asymptote(G),H)",
+      "coords": {}
+    },
+    {
+      "index": 1585,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1585.png",
+      "answer": "45/2",
+      "fact_expr": "A: Point;G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);LeftVertex(G) = A;B: Point;PointOnCurve(B, G) = True;PointOnCurve(C, yAxis) = True;C: Point;Coordinate(C) = (0, 4)",
+      "coords": {
+        "C": [
+          0.0,
+          4.0
+        ]
+      }
+    },
+    {
+      "index": 1586,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1586.png",
+      "answer": "4/5",
+      "fact_expr": "G: Ellipse;m: Number;m>0;Expression(G)=(y^2/9 + x^2/m^2 = 1);Coordinate(OneOf(Focus(G)))=(4,0)",
+      "coords": {}
+    },
+    {
+      "index": 1587,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 2.8284271247461903,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1587.png",
+      "answer": "12*sqrt(6)",
+      "fact_expr": "C: Hyperbola;A: Point;P: Point;F: Point;Expression(C) = (x^2 - y^2/8 = 1);Coordinate(A) = (0, 6*sqrt(6));RightFocus(C) = F;PointOnCurve(P, LeftPart(C));WhenMin(Perimeter(TriangleOf(A,P,F)))",
+      "coords": {
+        "A": [
+          0.0,
+          14.696938456699067
+        ]
+      }
+    },
+    {
+      "index": 1588,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 3.0,
+        "direction": "down"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1588.png",
+      "answer": "(-6, -6)",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = -2*p*y);p: Number;p>0;F: Point;Coordinate(F) = (0, -3);Focus(G) = F;H: Line;Expression(H) = (y = x)",
+      "coords": {
+        "F": [
+          0.0,
+          -3.0
+        ]
+      }
+    },
+    {
+      "index": 1589,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1589.png",
+      "answer": "y = pm*sqrt(3)*x",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = e ;e: Number;e = 2",
+      "coords": {}
+    },
+    {
+      "index": 1590,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1590.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Parabola;O: Origin;P: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G)=F;PointOnCurve(P,G);Abs(LineSegmentOf(P, F)) = 3",
+      "coords": {}
+    },
+    {
+      "index": 1591,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1592,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1593,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          -4.0,
+          0.0
+        ],
+        "radius": 13.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/circle/problem_1593.png",
+      "answer": "x^2/64+y^2/48=1",
+      "fact_expr": "M: Circle;Expression(M) = (y^2 + (x - 4)^2 = 169);N: Circle;Expression(N) = (y^2 + (x + 4)^2 = 9);P: Circle;IsInTangent(P,M) = True;IsOutTangent(P,N) = True;P1: Point;Center(P) = P1",
+      "coords": {}
+    },
+    {
+      "index": 1594,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "major_axis": "x",
+        "x_coef": 6.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1594.png",
+      "answer": "-1",
+      "fact_expr": "F1: Point;F2: Point;G: Ellipse;Expression(G) = (x^2/6 + y^2/3 = 1);Focus(G) = {F1, F2};A: Point;B: Point;LeftVertex(G) = A;UpperVertex(G) = B;P: Point;PointOnCurve(P, LineSegmentOf(A, B)) = True",
+      "coords": {}
+    },
+    {
+      "index": 1595,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "y",
+        "x_coef": 3.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1595.png",
+      "answer": "8",
+      "fact_expr": "C: Ellipse;G: Line;F2: Point;M: Point;N: Point;F1: Point;Expression(C) = (x^2/3 + y^2/4 = 1);Focus(C) = {F1, F2};PointOnCurve(F1,G);Intersection(G,C) = {M,N}",
+      "coords": {}
+    },
+    {
+      "index": 1596,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1596.png",
+      "answer": "5",
+      "fact_expr": "A: Point;Coordinate(A) = (-1, 1);C: Ellipse;Expression(C) = (x^2/4 + y^2/3 = 1);M: Point;PointOnCurve(M, C) = True;F: Point;RightFocus(C) = F",
+      "coords": {
+        "A": [
+          -1.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 1597,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1598,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 2.23606797749979,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 5.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1598.png",
+      "answer": "x^2/3-y^2=1",
+      "fact_expr": "C: Hyperbola;G: Ellipse;Expression(G) = (x^2/9 + y^2/5 = 1);Focus(C) = Focus(G);Expression(OneOf(Asymptote(C))) = (x + sqrt(3)*y = 0)",
+      "coords": {}
+    },
+    {
+      "index": 1599,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1600,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1600.png",
+      "answer": "(3 + pm*sqrt(5))/4",
+      "fact_expr": "G: Parabola;p: Number;H: Line;A: Point;B: Point;p>0;Expression(G) = (y^2 = 2*(p*x));PointOnCurve(Focus(G), H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = 3;YCoordinate(MidPoint(LineSegmentOf(A, B))) = 1/2",
+      "coords": {}
+    },
+    {
+      "index": 1601,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1601.png",
+      "answer": "(1+sqrt(145))/18",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;G: Line;A: Point;B: Point;PointOnCurve(F2, G);Intersection(G, C) = {A, B};Abs(LineSegmentOf(A, B)) = Abs(LineSegmentOf(F1, F2));Abs(LineSegmentOf(F1, A)) = 1/2*Abs(LineSegmentOf(F1, B))",
+      "coords": {}
+    },
+    {
+      "index": 1602,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1602.png",
+      "answer": "(0,1/2)",
+      "fact_expr": "G: Parabola;Expression(G) = (y = x^2/2)",
+      "coords": {}
+    },
+    {
+      "index": 1603,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1604,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1604.png",
+      "answer": "4",
+      "fact_expr": "E: Ellipse;G: Line;m: Number;F1: Point;A: Point;B: Point;Expression(E) = (x^2/25 + y^2/9 = 1);Expression(G) = (x = m);LeftFocus(E) = F1;Intersection(G, E) = {A, B};WhenMax(Perimeter(TriangleOf(F1,A,B)))",
+      "coords": {}
+    },
+    {
+      "index": 1605,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1606,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1606.png",
+      "answer": "1/3",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;e: Number;H: Line;A: Point;P: Point;B: Point;C: Point;Q: Point;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = e ;e = 1/2;PointOnCurve(Center(G), H);Intersection(H, G) = {A, B};Quadrant(A) = 1;Z: Line;PointOnCurve(A, Z);IsPerpendicular(Z, xAxis);Intersection(Z, G) = C;PointOnCurve(A, LineOf(A, P));IsPerpendicular(LineOf(A, P), LineSegmentOf(A, B));Intersection(LineOf(A, P), G) = P;Intersection(LineSegmentOf(B, P), LineSegmentOf(A, C)) = Q",
+      "coords": {}
+    },
+    {
+      "index": 1607,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "y",
+        "x_coef": 16.0,
+        "y_coef": 25.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1607.png",
+      "answer": "(0,pm*3)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/25 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 1608,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1609,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1609.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "G: Hyperbola;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 - y^2 = 1);Focus(G) ={F1,F2};PointOnCurve(P, G);AngleOf(F1,P,F2)=ApplyUnit(60,degree)",
+      "coords": {}
+    },
+    {
+      "index": 1610,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1610.png",
+      "answer": "16",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);Focus(C)=Focus(G);Expression(LeftDirectrix(C))=(x=-1);P: Point;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P,RightPart(C))",
+      "coords": {}
+    },
+    {
+      "index": 1611,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1611.png",
+      "answer": "[2*sqrt(2) - 3, 56/9]",
+      "fact_expr": "C: Circle;Expression(C) = (y^2 + (x - 1)^2 = 1);G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);P: Point;PointOnCurve(P, G);L1: Line;L2: Line;TangentPoint(L1, C) = A;TangentPoint(L2, C) = B;A: Point;B: Point;TangentOfPoint(P, C) = {L1, L2}",
+      "coords": {}
+    },
+    {
+      "index": 1612,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1612.png",
+      "answer": "x^2/4-y^2=1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;FocalLength(G) = 2*sqrt(5);Length(RealAxis(G)) = 2*Length(ImageinaryAxis(G))",
+      "coords": {}
+    },
+    {
+      "index": 1613,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "vertical"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1613.png",
+      "answer": "y=pm*(x/2)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/16 + y^2/4 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 1614,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1614.png",
+      "answer": "[sqrt(7)/4, 1)",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/9 + x^2/a = 1);a: Number;a>9;H: Line;Expression(H) = (x - sqrt(3)*y = 0);Negation(Distance(OneOf(Vertex(MajorAxis(G))),H) < 2)",
+      "coords": {}
+    },
+    {
+      "index": 1615,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1615.png",
+      "answer": "5/6",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x);F: Point;Focus(G) = F;H: Line;A: Point;B: Point;PointOnCurve(F, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = 25/12;Abs(LineSegmentOf(A, F)) < Abs(LineSegmentOf(B, F))",
+      "coords": {}
+    },
+    {
+      "index": 1616,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1616.png",
+      "answer": "sqrt(2)+1",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(OneOf(Focus(G))) = (m, 0);Coordinate(P) = (m, 2*m);PointOnCurve(P, G);m:Number;m>0",
+      "coords": {}
+    },
+    {
+      "index": 1617,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1617.png",
+      "answer": "9",
+      "fact_expr": "G: Hyperbola;m: Real;Expression(G) = (x^2/7 - y^2/m = 1);PointOnCurve(Focus(G), xAxis);FocalLength(G) = 8",
+      "coords": {}
+    },
+    {
+      "index": 1618,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1618.png",
+      "answer": "6",
+      "fact_expr": "G: Ellipse;M: Point;Expression(G) = (x^2/25 + y^2/16 = 1);PointOnCurve(M, G);Distance(M, RightDirectrix(G)) = 20/3",
+      "coords": {}
+    },
+    {
+      "index": 1619,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1620,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1620.png",
+      "answer": "3",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;P: Point;O: Origin;Negation(P=O);PointOnCurve(P, G);H: Line;PointOnCurve(P, H);Z: Parabola;Expression(Z) = (y^2 = 8*p*x);A: Point;B: Point;Intersection(H, Z)={A, B};Q: Point;Intersection(LineOf(O,P), Z)={P, Q}",
+      "coords": {}
+    },
+    {
+      "index": 1621,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1621.png",
+      "answer": "5",
+      "fact_expr": "l: Line;C: Hyperbola;P: Point;F1: Point;Q: Point;Expression(C) = (x^2/4 - y^2 = 1);PointOnCurve(P, RightPart(C));OneOf(Asymptote(C))=l;Projection(P,l)=Q;LeftFocus(C)=F1",
+      "coords": {}
+    },
+    {
+      "index": 1622,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.4149149626832767,
+        "b": 2.450642358201144,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 1.0319236343469396e-05,
+        "converged": true,
+        "iterations": 94
+      },
+      "output_path": "results/test/hyperbola/problem_1622.png",
+      "answer": "(pm*2, 0)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/b^2 = 1);b: Number;b>0;Expression(OneOf(Asymptote(G))) = (y = sqrt(3)*x)",
+      "coords": {}
+    },
+    {
+      "index": 1623,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1623.png",
+      "answer": "32/15",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);A: Point;RightVertex(G) = A;RightFocus(G) = F;F: Point;PointOnCurve(F, H) = True;IsParallel(H, OneOf(Asymptote(G))) = True;H: Line;Intersection(H, G) = B;B: Point",
+      "coords": {}
+    },
+    {
+      "index": 1624,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1624.png",
+      "answer": "x^2/16+y^2/12=1",
+      "fact_expr": "O: Origin;Center(H) = O;H: Ellipse;PointOnCurve(I, H);I: Point;Coordinate(I) = (2, -3);G: Parabola;Expression(G) = (y^2 = 8*x);OneOf(Focus(H)) = Focus(G)",
+      "coords": {
+        "I": [
+          2.0,
+          -3.0
+        ]
+      }
+    },
+    {
+      "index": 1625,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1626,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1627,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 0.5773502691896257,
+        "major_axis": "x",
+        "x_coef": 1.0,
+        "y_coef": 0.3333333333333333
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1627.png",
+      "answer": "2",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2 + 3*y^2 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 1628,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.7320508075688772,
+        "b": 1.4142135623730951,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1628.png",
+      "answer": "y^2/12-x^2/18=1",
+      "fact_expr": "G: Hyperbola;C:Hyperbola;M: Point;Expression(G) = (x^2/3 - y^2/2 = 1);Coordinate(M) = (2*sqrt(3), 2*sqrt(5));Asymptote(G) = Asymptote(C);PointOnCurve(M,C)",
+      "coords": {
+        "M": [
+          3.4641016151377544,
+          4.47213595499958
+        ]
+      }
+    },
+    {
+      "index": 1629,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1629.png",
+      "answer": "3",
+      "fact_expr": "G: Parabola;p: Number;E: Point;O: Origin;M: Point;F: Point;l: Line;p>0;Expression(G) = (y^2 = 2*(p*x));PointOnCurve(M, G);Focus(G) = F;L:Line;PointOnCurve(M, L);IsPerpendicular(L, l);FootPoint(L, l)=E;Abs(LineSegmentOf(E, O)) = Abs(LineSegmentOf(M, F));XCoordinate(M) = 3;Directrix(G)=l",
+      "coords": {}
+    },
+    {
+      "index": 1630,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1630.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/b^2 - y^2/a^2 = 1);b: Number;a: Number;l1: Line;l2: Line;Asymptote(G) = {l1, l2};IsPerpendicular(l1, l2)",
+      "coords": {}
+    },
+    {
+      "index": 1631,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 2.23606797749979,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1631.png",
+      "answer": "14/3",
+      "fact_expr": "G: Hyperbola;Expression(G) = (5*x^2 - 4*y^2 = 20);P: Point;PointOnCurve(P, G) = True;Distance(P, RightFocus(G)) = 3",
+      "coords": {}
+    },
+    {
+      "index": 1632,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_focus_mismatch",
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1632.png",
+      "answer": "[3/5, 3/4]",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/b^2 + y^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;Coordinate(F) = (0, -3);OneOf(Focus(G)) = F;A: Point;Coordinate(A) = (0, 2);P: Point;PointOnCurve(P, G);Abs(LineSegmentOf(P, A)) + Abs(LineSegmentOf(P, F)) = 9",
+      "coords": {
+        "F": [
+          0.0,
+          -3.0
+        ],
+        "A": [
+          0.0,
+          2.0
+        ]
+      },
+      "validation_reasons": [
+        "ellipse_focus_mismatch"
+      ]
+    },
+    {
+      "index": 1633,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1633.png",
+      "answer": "100/3",
+      "fact_expr": "H: Ellipse;Expression(H) = (x^2/25 + y^2/16 = 1);G: Parabola;Center(H) = Vertex(G);LeftDirectrix(H) = Directrix(G);Intersection(G,RightDirectrix(H)) = {A,B};A: Point;B: Point",
+      "coords": {}
+    },
+    {
+      "index": 1634,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 2.23606797749979,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1634.png",
+      "answer": "sqrt(6)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/5 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 1635,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1635.png",
+      "answer": "x^2/4-y^2/12=1",
+      "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/25 + y^2/9 = 1);Focus(G)=Focus(H);Eccentricity(G)=2",
+      "coords": {}
+    },
+    {
+      "index": 1636,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1636.png",
+      "answer": "8",
+      "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/25 + y^2/9 = 1);PointOnCurve(P, G);Distance(P, LeftFocus(G)) = 2",
+      "coords": {}
+    },
+    {
+      "index": 1637,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_point_off_curve",
+      "params": {
+        "a": 1.9960020780563354,
+        "b": 1.4970015287399292,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "iterations": 2
+      },
+      "output_path": "results/test/hyperbola/problem_1637.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;Q: Point;F1: Point;M: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(P) = (-sqrt(2), -sqrt(3));Coordinate(Q) = (sqrt(15)/3, sqrt(2));PointOnCurve(P,C);PointOnCurve(Q,C);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(M,C);AngleOf(F1, M, F2) = ApplyUnit(120, degree)",
+      "coords": {
+        "P": [
+          -1.4142135623730951,
+          -1.7320508075688772
+        ],
+        "Q": [
+          1.2909944487358056,
+          1.4142135623730951
+        ]
+      },
+      "validation_reasons": [
+        "hyperbola_point_off_curve"
+      ]
+    },
+    {
+      "index": 1638,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1638.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "C1: Hyperbola;Expression(C1) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C1) = F1;RightFocus(C1) = F2;PointOnCurve(F1, l) = True;Intersection(l, RightPart(C1)) = P;l: Line;P: Point;IsPerpendicular(LineSegmentOf(P, F2), xAxis) = True;Sin(AngleOf(P, F1, F2)) = 1/3",
+      "coords": {}
+    },
+    {
+      "index": 1639,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1639.png",
+      "answer": "8*sqrt(5)/5",
+      "fact_expr": "P: Point;G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);PointOnCurve(P, G) = True;l: Line;Expression(l) = (x + 2*y - 12 = 0)",
+      "coords": {}
+    },
+    {
+      "index": 1640,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          -0.0,
+          2.0
+        ],
+        "radius": 6.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/circle/problem_1640.png",
+      "answer": "y^2/9 + x^2/5 = 1",
+      "fact_expr": "C1: Circle;Expression(C1) = (-4*y + x^2 + y^2 - 32 = 0);B: Point;PointOnCurve(B, C1);A: Point;Coordinate(A) = (0, -2);C: Point;Center(C1) = C;P: Point;Intersection(PerpendicularBisector(LineSegmentOf(A,B)), LineSegmentOf(B, C)) = P;Locus(P) = E;E: Curve",
+      "coords": {}
+    },
+    {
+      "index": 1641,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1641.png",
+      "answer": "sqrt(3)/3",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F2: Point;F1: Point;Z: Line;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;PointOnCurve(F1, Z);IsPerpendicular(Z, xAxis);Intersection(Z, G) = P;RightFocus(G) = F2;AngleOf(F1, P, F2) = ApplyUnit(60, degree)",
+      "coords": {}
+    },
+    {
+      "index": 1642,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1642.png",
+      "answer": "1/2\n{((2*sqrt(2))*x-y-4*sqrt(2)=0), ((2*sqrt(2))*x+y-4*sqrt(2)=0)}",
+      "fact_expr": "E: Parabola;Expression(E) = (y^2 = 8*x);F: Point;Focus(E) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, E) = {A, B};VectorOf(A,B) = 3*VectorOf(F,B)",
+      "coords": {}
+    },
+    {
+      "index": 1643,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1643.png",
+      "answer": "8*x+25*y-58=0",
+      "fact_expr": "M: Point;Coordinate(M) = (1, 2);l: Line;PointOnCurve(M, l);G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);A: Point;B: Point;Intersection(l, G) = {A, B};MidPoint(LineSegmentOf(A, B)) = M",
+      "coords": {
+        "M": [
+          1.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 1644,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1645,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.6457513110645907,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 7.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1645.png",
+      "answer": "3",
+      "fact_expr": "G: Ellipse;F1: Point;M: Point;F2: Point;O: Origin;Expression(G) = (x^2/7 + y^2/3 = 1);PointOnCurve(M, G);Focus(G) = {F1, F2};2*LineSegmentOf(M, O) = LineSegmentOf(F1, F2)",
+      "coords": {}
+    },
+    {
+      "index": 1646,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          4.0,
+          0.0
+        ],
+        "radius": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/circle/problem_1646.png",
+      "answer": "9",
+      "fact_expr": "G: Circle;C:Circle;H: Curve;P: Point;M: Point;N: Point;Expression(H)=(sqrt((x+4)^2+y^2)+sqrt((x-4)^2+y^2)=12);Expression(G) = (y^2 + (x + 4)^2 = 4);Expression(C) = (y^2 + (x - 4)^2 = 1);PointOnCurve(P,H);PointOnCurve(M, G);PointOnCurve(N, C)",
+      "coords": {}
+    },
+    {
+      "index": 1647,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1647.png",
+      "answer": "1/4",
+      "fact_expr": "H: Parabola;Expression(H) = (y^2 = 2*p*x);p: Number;p>0;M: Point;Coordinate(M) = (1, m);m: Number;PointOnCurve(M, H);Distance(M, Focus(H)) = 5;G: Hyperbola;Expression(G) = (x^2 - y^2/a = 1);a: Real;A: Point;LeftVertex(G) = A;IsPerpendicular(OneOf(Asymptote(G)), LineOf(A, M))",
+      "coords": {}
+    },
+    {
+      "index": 1648,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1648.png",
+      "answer": "(4,2)",
+      "fact_expr": "H: Line;Expression(H) = (x - y = 2);G: Parabola;Expression(G) = (y^2 = 4*x);A: Point;B: Point;Intersection(H, G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 1649,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1650,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1650.png",
+      "answer": "{2,-14}",
+      "fact_expr": "G: Parabola;a: Real;H: Circle;Expression(G) = (y^2 = 2*(a*x));Expression(H) = (y^2 + (x - 3)^2 = 16);IsTangent(Directrix(G),H);Negation(a=0)",
+      "coords": {}
+    },
+    {
+      "index": 1651,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1651.png",
+      "answer": "x^2/4-y^2=1",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;A: Point;O:Origin;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);L:Line;PointOnCurve(RightVertex(C),L);IsPerpendicular(xAxis,L);Intersection(L,OneOf(Asymptote(C)))=A;RightFocus(C)=Center(G);Radius(G)=4;PointOnCurve(A,G);PointOnCurve(O,G)",
+      "coords": {}
+    },
+    {
+      "index": 1652,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1653,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1653.png",
+      "answer": "4/3",
+      "fact_expr": "G: Parabola;Q: Point;P: Point;F: Point;H:Line;Expression(G) = (y^2 = 4*x);Focus(G)=F;PointOnCurve(P,G);PointOnCurve(P,H);IsPerpendicular(H,Directrix(G));FootPoint(H,Directrix(G))=Q;AngleOf(Q, P, F) = ApplyUnit(120, degree)",
+      "coords": {}
+    },
+    {
+      "index": 1654,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1654.png",
+      "answer": "a^2*b^2/(b^2-a^2)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;b > a;a > 0;A: Point;B: Point;PointOnCurve(A, G);PointOnCurve(B, G);O: Origin;IsPerpendicular(LineSegmentOf(O, A), LineSegmentOf(O, B))",
+      "coords": {}
+    },
+    {
+      "index": 1655,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1655.png",
+      "answer": "1/2",
+      "fact_expr": "O: Point;Vertex(G)=O;F: Point;Focus(G) =F;G: Parabola;Expression(G) = (y = x^2/2);H: Line;PointOnCurve(F,H);A: Point;B: Point;Intersection(H,G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 1656,
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+        "converged": true,
+        "iterations": 95
+      },
+      "output_path": "results/test/hyperbola/problem_1656.png",
+      "answer": "x^2/9-y^2/16=1",
+      "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/49 + y^2/24 = 1);Focus(G)=Focus(H);Expression(Asymptote(G))=(y=pm*(4/3)*x)",
+      "coords": {}
+    },
+    {
+      "index": 1657,
+      "success": true,
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+        "b": 1.4142135623730951,
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+        "y_coef": 2.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1657.png",
+      "answer": "x^2+y^2=4/3",
+      "fact_expr": "l: Line;Gamma: Ellipse;H: Circle;O: Origin;P: Point;Q: Point;Expression(Gamma)=(x^2/4+y^2/2=1);Intersection(l,Gamma) = {P, Q};IsPerpendicular(LineSegmentOf(O, P), LineSegmentOf(O, Q));Center(H)=O;IsTangent(l,H)",
+      "coords": {}
+    },
+    {
+      "index": 1658,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
+      },
+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1658.png",
+      "answer": "(1/2,2)",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);Focus(G) = F;F: Point;A: Point;Coordinate(A) = (1, 2);P: Point;PointOnCurve(P, G) = True;WhenMin(Perimeter(TriangleOf(P,A,F))) = True",
+      "coords": {
+        "A": [
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+          2.0
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+      }
+    },
+    {
+      "index": 1659,
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+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 2.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1659.png",
+      "answer": "pm*1",
+      "fact_expr": "G: Hyperbola;H: Line;m: Number;A: Point;B: Point;Expression(G) = (x^2 - y^2/4 = 1);Expression(H) = (y = m + x);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = sqrt(2)*(8/3)",
+      "coords": {}
+    },
+    {
+      "index": 1660,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1661,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.7224669456481934,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 2.391175712546101e-06,
+        "converged": true,
+        "iterations": 83
+      },
+      "output_path": "results/test/hyperbola/problem_1661.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/3 + x^2/a^2 = 1);a: Number;a>0;Eccentricity(C) = sqrt(2);G: Parabola;Expression(G) = (y^2 = 8*x)",
+      "coords": {}
+    },
+    {
+      "index": 1662,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 6.0,
+        "y_coef": 2.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1662.png",
+      "answer": "y=pm*sqrt(3)*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (n*y^2 + x^2 = 1);n: Real;H: Ellipse;Expression(H) = (x^2/6 + y^2/2 = 1);Focus(G) = Focus(H)",
+      "coords": {}
+    },
+    {
+      "index": 1663,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1664,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1664.png",
+      "answer": "3",
+      "fact_expr": "G: Hyperbola;a: Number;b: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(Asymptote(G)) = (y = pm*2*sqrt(2)*x)",
+      "coords": {}
+    },
+    {
+      "index": 1665,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "vertical"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1665.png",
+      "answer": "y=pm*sqrt(2)*x",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-x^2/b^2 + y^2/a^2 = 1);Length(ImageinaryAxis(G))=2;FocalLength(G) = 2*sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 1666,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1666.png",
+      "answer": "x^2/4+y^2=1",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;A: Point;B: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x + 2*y - 2 = 0);Coordinate(MidPoint(LineSegmentOf(A, B))) = (m, 1/2);Intersection(G, H) = {A, B};Abs(LineSegmentOf(A, B)) = sqrt(5);m:Number",
+      "coords": {}
+    },
+    {
+      "index": 1667,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1668,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1668.png",
+      "answer": "sqrt(2)-1",
+      "fact_expr": "C: Parabola;p: Number;lambda:Number;T: Point;P: Point;F: Point;Q: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;Intersection(Directrix(C),xAxis)=T;PointOnCurve(P,C);PointOnCurve(Q,C);IsTangent(LineOf(T,P),C);VectorOf(T, P) = lambda*VectorOf(F, Q);lambda>0",
+      "coords": {}
+    },
+    {
+      "index": 1669,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 8.0,
+        "y_coef": 4.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1669.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;H: Ellipse;Expression(H) = (x^2/8 + y^2/4 = 1);Focus(G) = RightFocus(H)",
+      "coords": {}
+    },
+    {
+      "index": 1670,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1671,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 2.85904548036342e-06,
+        "converged": true,
+        "iterations": 88
+      },
+      "output_path": "results/test/parabola/problem_1671.png",
+      "answer": "4\nx=-2",
+      "fact_expr": "G: Parabola;p: Number;H: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(H) = (2, -4);PointOnCurve(H, G)",
+      "coords": {
+        "H": [
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+          -4.0
+        ]
+      }
+    },
+    {
+      "index": 1672,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1673,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1674,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1674.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 2*(p*y));p: Number;p>0;H: Point;m: Number;Coordinate(H) = (m, 3);PointOnCurve(H, G);Distance(H, Focus(G)) = 5",
+      "coords": {}
+    },
+    {
+      "index": 1675,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
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+          0.0
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+        "radius": 1.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/circle/problem_1675.png",
+      "answer": "(4*x^2-4*y^2/15=1)&(x<=-1/2)",
+      "fact_expr": "M: Circle;C1:Circle;C2:Circle;Expression(C1)=((x+2)^2+y^2=1);Expression(C2)=((x-2)^2+y^2=4);IsOutTangent(M,C1);IsOutTangent(M,C2)",
+      "coords": {}
+    },
+    {
+      "index": 1676,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1676.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "G: Parabola;H: Line;O: Origin;P: Point;Q: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(Focus(G), H);Inclination(H)=(3/4)*pi;Intersection(H, G) = {P, Q}",
+      "coords": {}
+    },
+    {
+      "index": 1677,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1678,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1678.png",
+      "answer": "[1/2, 1)",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b + x^2/a = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(C) = F;K: Point;PointOnCurve(K, C);d: Number;Max(Distance(K, F)) = d;P: Point;PointOnCurve(P, RightDirectrix(C));LineSegmentOf(P, F) = d",
+      "coords": {}
+    },
+    {
+      "index": 1679,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1679.png",
+      "answer": "4",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);P: Point;PointOnCurve(P, G) = True;F1: Point;OneOf(Focus(G)) = F1;Distance(P, F1) = 2;M: Point;MidPoint(LineSegmentOf(P, F1)) = M;O: Origin",
+      "coords": {}
+    },
+    {
+      "index": 1680,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1680.png",
+      "answer": "{5/3, 5/4}",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/n + x^2/m = 1);m: Number;n: Number;Expression(OneOf(Asymptote(G))) = (y = (4/3)*x);e: Number;Eccentricity(G) = e",
+      "coords": {}
+    },
+    {
+      "index": 1681,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1681.png",
+      "answer": "(1, 2]",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F2: Point;F1: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, RightPart(G));Abs(LineSegmentOf(P, F1)) = 3*Abs(LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 1682,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1683,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1683.png",
+      "answer": "5*sqrt(3)",
+      "fact_expr": "l: Line;C: Parabola;p: Number;M: Point;N: Point;F: Point;O: Origin;p>0;Expression(C) = (y^2 = 2*p*x);Focus(C)=F;PointOnCurve(F,l);Intersection(l,C)={M,N};VectorOf(M, F) = 4*VectorOf(F, N);Area(TriangleOf(O,M,N))=sqrt(3)*Abs(LineSegmentOf(M,N))",
+      "coords": {}
+    },
+    {
+      "index": 1684,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1684.png",
+      "answer": "y=-1",
+      "fact_expr": "G: Parabola;Expression(G) = (y = x^2/4)",
+      "coords": {}
+    },
+    {
+      "index": 1685,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 2.8284271247461903,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 8.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1685.png",
+      "answer": "-12",
+      "fact_expr": "l: Line;G: Ellipse;B: Point;A: Point;M: Point;Expression(G) = (x^2/16 + y^2/8 = 1);Coordinate(M) = (1, 1);Intersection(l, G) = {A, B};MidPoint(LineSegmentOf(A,B)) = M;k:Number;Slope(l)=k",
+      "coords": {
+        "M": [
+          1.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 1686,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1686.png",
+      "answer": "sqrt(3)/3",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;L: Line;F2: Point;P: Point;Q: Point;F1: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1,L);Intersection(L, G) = {P, Q};TangentPoint(InscribedCircle(TriangleOf(P,F2,Q)),LineSegmentOf(P,F2))=MidPoint(LineSegmentOf(P,F2));TangentPoint(InscribedCircle(TriangleOf(P,F2,Q)),LineSegmentOf(P,Q))=F1",
+      "coords": {}
+    },
+    {
+      "index": 1687,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_focus_mismatch",
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1687.png",
+      "answer": "x^2/4+y^2/2=1",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;Coordinate(F) = (sqrt(2), 0);RightFocus(C) = F;G: Line;PointOnCurve(F, G);IsPerpendicular(G, xAxis);Length(InterceptChord(G, C)) = 2",
+      "coords": {
+        "F": [
+          1.4142135623730951,
+          0.0
+        ]
+      },
+      "validation_reasons": [
+        "ellipse_focus_mismatch"
+      ]
+    },
+    {
+      "index": 1688,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1688.png",
+      "answer": "[3/8,3/4]",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2/3 = 1);A1: Point;A2: Point;LeftVertex(C) = A1;RightVertex(C) = A2;P: Point;PointOnCurve(P, C) = True;Range(Slope(LineOf(P, A1))) = [-2, -1]",
+      "coords": {}
+    },
+    {
+      "index": 1689,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1689.png",
+      "answer": "sqrt(2) + 1",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;O: Origin;Q: Point;F: Point;x0: Number;y0: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(Q) = (x0, y0);RightFocus(C) = F;Quadrant(P) = 1;PointOnCurve(P, C);a*y0 + b*x0 = 0;PerpendicularBisector(LineSegmentOf(O, P)) = LineSegmentOf(Q, F);PerpendicularBisector(LineSegmentOf(Q, F)) = LineSegmentOf(O, P)",
+      "coords": {}
+    },
+    {
+      "index": 1690,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1690.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/9 + x^2/a^2 = 1);a: Number;a>0;Expression(Asymptote(G)) = (3*x + pm*2*y = 0)",
+      "coords": {}
+    },
+    {
+      "index": 1691,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1691.png",
+      "answer": "8",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;PointOnCurve(F, l) = True;Slope(l) = -1;l: Line;Intersection(l, C) = {A, B};A: Point;B: Point",
+      "coords": {}
+    },
+    {
+      "index": 1692,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1692.png",
+      "answer": "2*sqrt(3)/3",
+      "fact_expr": "G: Hyperbola;a: Number;H: Parabola;Expression(G) = (-y^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 8*x);Focus(H) = OneOf(Focus(G))",
+      "coords": {}
+    },
+    {
+      "index": 1693,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1694,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1694.png",
+      "answer": "(x-1)^2+y^2=4",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);H: Circle;Focus(G) = Center(H);Radius(H) = 2",
+      "coords": {}
+    },
+    {
+      "index": 1695,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1695.png",
+      "answer": "6",
+      "fact_expr": "l: Line;C: Parabola;A: Point;B: Point;F: Point;O: Origin;Coordinate(F) = (1, 0);Vertex(C) = O;Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};Coordinate(MidPoint(LineSegmentOf(A, B)))= (2,m);IsChordOf(LineSegmentOf(A, B),C);m:Number",
+      "coords": {
+        "F": [
+          1.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 1696,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1697,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1697.png",
+      "answer": "19/2",
+      "fact_expr": "G: Parabola;Expression(G) = (y = x^2/2);P: Point;PointOnCurve(P, G);M: Point;Projection(P, xAxis) = M;A: Point;Coordinate(A) = (6, 17/2)",
+      "coords": {
+        "A": [
+          6.0,
+          8.5
+        ]
+      }
+    },
+    {
+      "index": 1698,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.366584062576294,
+        "b": 2.364173412322998,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.00017838670464698225,
+        "converged": true,
+        "iterations": 89
+      },
+      "output_path": "results/test/hyperbola/problem_1698.png",
+      "answer": "8",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;Eccentricity(G) = 2;M: Point;PointOnCurve(M,LeftPart(G)) = True;L: Line;PointOnCurve(M,L) = True;IsParallel(L,xAxis) = True;Intersection(L,Asymptote(G)) = {P,Q};P: Point;Q: Point;Abs(LineSegmentOf(M, Q))*Abs(LineSegmentOf(P, M)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 1699,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.23606797749979,
+        "major_axis": "y",
+        "x_coef": 5.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1699.png",
+      "answer": "10",
+      "fact_expr": "C: Ellipse;M: Point;F1: Point;F2: Point;Expression(C) = (x^2/5 + y^2/9 = 1);PointOnCurve(M, C);Focus(C) = {F1, F2}",
+      "coords": {}
+    },
+    {
+      "index": 1700,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1701,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1701.png",
+      "answer": "(sqrt(6)-sqrt(2))/4",
+      "fact_expr": "G: Ellipse;A: Point;B: Point;P: Point;Expression(G) = (x^2/4 + y^2 = 1);Coordinate(A) = (-2, 1);Coordinate(B) = (2, 1);PointOnCurve(P, G)",
+      "coords": {
+        "A": [
+          -2.0,
+          1.0
+        ],
+        "B": [
+          2.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 1702,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1702.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;e: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) =F1;RightFocus(G)=F2;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2));Area(TriangleOf(P, F1, F2)) = 2*a*b;Eccentricity(G) = e",
+      "coords": {}
+    },
+    {
+      "index": 1703,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1704,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1705,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1705.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Z1: Line;Z2: Line;OneOf(Asymptote(C)) = Z1;OneOf(Asymptote(C)) = Z2;Negation(Z1=Z2);IsPerpendicular(Z1, Z2);Distance(Vertex(C), Z1) = 1",
+      "coords": {}
+    },
+    {
+      "index": 1706,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1706.png",
+      "answer": "x^2/4-y^2/12=1",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;FocalLength(C) = 8;M: Point;Coordinate(M) = (1, sqrt(3));PointOnCurve(M, Asymptote(C))",
+      "coords": {
+        "M": [
+          1.0,
+          1.7320508075688772
+        ]
+      }
+    },
+    {
+      "index": 1707,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1707.png",
+      "answer": "sqrt(3)/2",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;A: Point;B: Point;LeftVertex(C) = A;RightVertex(C) = B;P: Point;Negation(P=A);Negation(P=B);PointOnCurve(P, C);m: Number;n: Number;Slope(LineOf(A, P)) = m;Slope(LineOf(B, P)) = n;WhenMin((a/b)*(3 - 2/(3*m*n)) + 2/(m*n) + 3*(ln(Abs(m)) + ln(Abs(n))))",
+      "coords": {}
+    },
+    {
+      "index": 1708,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.366584062576294,
+        "b": 2.364173412322998,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.00017838670464698225,
+        "converged": true,
+        "iterations": 89
+      },
+      "output_path": "results/test/hyperbola/problem_1708.png",
+      "answer": "3",
+      "fact_expr": "E: Hyperbola;Expression(E) = (x^2/4 - y^2/b^2 = 1);Eccentricity(E) = 2;b: Number;b>0;P: Point;PointOnCurve(P, E) = True;A1: Line;A2: Line;Asymptote(E) = {A1, A2}",
+      "coords": {}
+    },
+    {
+      "index": 1709,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1710,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1710.png",
+      "answer": "25",
+      "fact_expr": "G: Ellipse;m: Number;P: Point;F1:Point;F2:Point;Expression(G) = (y^2/16 + x^2/m = 1);Distance(P, F1) = 3;Distance(P, F2) = 7;PointOnCurve(P, G);OneOf(Focus(G))=F1;OneOf(Focus(G))=F2;Negation(F1=F2)",
+      "coords": {}
+    },
+    {
+      "index": 1711,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1711.png",
+      "answer": "x=-1\n{(4,4),(4,-4)}",
+      "fact_expr": "G: Parabola;P: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(P, G);Abs(LineSegmentOf(P, F)) = 5",
+      "coords": {}
+    },
+    {
+      "index": 1712,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1713,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1714,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1714.png",
+      "answer": "{x+3*y-2=0,x-y+2=0}",
+      "fact_expr": "m: Line;G: Parabola;A: Point;B: Point;P: Point;Expression(G) = (x^2 = y);Coordinate(P) = (-1, 1);PointOnCurve(P, m);Intersection(m, xAxis) = A;PointOnCurve(B,G);IsPerpendicular(LineSegmentOf(P,A),LineSegmentOf(P,B));IsTangent(LineSegmentOf(A, B), G)",
+      "coords": {
+        "P": [
+          -1.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 1715,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1716,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1717,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1717.png",
+      "answer": "sqrt(13)/2",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 6*x);F: Point;Focus(C) = F;l: Line;PointOnCurve(F, l);A: Point;Intersection(l, C) = A;B: Point;Intersection(l, Directrix(C)) = B;VectorOf(F, B) = 3*VectorOf(F, A);O: Origin",
+      "coords": {}
+    },
+    {
+      "index": 1718,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1718.png",
+      "answer": "sqrt(3)/3",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;A: Point;B: Point;F2: Point;F1: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;UpperVertex(G) = A;Intersection(PerpendicularBisector(LineSegmentOf(A, F2)), G) = B;PointOnCurve(F1,LineSegmentOf(A,B))",
+      "coords": {}
+    },
+    {
+      "index": 1719,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1720,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.4641016151377544,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 12.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1720.png",
+      "answer": "x^2/4-y^2/5=1",
+      "fact_expr": "C: Hyperbola;G: Ellipse;Expression(G) = (x^2/12 + y^2/3 = 1);Focus(G) = Focus(C);Expression(OneOf(Asymptote(C))) = (y = x*(sqrt(5)/2))",
+      "coords": {}
+    },
+    {
+      "index": 1721,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1721.png",
+      "answer": "10",
+      "fact_expr": "C: Hyperbola;A: Point;P: Point;F: Point;Expression(C) = (x^2 - y^2/3 = 1);Coordinate(A) = (0, 2*sqrt(3));RightFocus(C) = F;PointOnCurve(P, LeftPart(C))",
+      "coords": {
+        "A": [
+          0.0,
+          3.4641016151377544
+        ]
+      }
+    },
+    {
+      "index": 1722,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1723,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1724,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.23606797749979,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1724.png",
+      "answer": "x^2/10+y^2/4=1",
+      "fact_expr": "G: Hyperbola;H: Ellipse;I: Point;Expression(G) = (x^2/5 - y^2 = 1);Coordinate(I) = (0, -2);Focus(G)=Focus(H);PointOnCurve(I, H)",
+      "coords": {
+        "I": [
+          0.0,
+          -2.0
+        ]
+      }
+    },
+    {
+      "index": 1725,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1725.png",
+      "answer": "2*pi/3\n2*sqrt(3)",
+      "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/9 + y^2/2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 1726,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1726.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 1727,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.463192105293274,
+        "b": 2.070901870727539,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 1.4661259228887502e-06,
+        "converged": true,
+        "iterations": 106
+      },
+      "output_path": "results/test/hyperbola/problem_1727.png",
+      "answer": "pm*1",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;H: Line;m: Number;B: Point;A: Point;a>0;b>0;a=1;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2 + y^2 = 5);Expression(H) = (m + x - y = 0);Eccentricity(C) = sqrt(3);Intersection(H, C) = {A, B};PointOnCurve(MidPoint(LineSegmentOf(A,B)),G);Negation(A=B)",
+      "coords": {}
+    },
+    {
+      "index": 1728,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "down"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1728.png",
+      "answer": "y = 2",
+      "fact_expr": "G: Parabola;Expression(G) = (y = -x^2/8)",
+      "coords": {}
+    },
+    {
+      "index": 1729,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1729.png",
+      "answer": "x^2/16+y^2/9=1",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Length(MajorAxis(C)) = 8;F1: Point;F2: Point;Focus(C) = {F1, F2};P: Point;PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2));Area(TriangleOf(P, F1, F2)) = 9",
+      "coords": {}
+    },
+    {
+      "index": 1730,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1730.png",
+      "answer": "",
+      "fact_expr": "l: Line;G: Parabola;F: Point;M: Point;N: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;Intersection(Directrix(G), xAxis) = A;PointOnCurve(A, l);Intersection(l, G) = {M, N};A:Point",
+      "coords": {}
+    },
+    {
+      "index": 1731,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1731.png",
+      "answer": "(sqrt(5)+1)/2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;A: Point;B: Point;O: Origin;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(A,G);PointOnCurve(B,G);c:Number;IsPerpendicular(LineSegmentOf(O,A),LineSegmentOf(O,B));Distance(O, LineOf(A,B)) = c",
+      "coords": {}
+    },
+    {
+      "index": 1732,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.4142135623730951,
+        "major_axis": "y",
+        "x_coef": 2.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1732.png",
+      "answer": "(0, pm*sqrt(2))",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/2 + y^2/4 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 1733,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 7.0,
+        "b": 6.324555320336759,
+        "major_axis": "x",
+        "x_coef": 49.0,
+        "y_coef": 40.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1733.png",
+      "answer": "sqrt(15)",
+      "fact_expr": "G: Ellipse;A: Point;M: Point;P: Point;Expression(G) = (x^2/49 + y^2/40 = 1);PointOnCurve(P,G);Coordinate(A) = (3, 0);Abs(VectorOf(A, M)) = 1;DotProduct(VectorOf(P, M), VectorOf(A, M)) = 0",
+      "coords": {
+        "A": [
+          3.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 1734,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_point_off_curve",
+      "params": {
+        "a": 2.825101613998413,
+        "b": 2.34489402930029,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 1.7647765222287931e-06,
+        "converged": true,
+        "iterations": 118
+      },
+      "output_path": "results/test/ellipse/problem_1734.png",
+      "answer": "1/2",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;A: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (2, 3);Coordinate(OneOf(Focus(C))) = (2,0);PointOnCurve(A, C)",
+      "coords": {
+        "A": [
+          2.0,
+          3.0
+        ]
+      },
+      "validation_reasons": [
+        "ellipse_point_off_curve"
+      ]
+    },
+    {
+      "index": 1735,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1735.png",
+      "answer": "3/8",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(C) = F;l: Line;PointOnCurve(F, l) = True;Expression(l) = (y = sqrt(3)*(-2*a + x));A: Point;Intersection(l,RightPart(C)) = A;B: Point;Intersection(l,yAxis) = B;Area(TriangleOf(O, B, F)) = 8*sqrt(3);O: Origin",
+      "coords": {}
+    },
+    {
+      "index": 1736,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1736.png",
+      "answer": "1/2",
+      "fact_expr": "G: Parabola;H: Line;F: Point;A: Point;B: Point;C: Point;D: Point;P: Point;Expression(G) = (y^2 = 4*x);Coordinate(P) = (2, 0);Focus(G) = F;PointOnCurve(P, H);Intersection(H, G) = {A, B};Intersection(LineOf(A, F), G) = C;Intersection(LineOf(B, F), G) = D;Slope(LineOf(A, B)) = k1;Slope(LineOf(C, D)) = k2;k1: Number;k2: Number",
+      "coords": {
+        "P": [
+          2.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 1737,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1738,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1738.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);P: Point;PointOnCurve(P, G);l1: Line;Expression(l1) = (4*x - 3*y - 7 = 0);l2: Line;Expression(l2) = (y = -1)",
+      "coords": {}
+    },
+    {
+      "index": 1739,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 3.4641016151377544,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1739.png",
+      "answer": "(pm*2, 0)\ny = pm*sqrt(3)*x",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Ellipse;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2/25 + y^2/9 = 1);Eccentricity(G) = 2;Focus(G) = Focus(H)",
+      "coords": {}
+    },
+    {
+      "index": 1740,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1740.png",
+      "answer": "(0, 2]+[8, +oo)",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2/m = 1);m: Number;F1: Point;F2: Point;Focus(C) = {F1, F2};P: Point;PointOnCurve(P, C);AngleOf(F1, P, F2) = ApplyUnit(90, degree)",
+      "coords": {}
+    },
+    {
+      "index": 1741,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1741.png",
+      "answer": "5",
+      "fact_expr": "M: Point;PointOnCurve(M, G) = True;G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;A: Point;PointOnCurve(A, H) = True;H: Circle;Expression(H) = ((x - 5)^2 + (y + 1)^2 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 1742,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1742.png",
+      "answer": "2*sqrt(2)/3",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);P: Point;Intersection(Directrix(G), xAxis) = P;H: Line;PointOnCurve(P, H);k: Number;Slope(H) = k;k>0;A: Point;B: Point;Intersection(H, G) = {A, B};F: Point;Focus(G) = F;Abs(LineSegmentOf(F, A)) = 2*Abs(LineSegmentOf(F, B))",
+      "coords": {}
+    },
+    {
+      "index": 1743,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1743.png",
+      "answer": "-b^2/a^2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;M: Point;N: Point;Endpoint(RealAxis(G)) = {M, N};Q: Point;PointOnCurve(Q, G);Negation(Q=M);Negation(Q=N);alpha: Number;AngleOf(Q, M, N) = alpha;beta: Number;AngleOf(Q, N, M) = beta",
+      "coords": {}
+    },
+    {
+      "index": 1744,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1744.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "G: Ellipse;O: Origin;A: Point;F2: Point;F1: Point;a: Number;b: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;Quadrant(A) = 1;PointOnCurve(A, G);DotProduct(VectorOf(O, A), VectorOf(O, F2)) = Abs(VectorOf(O, F2))^2;Expression(LineOf(O, A)) = (y = x*(sqrt(2)/2))",
+      "coords": {}
+    },
+    {
+      "index": 1745,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          2.0,
+          2.0
+        ],
+        "radius": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/circle/problem_1745.png",
+      "answer": "16",
+      "fact_expr": "C1: Circle;Expression(C1) = ((x - 2)^2 + (y - 2)^2 = 4);C2: Circle;Expression(C2) = ((x + 2)^2 + (y + 1)^2 = 2);A: Point;B: Point;P: Point;PointOnCurve(P, C1);IsChordOf(LineSegmentOf(A, B), C2);Abs(LineSegmentOf(A, B)) = 2",
+      "coords": {}
+    },
+    {
+      "index": 1746,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1746.png",
+      "answer": "1/2",
+      "fact_expr": "C: Hyperbola;O: Origin;N: Point;M: Point;Expression(C) = (x^2 - y^2 = 1);PointOnCurve(M, C);L:Line;PointOnCurve(M, L);IsPerpendicular(L, OneOf(Asymptote(C)));FootPoint(L, OneOf(Asymptote(C)))=N",
+      "coords": {}
+    },
+    {
+      "index": 1747,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1748,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1749,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1750,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 6.0,
+        "b": 8.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1750.png",
+      "answer": "26",
+      "fact_expr": "G: Hyperbola;P: Point;F2: Point;F1: Point;Expression(G) = (x^2/36 - y^2/64 = 1);PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G) = F2;Abs(LineSegmentOf(P, F2)) = 14",
+      "coords": {}
+    },
+    {
+      "index": 1751,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1751.png",
+      "answer": "y = pm*(3/2)*x",
+      "fact_expr": "l: Line;G: Hyperbola;b: Number;a: Number;H: Circle;F: Point;A: Point;B: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2 + y^2 = a^2);LeftFocus(G) = F;PointOnCurve(F, l);TangentPoint(l, H) = A;Intersection(l, RightPart(G)) = B;VectorOf(F, A) = (1/3)*VectorOf(F, B)",
+      "coords": {}
+    },
+    {
+      "index": 1752,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.7320508075688772,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1752.png",
+      "answer": "1",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2/3 - y^2 = 1);F: Point;OneOf(Focus(C)) = F",
+      "coords": {}
+    },
+    {
+      "index": 1753,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 4.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1753.png",
+      "answer": "3",
+      "fact_expr": "F: Point;Focus(C) = F;C: Parabola;Expression(C) = (x^2 = 16*y);l: Line;Expression(l) = (y = -1);A: Point;PointOnCurve(A, C) = True;PointOnCurve(A,LineSegmentOf(A, P)) = True;IsPerpendicular(LineSegmentOf(A,P),l) = True;FootPoint(LineSegmentOf(A,P),l) = P;P: Point",
+      "coords": {}
+    },
+    {
+      "index": 1754,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1755,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1756,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1756.png",
+      "answer": "-1",
+      "fact_expr": "C: Parabola;p: Number;A: Point;F: Point;Expression(C) = (y^2 = 2*(p*x));Coordinate(A) = (-2, 4);PointOnCurve(A, Directrix(C));Focus(C) = F",
+      "coords": {
+        "A": [
+          -2.0,
+          4.0
+        ]
+      }
+    },
+    {
+      "index": 1757,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1758,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.6277244091033936,
+        "b": 1.818325400352478,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 6.648646149187698e-07,
+        "converged": true,
+        "iterations": 99
+      },
+      "output_path": "results/test/hyperbola/problem_1758.png",
+      "answer": "y=pm*sqrt(5)*x/2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;Eccentricity(G) = 3/2",
+      "coords": {}
+    },
+    {
+      "index": 1759,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1760,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1760.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;p: Number;P: Point;F: Point;p>0;Expression(G)=(y^2 = 2*p*x);Coordinate(P)=(6, y1);PointOnCurve(P,G);Focus(G)=F;Distance(P,F) = 8;y1:Number",
+      "coords": {}
+    },
+    {
+      "index": 1761,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1761.png",
+      "answer": "2*sqrt(5)/5",
+      "fact_expr": "G: Hyperbola;A: Point;Expression(G) = (x^2/4 - y^2 = 1);Coordinate(A) = (0, 1)",
+      "coords": {
+        "A": [
+          0.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 1762,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1762.png",
+      "answer": "sqrt(3) + 1",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Focus(C) = {F1, F2};PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2));AngleOf(P, F1, F2) = ApplyUnit(30, degree)",
+      "coords": {}
+    },
+    {
+      "index": 1763,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1763.png",
+      "answer": "5",
+      "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (y = k*(x - 1));Coordinate(MidPoint(LineSegmentOf(A, B))) = (3/2, 1);Intersection(H, G) = {A, B};k:Number",
+      "coords": {}
+    },
+    {
+      "index": 1764,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1764.png",
+      "answer": "2/3",
+      "fact_expr": "F: Point;A: Point;LeftFocus(Gamma) = F;LowerVertex(Gamma) = A;Gamma: Ellipse;Expression(Gamma) = (x^2/2 + y^2 = 1);M: Point;x0: Number;y0: Number;Coordinate(M) = (x0, y0);Quadrant(M) = 1;PointOnCurve(M, Gamma);N: Point;Coordinate(N) = (0, 1/y0);H: Point;PointOnCurve(H, LineSegmentOf(F, M));Abs(VectorOf(F, H)) = Abs(VectorOf(A, H));DotProduct(VectorOf(F, H), VectorOf(N, H)) = 0",
+      "coords": {}
+    },
+    {
+      "index": 1765,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.4142135623730951,
+        "major_axis": "y",
+        "x_coef": 2.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1765.png",
+      "answer": "2*x+y-3=0",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/2 + y^2/4 = 1);P: Point;Coordinate(P) = (1, 1);H: LineSegment;PointOnCurve(P, H);IsChordOf(H, G);MidPoint(H) = P",
+      "coords": {
+        "P": [
+          1.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 1766,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1766.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;l1: Line;l2: Line;Expression(G) = (y = x^2/4);Expression(l1) = (3*x - 4*y - 6 = 0);Expression(l2) = (y = -1);P: Point;PointOnCurve(P, G)",
+      "coords": {}
+    },
+    {
+      "index": 1767,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1767.png",
+      "answer": "3/2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;L: Line;PointOnCurve(F, L);Intersection(L, G) = {P, Q};P: Point;Q: Point;Length(LineSegmentOf(P, F)) = 3",
+      "coords": {}
+    },
+    {
+      "index": 1768,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1768.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;H: Line;k: Number;A: Point;B: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (y = k*x - k);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = 4;IsChordOf(LineSegmentOf(A, B), G)",
+      "coords": {}
+    },
+    {
+      "index": 1769,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.6457513110645907,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 7.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1769.png",
+      "answer": "7/2",
+      "fact_expr": "G: Ellipse;A: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 + y^2/7 = 1);Focus(G) = {F1, F2};PointOnCurve(A, G);AngleOf(A, F1, F2) = ApplyUnit(45, degree)",
+      "coords": {}
+    },
+    {
+      "index": 1770,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.7320508075688772,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1770.png",
+      "answer": "sqrt(3)/3",
+      "fact_expr": "l: Line;G: Hyperbola;M: Point;N: Point;F2: Point;F1: Point;Expression(G) = (x^2/3 - y^2 = 1);RightFocus(G) = F2;LeftFocus(G)=F1;PointOnCurve(F1, l);Inclination(l) = ApplyUnit(60, degree);Intersection(l, G) = {M, N}",
+      "coords": {}
+    },
+    {
+      "index": 1771,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1772,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1772.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;N: Point;Coordinate(N) = (4, 0);l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, C) = {A, B};G: Circle;IsDiameter(LineSegmentOf(N, F), G);M: Point;Intersection(l, G) = M;Negation(M=F);MidPoint(LineSegmentOf(A, B)) = M",
+      "coords": {
+        "N": [
+          4.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 1773,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1773.png",
+      "answer": "1+sqrt(2)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;F2: Point;RightFocus(G) = F2;P: Point;Q: Point;IsChordOf(LineSegmentOf(P,Q),G) = True;PointOnCurve(F2,LineSegmentOf(P,Q)) = True;IsPerpendicular(LineSegmentOf(P,Q),RealAxis(G)) = True;F1: Point;LeftFocus(G) = F1;AngleOf(P, F1, Q) = ApplyUnit(90, degree)",
+      "coords": {}
+    },
+    {
+      "index": 1774,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1774.png",
+      "answer": "9/2",
+      "fact_expr": "C1: Ellipse;Expression(C1) = (x^2/a1^2 + y^2/b1^2 = 1);a1: Number;b1: Number;a1 > 0;b1 > 0;C2: Hyperbola;Expression(C2) = (x^2/a2^2 - y^2/b2^2 = 1);a2: Number;b2: Number;a2 > 0;b2 > 0;F1: Point;F2: Point;Focus(C1) = {F1, F2};Focus(C2) = {F1, F2};OneOf(Intersection(C1, C2)) = P;P: Point;e1: Number;e2: Number;Eccentricity(C1) = e1;Eccentricity(C2) = e2;IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2)) = True",
+      "coords": {}
+    },
+    {
+      "index": 1775,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1775.png",
+      "answer": "4",
+      "fact_expr": "G: Ellipse;P: Point;F: Point;O: Origin;M: Point;x1:Number;y1:Number;Expression(G) = (x^2/25 + y^2/16 = 1);Coordinate(P) = (x1, y1);Coordinate(F) = (3, 0);PointOnCurve(P, G);Abs(LineSegmentOf(P, F)) = 2;MidPoint(LineSegmentOf(P, F)) = M",
+      "coords": {
+        "F": [
+          3.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 1776,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1776.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;l1: Line;l2: Line;P: Point;Expression(G) = (y = x^2/4);PointOnCurve(P, G);Expression(l1) = (4*x - 3*y = 7);Expression(l2) = (y = -1)",
+      "coords": {}
+    },
+    {
+      "index": 1777,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1777.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;G: Line;PointOnCurve(F2, G);Slope(G) = b/a;P: Point;Intersection(G, C) = P;DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0",
+      "coords": {}
+    },
+    {
+      "index": 1778,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1778.png",
+      "answer": "pi/2",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;M: Point;Intersection(Directrix(C), xAxis) = M;P: Point;Q: Point;l1: Line;l2: Line;TangentOfPoint(M, C) = {l1, l2};TangentPoint(l1, C) = P;TangentPoint(l2, C) = Q",
+      "coords": {}
+    },
+    {
+      "index": 1779,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1779.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "Gamma: Hyperbola;l: Line;a: Number;b: Number;a>0;b>0;A: Point;B: Point;Expression(l) = (x - sqrt(3)*y = 0);Expression(Gamma) = (-y^2/b^2 + x^2/a^2 = 1);Intersection(l, Gamma) = {A, B};C: Point;PointOnCurve(A, LineOf(A, C));IsPerpendicular(LineOf(A, C), l);Intersection(LineOf(A, C), Gamma) = C;AngleOf(A, B, C) = ApplyUnit(60, degree)",
+      "coords": {}
+    },
+    {
+      "index": 1780,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1780.png",
+      "answer": "1/4",
+      "fact_expr": "l: Line;G: Parabola;M: Point;A: Point;B: Point;Expression(G) = (y^2 = 4*x);Coordinate(M) = (2, 0);Negation(IsPerpendicular(l,xAxis));PointOnCurve(M, l);Intersection(l, G) = {A, B}",
+      "coords": {
+        "M": [
+          2.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 1781,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1781.png",
+      "answer": "10",
+      "fact_expr": "G: Ellipse;A: Point;F: Point;P: Point;Expression(G) = (x^2/4 + y^2 = 1);PointOnCurve(P, G);RightFocus(G) = F;Coordinate(A) = (0, sqrt(6))",
+      "coords": {
+        "A": [
+          0.0,
+          2.449489742783178
+        ]
+      }
+    },
+    {
+      "index": 1782,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1782.png",
+      "answer": "16",
+      "fact_expr": "M: Parabola;Expression(M) = (y^2 = 4*x);F: Point;Focus(M) = F;A: Point;B: Point;C: Point;D: Point;PointOnCurve(F, LineSegmentOf(A, B));PointOnCurve(F, LineSegmentOf(C, D));IsChordOf(LineSegmentOf(A, B), M);IsChordOf(LineSegmentOf(C, D), M);IsPerpendicular(LineSegmentOf(A, B), LineSegmentOf(C, D));Intersection(LineSegmentOf(A, B), M) = {A, B};Intersection(LineSegmentOf(C, D), M) = {C, D}",
+      "coords": {}
+    },
+    {
+      "index": 1783,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1783.png",
+      "answer": "8",
+      "fact_expr": "l: Line;G: Ellipse;A: Point;B: Point;F2: Point;F1: Point;Expression(G) = (x^2/4 + y^2/3 = 1);LeftFocus(G)=F1;RightFocus(G) = F2;PointOnCurve(F1, l);Intersection(l,G) ={A,B}",
+      "coords": {}
+    },
+    {
+      "index": 1784,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1784.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;Q: Point;F: Point;P: Point;O:Origin;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F;PointOnCurve(O,l);Intersection(l,LeftPart(C))=P;Intersection(l,RightPart(C))=Q;Abs(LineSegmentOf(Q,F))-Abs(LineSegmentOf(P,F))=Length(ImageinaryAxis(C))",
+      "coords": {}
+    },
+    {
+      "index": 1785,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1785.png",
+      "answer": "3",
+      "fact_expr": "C: Parabola;p: Number;P: Point;F: Point;Q: Point;O: Origin;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C)=F;PointOnCurve(P,C);IsPerpendicular(LineSegmentOf(P,F),xAxis);PointOnCurve(Q, xAxis);IsPerpendicular(LineSegmentOf(P, Q), LineSegmentOf(O, P));Abs(LineSegmentOf(F, Q)) = 6",
+      "coords": {}
+    },
+    {
+      "index": 1786,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1787,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1787.png",
+      "answer": "1",
+      "fact_expr": "C: Parabola;M: Point;O:Origin;P: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(P, C);Quadrant(P)=1;MidPoint(LineSegmentOf(P, F)) = M;Vertex(C)=O",
+      "coords": {}
+    },
+    {
+      "index": 1788,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 4.0,
+        "direction": "left"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1788.png",
+      "answer": "4*sqrt(13)",
+      "fact_expr": "G: Parabola;A: Point;F: Point;P: Point;O: Origin;Expression(G) = (y^2 = -16*x);Focus(G) = F;PointOnCurve(P, Directrix(G));PointOnCurve(A, G);Abs(LineSegmentOf(A, F)) = 8",
+      "coords": {}
+    },
+    {
+      "index": 1789,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1790,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1791,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1792,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1792.png",
+      "answer": "sqrt(3)/3",
+      "fact_expr": "l: Line;E: Ellipse;b: Number;a: Number;P: Point;F2: Point;Q: Point;F1: Point;a > b;b > 0;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(E) = F1;RightFocus(E) = F2;PointOnCurve(F2,l);Intersection(l, E) = {P, Q};Abs(LineSegmentOf(P, F2)) = 2*Abs(LineSegmentOf(Q,F2));Abs(LineSegmentOf(Q,F1))=3*Abs(LineSegmentOf(Q,F2))",
+      "coords": {}
+    },
+    {
+      "index": 1793,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1793.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;H: Line;D: Point;C: Point;A: Point;M: Point;B: Point;F: Point;k2: Number;k1: Number;Expression(G) = (y^2 = 4*x);Coordinate(M) = (4, 0);Focus(G) = F;PointOnCurve(F, H);Slope(H) = k1;Intersection(H, G) = {A, B};Intersection(OverlappingLine(LineSegmentOf(A, M)), G) = C;Intersection(OverlappingLine(LineSegmentOf(B, M)), G) = D;Slope(LineOf(C, D)) = k2",
+      "coords": {
+        "M": [
+          4.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 1794,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1794.png",
+      "answer": "4+pm*2*sqrt(3)",
+      "fact_expr": "C: Parabola;P: Point;F: Point;H: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;Intersection(Directrix(C), xAxis) = H;PointOnCurve(P, C);Abs(LineSegmentOf(P, H)) = (sqrt(5)/2)*Abs(LineSegmentOf(P, F))",
+      "coords": {}
+    },
+    {
+      "index": 1795,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1795.png",
+      "answer": "(sqrt(2) + 1, +oo)",
+      "fact_expr": "C: Hyperbola;a: Number;b: Number;N: Point;M: Point;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;H: Line;PointOnCurve(F2,H);IsPerpendicular(H,xAxis);Intersection(H, C) = {M, N};DotProduct(VectorOf(N, F1), VectorOf(M, F1)) < 0",
+      "coords": {}
+    },
+    {
+      "index": 1796,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1796.png",
+      "answer": "y^2=16*x",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;O: Origin;M: Point;PointOnCurve(M, G) = True;Abs(LineSegmentOf(M, F)) = 3*Abs(LineSegmentOf(O, F));Area(TriangleOf(M, F, O)) = 16*sqrt(2)",
+      "coords": {}
+    },
+    {
+      "index": 1797,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.2135674953460693,
+        "b": 2.4270472526550293,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 4.453341261978494e-06,
+        "converged": true,
+        "iterations": 125
+      },
+      "output_path": "results/test/hyperbola/problem_1797.png",
+      "answer": "1",
+      "fact_expr": "C: Hyperbola;a: Number;b: Number;F1: Point;P: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);Eccentricity(C)=sqrt(5);IsPerpendicular(LineSegmentOf(F1, P), LineSegmentOf(F2, P));Area(TriangleOf(P, F1, F2)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 1798,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "vertical"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1798.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "C1: Hyperbola;a: Number;C2: Hyperbola;b: Number;a>0;b>0;Expression(C1) = (y^2/a^2 - x^2/b^2 = 1);Expression(C2) = (x^2/4 - y^2 = 1);Asymptote(C1) = Asymptote(C2)",
+      "coords": {}
+    },
+    {
+      "index": 1799,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1799.png",
+      "answer": "(11/10,0)",
+      "fact_expr": "A: Point;B: Point;A1: Point;F: Point;G:Curve;C:Point;Expression(G)=(x^2/16-y^2/9=1);IsChordOf(LineSegmentOf(A,B),G);PointOnCurve(F,LineSegmentOf(A,B));RightFocus(G)=F;PointOnCurve(A,LineSegmentOf(A,A1));IsPerpendicular(RightDirectrix(G),LineSegmentOf(A,A1));FootPoint(RightDirectrix(G),LineSegmentOf(A,A1))=A1;Intersection(LineSegmentOf(B,A1),xAxis)=C",
+      "coords": {}
+    },
+    {
+      "index": 1800,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.9876673221588135,
+        "b": 1.5893483050269506,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 3.171300880237459e-05,
+        "converged": true,
+        "iterations": 68
+      },
+      "output_path": "results/test/ellipse/problem_1800.png",
+      "answer": "16",
+      "fact_expr": "G: Ellipse;k: Number;Expression(G) = (x^2/25 + y^2/k = 1);PointOnCurve(Focus(G), xAxis);Eccentricity(G) = 3/5",
+      "coords": {}
+    },
+    {
+      "index": 1801,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1802,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.23606797749979,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 5.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1802.png",
+      "answer": "pm*sqrt(3)",
+      "fact_expr": "l: Line;G: Ellipse;A: Point;F: Point;B: Point;k: Number;Expression(G) = (x^2/9 + y^2/5 = 1);RightFocus(G) = F;PointOnCurve(F, l);Slope(l) = k;Intersection(l, G) = {A, B};VectorOf(A, F) = 2*VectorOf(F, B)",
+      "coords": {}
+    },
+    {
+      "index": 1803,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          1.0,
+          0.0
+        ],
+        "radius": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/circle/problem_1803.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;p: Number;H: Circle;Expression(G) = (y^2 = -2*p*x);Expression(H) = (y^2 + (x + 1)^2 = 1);Center(H) = Focus(G)",
+      "coords": {}
+    },
+    {
+      "index": 1804,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.1532304286956787,
+        "b": 2.5773532390594482,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 8.412490387854632e-06,
+        "converged": true,
+        "iterations": 114
+      },
+      "output_path": "results/test/hyperbola/problem_1804.png",
+      "answer": "sqrt(6)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Expression(OneOf(Asymptote(G))) = (y = sqrt(5)*x)",
+      "coords": {}
+    },
+    {
+      "index": 1805,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1805.png",
+      "answer": "[1,3)+(16/3,16]",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/4 + x^2/a = 1);a: Number;e: Number;Eccentricity(G) = e;In(e,(1/2, sqrt(3)/2])",
+      "coords": {}
+    },
+    {
+      "index": 1806,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1807,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.4142135623730951,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1807.png",
+      "answer": "x^2/2-y^2/4=1",
+      "fact_expr": "G: Hyperbola;H: Point;C:Hyperbola;Expression(G) = (x^2 - y^2/2 = 1);Coordinate(H) = (2, 2);Asymptote(G) = Asymptote(C);PointOnCurve(H, C)",
+      "coords": {
+        "H": [
+          2.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 1808,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1808.png",
+      "answer": "",
+      "fact_expr": "G: Hyperbola;m: Number;n: Number;H: Parabola;Expression(G) = (m*x^2 + n*y^2 = 1);Expression(H) = (y = x^2/8);OneOf(Focus(G)) = Focus(H);Eccentricity(G)=2",
+      "coords": {}
+    },
+    {
+      "index": 1809,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1809.png",
+      "answer": "2*sqrt(13)",
+      "fact_expr": "C: Parabola;A: Point;F: Point;P: Point;O: Origin;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(A, C);PointOnCurve(P, Directrix(C));Abs(LineSegmentOf(A, F)) = 5",
+      "coords": {}
+    },
+    {
+      "index": 1810,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1810.png",
+      "answer": "[sqrt(3)/2,1)",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;P: Point;F: Point;O: Origin;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);RightFocus(C)=F;PointOnCurve(P,C);Quadrant(P)=1;DotProduct(VectorOf(F, O), VectorOf(F, P)) = Abs(LineSegmentOf(P, F))^2",
+      "coords": {}
+    },
+    {
+      "index": 1811,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          2.0,
+          -0.0
+        ],
+        "radius": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/circle/problem_1811.png",
+      "answer": "y^2 = -16*x",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = -2*p*x);p: Number;p>0;H: Circle;Expression(H) = (-4*x + x^2 + y^2 = 0);IsTangent(H, Directrix(G)) = True",
+      "coords": {}
+    },
+    {
+      "index": 1812,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1812.png",
+      "answer": "4*sqrt(3)",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);Intersection(Directrix(C), xAxis) = M;M: Point;F: Point;Focus(C) = F;L: Line;Inclination(L) = ApplyUnit(60,degree);PointOnCurve(F,L) = True;Intersection(L,C) = {A,B};A: Point;B: Point",
+      "coords": {}
+    },
+    {
+      "index": 1813,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1813.png",
+      "answer": "y^2 = 2*x - 2",
+      "fact_expr": "G: Parabola;A: Point;B: Point;P: Point;Expression(G) = (y^2 = 4*x);Coordinate(B) = (2, 0);MidPoint(LineSegmentOf(A, B)) = P;PointOnCurve(A, G)",
+      "coords": {
+        "B": [
+          2.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 1814,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 2.8284271247461903,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1814.png",
+      "answer": "2*sqrt(5)",
+      "fact_expr": "G: Hyperbola;H: Line;Expression(G) = (x^2 - y^2/8 = 1);Expression(H) = (x + 2*y + 7 = 0)",
+      "coords": {}
+    },
+    {
+      "index": 1815,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1816,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1817,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1817.png",
+      "answer": "pm*3",
+      "fact_expr": "C: Parabola;p: Number;M: Point;F: Point;A: Point;O: Origin;Expression(C) = (y^2 = 2*p*x);PointOnCurve(A, C);Focus(C) = F;Distance(A, F) = 4;PointOnCurve(M, Directrix(C));VectorOf(M, F) = 3*VectorOf(F, A)",
+      "coords": {}
+    },
+    {
+      "index": 1818,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1818.png",
+      "answer": "[-1,0)",
+      "fact_expr": "x0: Number;x0 < 0;M: Point;Coordinate(M) = (x0, 0);O: Origin;C: Parabola;Expression(C) = (y^2 = 4*x);N: Point;PointOnCurve(N, C);AngleOf(O, M, N) = ApplyUnit(45, degree)",
+      "coords": {}
+    },
+    {
+      "index": 1819,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1819.png",
+      "answer": "7",
+      "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 + y^2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);PointOnCurve(MidPoint(LineSegmentOf(P,F1)), yAxis);Abs(LineSegmentOf(P, F1)) = lambda*Abs(LineSegmentOf(P, F2));lambda:Number",
+      "coords": {}
+    },
+    {
+      "index": 1820,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1820.png",
+      "answer": "sqrt(5)/2",
+      "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;O: Origin;Expression(G) = (y^2 = 2*x);Coordinate(A) = (-1/2, 0);Focus(G) = F;PointOnCurve(P, G);Abs(LineSegmentOf(A, P)) = sqrt(2)*Abs(LineSegmentOf(P, F))",
+      "coords": {
+        "A": [
+          -0.5,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 1821,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1821.png",
+      "answer": "20",
+      "fact_expr": "C: Hyperbola;b: Number;G: Circle;H: Circle;F2: Point;P: Point;F1: Point;A1: Point;A2: Point;b>0;Expression(C) = (x^2/5 - y^2/b^2 = 1);LeftVertex(C) = A1;RightVertex(C) = A2;LeftFocus(C)=F1;RightFocus(C)=F2;IsDiameter(LineSegmentOf(F1,F2),G);OneOf(Intersection(G,LeftPart(C)))=P;IsDiameter(LineSegmentOf(A1,A2),H);IsTangent(H,LineOf(P,F2))",
+      "coords": {}
+    },
+    {
+      "index": 1822,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1823,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1823.png",
+      "answer": "sqrt(2)-1",
+      "fact_expr": "N: Parabola;p: Number;M: Ellipse;a: Number;b: Number;P: Point;F1: Point;F2: Point;Expression(N) = (y^2 = 2*(p*x));a > b;b > 0;Expression(M) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(M) = F1;RightFocus(M) = F2;Focus(N) = F2;OneOf(Intersection(M,N))=P;Abs(LineSegmentOf(P, F1)) = sqrt(2)*Abs(LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 1824,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1824.png",
+      "answer": "x^2/4-y^2/3=1",
+      "fact_expr": "G: Ellipse;A: Point;M: Point;N: Point;B: Point;Expression(G) = (x^2/4 + y^2/3 = 1);LeftVertex(G) = A;RightVertex(G) = B;PointOnCurve(M,G);PointOnCurve(N,G);IsPerpendicular(LineSegmentOf(M,N),xAxis)",
+      "coords": {}
+    },
+    {
+      "index": 1825,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1825.png",
+      "answer": "-11",
+      "fact_expr": "G: Parabola;H: Line;F: Point;A: Point;B: Point;Expression(G) = (y^2 = 8*x);Expression(H) = (y = 2*x - 2);Intersection(H, G) = {A, B};Focus(G) = F",
+      "coords": {}
+    },
+    {
+      "index": 1826,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1826.png",
+      "answer": "{(-2/3, sqrt(2)/3), (-2/3, -sqrt(2)/3)}",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/2 + y^2 = 1);F: Point;LeftFocus(C) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, C) = {A, B};O: Origin;DotProduct(VectorOf(O, A), VectorOf(O, B)) = 0;Z: Line;PointOnCurve(O, Z);IsPerpendicular(Z, LineOf(A, B));H: Point;FootPoint(Z, LineOf(A, B)) = H",
+      "coords": {}
+    },
+    {
+      "index": 1827,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1827.png",
+      "answer": "1",
+      "fact_expr": "l: Line;C: Curve;O: Origin;P: Point;Q: Point;Expression(C) = (y = x^2/2);PointOnCurve(P, C);TangentOnPoint(P, C) = l;Z: Line;PointOnCurve(P, Z);IsPerpendicular(Z, l);Intersection(Z, C) = Q;IsPerpendicular(LineSegmentOf(O, P), LineSegmentOf(O, Q))",
+      "coords": {}
+    },
+    {
+      "index": 1828,
+      "success": false,
+      "conic_type": "parabola",
+      "error": "validation: parabola_point_off_curve",
+      "params": {
+        "p": 3.9731175899505615,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.00018243117665406317,
+        "converged": true,
+        "iterations": 127
+      },
+      "output_path": "results/test/parabola/problem_1828.png",
+      "answer": "5",
+      "fact_expr": "G: Parabola;a: Number;P: Point;F: Point;Expression(G) = (y^2 = a*x);Coordinate(P) = (1, 4);PointOnCurve(P, G);Focus(G) = F;Distance(P, F) = Abs(LineSegmentOf(P, F));Negation(a = 0)",
+      "coords": {
+        "P": [
+          1.0,
+          4.0
+        ]
+      },
+      "validation_reasons": [
+        "parabola_point_off_curve"
+      ]
+    },
+    {
+      "index": 1829,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_point_off_curve",
+      "params": {
+        "a": 1.9960020780563354,
+        "b": 1.4970015287399292,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "iterations": 2
+      },
+      "output_path": "results/test/hyperbola/problem_1829.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "C: Hyperbola;a: Number;b: Number;G: Circle;F1: Point;F2: Point;A: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (3, 4);PointOnCurve(A, C);LeftFocus(C) = F1;RightFocus(C) = F2;IsDiameter(LineSegmentOf(F1, F2), G);PointOnCurve(A, G)",
+      "coords": {
+        "A": [
+          3.0,
+          4.0
+        ]
+      },
+      "validation_reasons": [
+        "hyperbola_point_off_curve"
+      ]
+    },
+    {
+      "index": 1830,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1830.png",
+      "answer": "2",
+      "fact_expr": "G: Ellipse;O: Circle;H: Line;m: Number;n: Number;I: Point;L:Line;Expression(G) = (x^2/9 + y^2/4 = 1);Expression(O) = (x^2 + y^2 = 4);Expression(H) = (m*x + n*y = 4);Coordinate(I) = (m, n);NumIntersection(H, O)=0;PointOnCurve(I,L)",
+      "coords": {}
+    },
+    {
+      "index": 1831,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1832,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1833,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1834,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1834.png",
+      "answer": "sqrt(3)/3",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;H: Line;P: Point;F1: Point;Q: Point;F2: Point;a > b;b > 0;LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2, H);Intersection(H, G) = {P, Q};AngleOf(F1, P, Q) = ApplyUnit(60, degree);Abs(LineSegmentOf(P,F1))=Abs(LineSegmentOf(P,Q))",
+      "coords": {}
+    },
+    {
+      "index": 1835,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.23606797749979,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 5.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1835.png",
+      "answer": "5/13",
+      "fact_expr": "F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;G: Ellipse;Expression(G) = (x^2/9 + y^2/5 = 1);P: Point;PointOnCurve(P, G) = True;PointOnCurve(MidPoint(LineSegmentOf(P,F1)),yAxis) = True",
+      "coords": {}
+    },
+    {
+      "index": 1836,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.366584062576294,
+        "b": 2.364173412322998,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.00017838670464698225,
+        "converged": true,
+        "iterations": 89
+      },
+      "output_path": "results/test/hyperbola/problem_1836.png",
+      "answer": "pi/6",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Eccentricity(G) = 2*sqrt(3)/3;H: Line;Expression(H) = (y = x*(b/a))",
+      "coords": {}
+    },
+    {
+      "index": 1837,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1838,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1838.png",
+      "answer": "5/7",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;LeftFocus(C) = F;G: Line;O: Origin;PointOnCurve(O, G);A: Point;B: Point;Intersection(C, G) = {A,B};Abs(LineSegmentOf(A, B)) = 10;Abs(LineSegmentOf(A, F)) = 6;Cos(AngleOf(A, B, F)) = 4/5;e: Number;Eccentricity(C) = e",
+      "coords": {}
+    },
+    {
+      "index": 1839,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1839.png",
+      "answer": "(1, sqrt(2)]",
+      "fact_expr": "C1:Hyperbola;C2:Hyperbola;a:Number;b:Number;a>0;b>0;m:Number;n:Number;m>0;n>0;e1:Number;e2:Number;Expression(C1)=(x^2/a^2-y^2/b^2=1);Expression(C2)=(y^2/m^2-x^2/n^2=1);Eccentricity(C1)=e1;Eccentricity(C2)=e2;Asymptote(C1)=Asymptote(C2)",
+      "coords": {}
+    },
+    {
+      "index": 1840,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1840.png",
+      "answer": "3",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l: Line;PointOnCurve(F, l);k: Number;Slope(l) = k;P: Point;Q: Point;Intersection(l, C) = {P, Q};M: Point;Intersection(PerpendicularBisector(LineSegmentOf(P, Q)), xAxis) = M;Abs(LineSegmentOf(P, Q)) = 6",
+      "coords": {}
+    },
+    {
+      "index": 1841,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1841.png",
+      "answer": "",
+      "fact_expr": "G: Parabola;A: Point;P: Point;M: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (4, 6);PointOnCurve(P, G);Projection(P, yAxis) = M",
+      "coords": {
+        "A": [
+          4.0,
+          6.0
+        ]
+      }
+    },
+    {
+      "index": 1842,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1842.png",
+      "answer": "2*sqrt(5) - 1",
+      "fact_expr": "G: Parabola;H: Line;A: Point;m:Number;n:Number;Expression(G) = (x^2 = 4*y);Expression(H) = (x + 2*y + 8 = 0);PointOnCurve(A, G);Distance(A, xAxis) = m;Distance(A,H)=n",
+      "coords": {}
+    },
+    {
+      "index": 1843,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1843.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;G: Line;Slope(G) = sqrt(2);A: Point;B: Point;Intersection(G, C) = {A, B};O: Origin;P: Point;MidPoint(LineSegmentOf(A, B)) = P;Slope(LineOf(O, P)) = 2*sqrt(2)",
+      "coords": {}
+    },
+    {
+      "index": 1844,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1845,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1845.png",
+      "answer": "7",
+      "fact_expr": "l: Line;G: Parabola;B: Point;A: Point;M:Point;Expression(G) = (y^2 = 6*x);PointOnCurve(Focus(G),l);Intersection(l,G)={A,B};MidPoint(LineSegmentOf(A,B))=M;XCoordinate(M)=2",
+      "coords": {}
+    },
+    {
+      "index": 1846,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1846.png",
+      "answer": "(15*sqrt(41))/41",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);P: Point;PointOnCurve(P, G);H: Line;Expression(H) = (4*x - 5*y + 40 = 0)",
+      "coords": {}
+    },
+    {
+      "index": 1847,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.449489742783178,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 6.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1847.png",
+      "answer": "[5, 7]",
+      "fact_expr": "C: Ellipse;F2: Point;P: Point;F1: Point;Q: Point;F: Circle;Expression(C) = (x^2/9 + y^2/6 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;Center(F) = F2;Radius(F) = 1;PointOnCurve(P, C);PointOnCurve(Q, F)",
+      "coords": {}
+    },
+    {
+      "index": 1848,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1849,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1849.png",
+      "answer": "sqrt(5)/5",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;H: Line;PointOnCurve(F1, H);IsPerpendicular(H, xAxis);A: Point;B: Point;Intersection(H, G) = {A, B};C: Point;Intersection(LineOf(A, F2), G) = {A, C};VectorOf(A, F2) = 2*VectorOf(F2, C)",
+      "coords": {}
+    },
+    {
+      "index": 1850,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1850.png",
+      "answer": "",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/2 + y^2 = 1);H: LineSegment;IsChordOf(H,G) = True;Coordinate(MidPoint(H)) = (1/2, 1/2)",
+      "coords": {}
+    },
+    {
+      "index": 1851,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1851.png",
+      "answer": "pm*2*sqrt(2)/3",
+      "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;P: Point;Expression(G) = (y^2 = 4*x);Coordinate(P) = (-1, 0);PointOnCurve(P, H);Intersection(H, G) = {A, B};MidPoint(LineSegmentOf(P, B)) = A",
+      "coords": {
+        "P": [
+          -1.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 1852,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "vertical"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1852.png",
+      "answer": "",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-x^2/16 + y^2/9 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 1853,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1854,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1854.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;P: Point;F2: Point;F1: Point;Expression(G) = (x^2 - 4*y^2 = 4);Focus(G) = {F1, F2};PointOnCurve(P, G);DotProduct(VectorOf(P,F1),VectorOf(P,F2))=0",
+      "coords": {}
+    },
+    {
+      "index": 1855,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.24967060983181,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 6.555423624376999e-06,
+        "converged": true,
+        "iterations": 116
+      },
+      "output_path": "results/test/parabola/problem_1855.png",
+      "answer": "(0, 1/4)",
+      "fact_expr": "G: Parabola;m: Number;H: Point;Expression(G) = (x^2 = m*y);Coordinate(H) = (-1, 1);PointOnCurve(H, G)",
+      "coords": {
+        "H": [
+          -1.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 1856,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1857,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1858,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1858.png",
+      "answer": "6",
+      "fact_expr": "G: Parabola;P: Point;Expression(G) = (x^2 = 8*y);PointOnCurve(P, G);Distance(P, Focus(G)) = 8",
+      "coords": {}
+    },
+    {
+      "index": 1859,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1860,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1860.png",
+      "answer": "4/3",
+      "fact_expr": "C: Parabola;k: Number;k > 0;F: Point;A: Point;B: Point;A1: Point;B1: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;H: Line;PointOnCurve(F, H);Slope(H) = k;Intersection(H, C) = {A, B};l1: Line;l2: Line;PointOnCurve(A, l1);PointOnCurve(B, l2);IsPerpendicular(l1, Directrix(C));IsPerpendicular(l2, Directrix(C));FootPoint(l1, Directrix(C)) = A1;FootPoint(l2, Directrix(C)) = B1;Area(TriangleOf(A, B, B1))/Area(TriangleOf(A, B, A1)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 1861,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1861.png",
+      "answer": "",
+      "fact_expr": "G: Hyperbola;H: Line;A: Point;B: Point;Expression(G) = (x^2/2 - y^2 = 1);PointOnCurve(RightFocus(G), H);Inclination(H)=ApplyUnit(45, degree) ;Intersection(H, G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 1862,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 2.23606797749979,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1862.png",
+      "answer": "(x-3)^2+y^2=5",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/5 = 1);H: Circle;Center(H) = RightFocus(G);IsTangent(H, Asymptote(G))",
+      "coords": {}
+    },
+    {
+      "index": 1863,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1864,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1864.png",
+      "answer": "3*sqrt(2)",
+      "fact_expr": "P: Point;Coordinate(P) = (x1, y1);x1: Number;y1: Number;G: Ellipse;Expression(G) = (x^2/9 + y^2 = 1);PointOnCurve(P, G)",
+      "coords": {}
+    },
+    {
+      "index": 1865,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1865.png",
+      "answer": "y = pm*sqrt(2)*x",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;P: Point;F1: Point;F2: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2, H);IsPerpendicular(H, xAxis);OneOf(Intersection(H, G)) = P;AngleOf(P, F1, F2) = pi/6",
+      "coords": {}
+    },
+    {
+      "index": 1866,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1867,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1867.png",
+      "answer": "2",
+      "fact_expr": "l: Line;G: Parabola;B: Point;A: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;Intersection(l, G) = {A, B};Abs(LineSegmentOf(A, F)) + Abs(LineSegmentOf(B, F)) = 6",
+      "coords": {}
+    },
+    {
+      "index": 1868,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.2175992727279663,
+        "b": 2.434081792831421,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 3.452532837400213e-06,
+        "converged": true,
+        "iterations": 124
+      },
+      "output_path": "results/test/hyperbola/problem_1868.png",
+      "answer": "x^2/5-y^2/20=1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Expression(OneOf(Asymptote(G))) = (y = 2*x);H: Parabola;Expression(H) = (y^2 = 20*x);OneOf(Focus(G)) = Focus(H)",
+      "coords": {}
+    },
+    {
+      "index": 1869,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1870,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1871,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1871.png",
+      "answer": "(sqrt(5)-1)/2",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;M: Point;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(M, Asymptote(C));IsPerpendicular(LineSegmentOf(M, F1), LineSegmentOf(M, F2));Abs(LineSegmentOf(M, F1)) = 2*b + Abs(LineSegmentOf(M, F2))",
+      "coords": {}
+    },
+    {
+      "index": 1872,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1872.png",
+      "answer": "sqrt(3)-1",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/4 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;l: Line;Expression(l) = (x - sqrt(3)*y + 8 + 2*sqrt(3) = 0);P: Point;PointOnCurve(P, l);WhenMax(AngleOf(F1, P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 1873,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1873.png",
+      "answer": "x^2/9-y^2/3=1",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;E: Circle;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(C)=(-y^2/b^2 + x^2/a^2=1);Expression(E)=(x^2 + (y-1/2)^2=49/4);LeftFocus(C)=F1;RightFocus(C)=F2;PointOnCurve(P,RightPart(C));IsDiameter(LineSegmentOf(P,F1),E);PointOnCurve(F2,E)",
+      "coords": {}
+    },
+    {
+      "index": 1874,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1875,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1875.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "C: Hyperbola;P:Point;Expression(C) = (x^2/2 - y^2 = 1);PointOnCurve(P, C);l1:Line;l2:Line;Asymptote(C)={l1,l2};Distance(P,l1)=d1;Distance(P,l2)=d2;d1:Number;d2:Number",
+      "coords": {}
+    },
+    {
+      "index": 1876,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1877,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1877.png",
+      "answer": "1/2",
+      "fact_expr": "G: Parabola;A: Point;B: Point;C: Point;D: Point;lambda:Number;Expression(G) = (y^2 = 2*x);PointOnCurve(Focus(G),LineSegmentOf(A,B));PointOnCurve(Focus(G),LineSegmentOf(C,D));IsChordOf(LineSegmentOf(A,B),G);IsChordOf(LineSegmentOf(C,D),G);IsPerpendicular(LineSegmentOf(A,B),LineSegmentOf(C,D));Abs(LineSegmentOf(A, B)) + Abs(LineSegmentOf(C, D)) = (lambda*Abs(LineSegmentOf(A, B)))*Abs(LineSegmentOf(C, D))",
+      "coords": {}
+    },
+    {
+      "index": 1878,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1879,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1879.png",
+      "answer": "2\nApplyUnit(120, degree)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/2 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 1880,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1881,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.1622776601683795,
+        "b": 2.449489742783178,
+        "major_axis": "x",
+        "x_coef": 10.0,
+        "y_coef": 6.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1881.png",
+      "answer": "4",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/10 + y^2/6 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 1882,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1882.png",
+      "answer": "-6",
+      "fact_expr": "M: Point;H: Circle;Expression(H) = ((x - 6)^2 + (y - 4)^2 = 1);PointOnCurve(M, H);P: Point;G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);PointOnCurve(P, G);F: Point;Coordinate(F) = (-3, 0)",
+      "coords": {
+        "F": [
+          -3.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 1883,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1883.png",
+      "answer": "9/2",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;G: Line;PointOnCurve(F1, G);A: Point;B: Point;Intersection(G, LeftPart(C)) = {A, B};Abs(LineSegmentOf(A, F2)) = 3;Abs(LineSegmentOf(B, F2)) = 5;Abs(LineSegmentOf(A, B)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 1884,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1885,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1886,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.366584062576294,
+        "b": 2.364173412322998,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.00017838670464698225,
+        "converged": true,
+        "iterations": 89
+      },
+      "output_path": "results/test/hyperbola/problem_1886.png",
+      "answer": "y = pm*sqrt(3)*x",
+      "fact_expr": "G: Hyperbola;a: Number;a>0;Expression(G) = (-y^2/3^2 + x^2/a^2 = 1);Eccentricity(G) = 2",
+      "coords": {}
+    },
+    {
+      "index": 1887,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1887.png",
+      "answer": "2",
+      "fact_expr": "l: Line;C: Parabola;p: Number;M: Point;F: Point;B: Point;A: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Coordinate(M) = (-2, 0);PointOnCurve(M, l);Slope(l) = 2/3;Intersection(l, C) = {A, B};Focus(C)=F;Area(TriangleOf(M,F,B))=2*Area(TriangleOf(M,F,A))",
+      "coords": {
+        "M": [
+          -2.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 1888,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1888.png",
+      "answer": "(1,3]",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F2: Point;F1: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, LeftPart(G));Abs(LineSegmentOf(P, F2))^2/Abs(LineSegmentOf(P, F1)) = 8*a",
+      "coords": {}
+    },
+    {
+      "index": 1889,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1890,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1890.png",
+      "answer": "y^2 = 4*x",
+      "fact_expr": "E: Parabola;Expression(E) = (y^2 = 2*p*x);p: Number;M: Point;Coordinate(M) = (1, m);m: Number;PointOnCurve(M, E) = True;Distance(M, Focus(E)) = 2",
+      "coords": {}
+    },
+    {
+      "index": 1891,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1891.png",
+      "answer": "[2, 2*sqrt(2)]",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/2 + y^2 = 1);F1: Point;F2: Point;Focus(C) = {F1, F2};P: Point;x0: Number;y0: Number;Coordinate(P) = (x0, y0);x0^2/2 + y0^2 <= 1",
+      "coords": {}
+    },
+    {
+      "index": 1892,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1892.png",
+      "answer": "6",
+      "fact_expr": "l: Line;G: Hyperbola;O: Origin;c: Number;a: Number;b: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);HalfFocalLength(G) = c;Expression(l) = (a*x + b*y = a*b);Distance(O, l) = (1/3)*c + 1",
+      "coords": {}
+    },
+    {
+      "index": 1893,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1893.png",
+      "answer": "y=-x+1",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);F: Point;RightFocus(G) = F;l: Line;PointOnCurve(F, l) = True;Intersection(l, G) = {P, Q};Q: Point;P: Point;Quadrant(P) = 2;Q1: Point;PointOnCurve(Q, G) = True;PointOnCurve(Q1, G) = True;Coordinate(Q) = (xQ, yQ);xQ: Number;yQ: Number;Coordinate(Q1) = (xQ1, yQ1);xQ1: Number;yQ1: Number;yQ + yQ1 = 0;IsPerpendicular(LineSegmentOf(F, Q1), LineSegmentOf(P, Q)) = True",
+      "coords": {}
+    },
+    {
+      "index": 1894,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.4142135623730951,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1894.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;H: Parabola;p: Number;Expression(G) = (x^2 - y^2 = 2);Expression(H) = (y^2 = 2*(p*x));Directrix(H) = LeftDirectrix(G)",
+      "coords": {}
+    },
+    {
+      "index": 1895,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1895.png",
+      "answer": "[2, 3]",
+      "fact_expr": "E: Ellipse;F1: Point;P: Point;F2: Point;Expression(E) = (x^2/4 + y^2/3 = 1);Focus(E) = {F1, F2};PointOnCurve(P, E)",
+      "coords": {}
+    },
+    {
+      "index": 1896,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1896.png",
+      "answer": "x^2/4+3*y^2/4=1",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;B: Point;C: Point;A: Point;O: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (2, 0);Center(G) = O;OneOf(Endpoint(MajorAxis(G)))=A;PointOnCurve(O, LineSegmentOf(B, C));IsChordOf(LineSegmentOf(B, C),G);DotProduct(VectorOf(A, C), VectorOf(B, C)) = 0;Abs(-VectorOf(O, B) + VectorOf(O, C)) = 2*Abs(-VectorOf(B, A) + VectorOf(B, C))",
+      "coords": {
+        "A": [
+          2.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 1897,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1897.png",
+      "answer": "25/4",
+      "fact_expr": "l: Line;C: Parabola;A: Point;F: Point;B: Point;Expression(C)=(y^2 = 4*x);Focus(C)=F;PointOnCurve(F, l);Intersection(l, C) = {A, B};Abs(LineSegmentOf(A, F)) = 5",
+      "coords": {}
+    },
+    {
+      "index": 1898,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1898.png",
+      "answer": "sqrt(2)\n(pm*sqrt(2), 0)",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/3 + x^2/m = 1);m: Number;m>0;T:Point;OneOf(Focus(G))=T;Coordinate(T) = (0, 1);PointOnCurve(P, G) ;P: Point;F1: Point;F2: Point;Focus(G) = {F1, F2};Area(TriangleOf(P, F1, F2)) = sqrt(2)",
+      "coords": {
+        "T": [
+          0.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 1899,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1900,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1900.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Parabola;H: Line;A: Point;P: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (x = -1);Coordinate(A) = (0, 1);PointOnCurve(P, G)",
+      "coords": {
+        "A": [
+          0.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 1901,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1901.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Ellipse;J: Line;H: Line;A: Point;P: Point;B: Point;E: Point;F: Point;O: Origin;Expression(G) = (x^2/4 + y^2/2 = 1);Expression(H) = (x = 3);RightVertex(G) = P;PointOnCurve(O, J);Intersection(J, G) = {A, B};Intersection(LineOf(P, A), H) = E;Intersection(LineOf(P, B), H) = F",
+      "coords": {}
+    },
+    {
+      "index": 1902,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1902.png",
+      "answer": "3",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;P: Point;PointOnCurve(P, G);XCoordinate(P) = 2",
+      "coords": {}
+    },
+    {
+      "index": 1903,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1904,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.7320508075688772,
+        "b": 1.4142135623730951,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1904.png",
+      "answer": "pm*sqrt(210)/3",
+      "fact_expr": "l: Line;G: Hyperbola;Expression(G) = (x^2/3 - y^2/2 = 1);Length(InterceptChord(l,G))=4;Slope(l)=2",
+      "coords": {}
+    },
+    {
+      "index": 1905,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.449489742783178,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1905.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/6 - y^2/3 = 1);H: Circle;Expression(H) = (y^2 + (x - 3)^2 = r^2);r: Number;r>0;IsTangent(Asymptote(G), H)",
+      "coords": {}
+    },
+    {
+      "index": 1906,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 2.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1906.png",
+      "answer": "y = pm*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/4 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 1907,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          -2.0,
+          0.0
+        ],
+        "radius": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/circle/problem_1907.png",
+      "answer": "{2*sqrt(3)/3,2}",
+      "fact_expr": "C: Hyperbola;G: Circle;O: Origin;Expression(G) = (y^2 + (x - 2)^2 = 1);Center(C) = O;SymmetryAxis(C)=axis;IsTangent(Asymptote(C),G)",
+      "coords": {}
+    },
+    {
+      "index": 1908,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.7320508075688772,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1908.png",
+      "answer": "1/3",
+      "fact_expr": "G: Hyperbola;H: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/3 - y^2 = 1);Expression(H) = (x^2/6 + y^2/2 = 1);Focus(G)={F1,F2};Focus(H)={F1,F2};OneOf(Intersection(H,G))=P",
+      "coords": {}
+    },
+    {
+      "index": 1909,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1909.png",
+      "answer": "2*sqrt(38)/5",
+      "fact_expr": "G: Ellipse;H: Line;Expression(G) = (x^2 + 4*y^2 = 4);Expression(H) = (y = x - 1/2)",
+      "coords": {}
+    },
+    {
+      "index": 1910,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1910.png",
+      "answer": "sqrt(17)/3",
+      "fact_expr": "G: Hyperbola;a: Number;b: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(P,RightPart(G));DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0;Tan(AngleOf(P, F1, F2)) = 1/4",
+      "coords": {}
+    },
+    {
+      "index": 1911,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1911.png",
+      "answer": "y=pm*(sqrt(3)/3)*x",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;A: Point;F: Point;B: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;L1:Line;L2:Line;OneOf(Asymptote(C))=L1;OneOf(Asymptote(C))=L2;Negation(L1=L2);L:Line;PointOnCurve(F, L);IsPerpendicular(L1, L);FootPoint(L1, L)=A;Intersection(L,L2)=B;2*VectorOf(A, F) = VectorOf(F, B)",
+      "coords": {}
+    },
+    {
+      "index": 1912,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1912.png",
+      "answer": "1 + sqrt(2)",
+      "fact_expr": "G: Hyperbola;a: Number;b: Number;P: Point;Q: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Focus(G) = {F1, F2};IsChordOf(LineSegmentOf(P, Q), G);PointOnCurve(F1, LineSegmentOf(P, Q));IsPerpendicular(xAxis, LineSegmentOf(P, Q));AngleOf(P, F2, Q) = ApplyUnit(90, degree)",
+      "coords": {}
+    },
+    {
+      "index": 1913,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "vertical"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1913.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/16 + y^2/9 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 1914,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1914.png",
+      "answer": "8",
+      "fact_expr": "G: Ellipse;H: Circle;C:Circle;P: Point;N: Point;M: Point;Expression(G) = (x^2/25 + y^2/16 = 1);Expression(H) = (y^2 + (x + 3)^2 = 1);Expression(C) = (y^2 + (x - 3)^2 = 1);PointOnCurve(P, G);PointOnCurve(M, H);PointOnCurve(N, C)",
+      "coords": {}
+    },
+    {
+      "index": 1915,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1915.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;l: Line;P: Point;Expression(G) = (y^2 = 4*x);Expression(l) = (3*x - 4*y + 12 = 0);PointOnCurve(P, G);d1:Number;d2:Number;Distance(P, yAxis) = d1;Distance(P, l) = d2",
+      "coords": {}
+    },
+    {
+      "index": 1916,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1916.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;M: Circle;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(M) = (y^2 + (-a + x)^2 = b^2/4);IsTangent(OneOf(Asymptote(C)),M);WhenMax(a^2/(a^2*b^2+1)-(4/49)*a^2)",
+      "coords": {}
+    },
+    {
+      "index": 1917,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1917.png",
+      "answer": "y^2 = 8*x",
+      "fact_expr": "l: Line;Slope(l) = 2;G: Parabola;Expression(G) = (y^2 = a*x);a>0;a: Number;F: Point;PointOnCurve(F,l) = True;Focus(G) = F;Intersection(l,yAxis) = A;O: Origin;A: Point;Area(TriangleOf(O,A,F)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 1918,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1918.png",
+      "answer": "y=(pm*3/4)*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 1919,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1920,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1920.png",
+      "answer": "4+2*sqrt(2)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/2 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G) = True",
+      "coords": {}
+    },
+    {
+      "index": 1921,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1921.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;O: Circle;A: Point;C: Point;B: Point;F: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(O) = (x^2 + y^2 = a^2);LeftFocus(G) = F;l1: Line;l2: Line;TangentOfPoint(F, O) = {l1, l2};TangentPoint(l1,O)=A;TangentPoint(l2,O)=B;LeftVertex(G) = C;AngleOf(A, C, B) = ApplyUnit(120, degree)",
+      "coords": {}
+    },
+    {
+      "index": 1922,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1922.png",
+      "answer": "sqrt(17)-1",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);P: Point;PointOnCurve(P, C);G: Circle;Expression(G) = (x^2 + (y - 4)^2 = 1);Q: Point;PointOnCurve(Q, G);Distance(P, Directrix(C)) = d;d: Number",
+      "coords": {}
+    },
+    {
+      "index": 1923,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1923.png",
+      "answer": "(sqrt(5)+1)/2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;OneOf(Focus(G)) = F;F: Point;OneOf(Endpoint(ImageinaryAxis(G))) = B;B: Point;IsPerpendicular(LineOf(F,B),OneOf(Asymptote(G))) = True",
+      "coords": {}
+    },
+    {
+      "index": 1924,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 3.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1924.png",
+      "answer": "x^2=6*y-9",
+      "fact_expr": "G: Parabola;F: Point;M: Point;P: Point;Expression(G) = (x^2 = 12*y);Focus(G) = F;PointOnCurve(P, G);VectorOf(F, M) = VectorOf(M, P)",
+      "coords": {}
+    },
+    {
+      "index": 1925,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 2.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1925.png",
+      "answer": "x^2/3 - y^2/12 = 1",
+      "fact_expr": "G: Hyperbola;C:Hyperbola;H: Point;Expression(G) = (x^2 - y^2/4 = 1);Coordinate(H) = (2, 2);Asymptote(G) = Asymptote(C);PointOnCurve(H, C)",
+      "coords": {
+        "H": [
+          2.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 1926,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1926.png",
+      "answer": "(1, 2]",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;M: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(M, RightPart(G));Abs(LineSegmentOf(M, F1))/Abs(LineSegmentOf(M, F2)) = 3;Eccentricity(G)=e;e:Number",
+      "coords": {}
+    },
+    {
+      "index": 1927,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.366584062576294,
+        "b": 2.364173412322998,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.00017838670464698225,
+        "converged": true,
+        "iterations": 89
+      },
+      "output_path": "results/test/hyperbola/problem_1927.png",
+      "answer": "x^2-y^2/3=1",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y^2 + (-a + x)^2 = 3/4);Eccentricity(C) = 2;IsTangent(Asymptote(C),G)",
+      "coords": {}
+    },
+    {
+      "index": 1928,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1929,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1930,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1930.png",
+      "answer": "4*sqrt(2)/3",
+      "fact_expr": "G: Ellipse;H: Line;Expression(G) = (x^2/2 + y^2 = 1);Expression(H) = (y = x - 1)",
+      "coords": {}
+    },
+    {
+      "index": 1931,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1931.png",
+      "answer": "sqrt(5)/3",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;A: Point;P: Point;B: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftVertex(C) = A;RightVertex(C) = B;PointOnCurve(P, C);Slope(LineOf(P, A))=k1;Slope(LineOf(P, B)) = k2;k1:Number;k2:Number;k1*k2 = -4/9",
+      "coords": {}
+    },
+    {
+      "index": 1932,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "left"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1932.png",
+      "answer": "2",
+      "fact_expr": "C: Parabola;F: Point;P:Point;A:Point;Expression(C) = (y^2 = -4*x);Focus(C) = F;Coordinate(A) = (-1, 1);PointOnCurve(P,C)",
+      "coords": {
+        "A": [
+          -1.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 1933,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1934,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1934.png",
+      "answer": "[2*sqrt(2),+oo]",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Inclination(OneOf(Asymptote(G))) = ApplyUnit(60, degree)",
+      "coords": {}
+    },
+    {
+      "index": 1935,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1935.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;H: Circle;Expression(H) = ((x - sqrt(3))^2 + (y - 1)^2 = 1);IsTangent(Asymptote(G), H)",
+      "coords": {}
+    },
+    {
+      "index": 1936,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1937,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1937.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "l: Line;C: Ellipse;P: Point;F: Point;Expression(C) = (x^2/2 + y^2 = 1);Expression(l)=(x-y+2=0);LeftFocus(C) = F;PointOnCurve(P, C);Distance(P, l) = d;d:Number",
+      "coords": {}
+    },
+    {
+      "index": 1938,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1938.png",
+      "answer": "8",
+      "fact_expr": "F1: Point;F2: Point;Focus(G) = {F1, F2};G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);H: Line;PointOnCurve(F1, H);A: Point;B: Point;Intersection(H, G) = {A, B};Abs(LineSegmentOf(F2, A)) + Abs(LineSegmentOf(F2, B)) = 12",
+      "coords": {}
+    },
+    {
+      "index": 1939,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1939.png",
+      "answer": "sqrt(5)/3",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;P: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, C);Slope(LineSegmentOf(P, UpperVertex(C)))*Slope(LineSegmentOf(P, LowerVertex(C))) = -4/9",
+      "coords": {}
+    },
+    {
+      "index": 1940,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1940.png",
+      "answer": "{16,34}",
+      "fact_expr": "G: Ellipse;t: Number;Expression(G) = (x^2/25 + y^2/t = 1);Focus(G) = {F1,F2};F1: Point;F2: Point;Distance(F1,F2) = 6",
+      "coords": {}
+    },
+    {
+      "index": 1941,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1941.png",
+      "answer": "12",
+      "fact_expr": "G: Parabola;O: Origin;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(A, G);PointOnCurve(B, G);IsPerpendicular(LineSegmentOf(O, A), LineSegmentOf(O, B));Focus(G) = F",
+      "coords": {}
+    },
+    {
+      "index": 1942,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1942.png",
+      "answer": "9/4",
+      "fact_expr": "G: Parabola;H: Line;A: Point;F: Point;B: Point;Expression(G) = (y^2 = 2*x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};Quadrant(A) = 1;VectorOf(A, F) = 2*VectorOf(F, B)",
+      "coords": {}
+    },
+    {
+      "index": 1943,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1944,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 0.4148966481295064,
+        "b": 2.904512927119681,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 6.037912459509782e-07,
+        "converged": true,
+        "iterations": 470
+      },
+      "output_path": "results/test/hyperbola/problem_1944.png",
+      "answer": "7",
+      "fact_expr": "C: Hyperbola;b: Number;b>0;Expression(C)=(x^2-y^2/b^2 = 1);Expression(Asymptote(C))=(y=pm*7*x)",
+      "coords": {}
+    },
+    {
+      "index": 1945,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1945.png",
+      "answer": "-2",
+      "fact_expr": "T: Ellipse;Expression(T) = (x^2/a^2 + y^2/b^2 = 1);a: Number;b: Number;a > b;b > 0;Eccentricity(T) = sqrt(2)/2;F: Point;Coordinate(F) = (1, 0);RightFocus(T) = F;A: Point;B: Point;C: Point;PointOnCurve(Vertex(TriangleOf(A, B, C)), T) = True;MidPoint(LineSegmentOf(A, B)) = D;MidPoint(LineSegmentOf(B, C)) = E;MidPoint(LineSegmentOf(A, C)) = M;D: Point;E: Point;M: Point;k1: Number;k2: Number;k3: Number;Slope(OverlappingLine(LineSegmentOf(A, B))) = k1;Slope(OverlappingLine(LineSegmentOf(B, C))) = k2;Slope(OverlappingLine(LineSegmentOf(A, C))) = k3;Negation(k1 = 0);Negation(k2 = 0);Negation(k3 = 0);O: Origin;Slope(LineOf(O, D)) + Slope(LineOf(O, E)) + Slope(LineOf(O, M)) = 1",
+      "coords": {
+        "F": [
+          1.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 1946,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1947,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.75,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1947.png",
+      "answer": "-27/16",
+      "fact_expr": "G: Parabola;H: Line;O: Origin;A: Point;B: Point;Expression(G) = (y^2 = 3*x);PointOnCurve(Focus(G),H);Intersection(H,G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 1948,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_point_off_curve",
+      "params": {
+        "a": 0.929953396320343,
+        "b": 2.629857301712036,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 4.594453912432073e-06,
+        "converged": true,
+        "iterations": 175
+      },
+      "output_path": "results/test/hyperbola/problem_1948.png",
+      "answer": "4*sqrt(5)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(H) = (sqrt(3), 2);PointOnCurve(H, G);Eccentricity(G)=3",
+      "coords": {
+        "H": [
+          1.7320508075688772,
+          2.0
+        ]
+      },
+      "validation_reasons": [
+        "hyperbola_point_off_curve"
+      ]
+    },
+    {
+      "index": 1949,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1950,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1950.png",
+      "answer": "sqrt(17)/3",
+      "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;A: Point;F2: Point;B: Point;F1: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F2, l);Intersection(l,RightPart(C))={A, B};DotProduct(VectorOf(A,F1),VectorOf(A,F2))=0;Abs(LineSegmentOf(A, F2))/Abs(LineSegmentOf(B, F2)) = 1/2",
+      "coords": {}
+    },
+    {
+      "index": 1951,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1951.png",
+      "answer": "6",
+      "fact_expr": "G: Parabola;Expression(G) = (y = x^2/4);H: Line;PointOnCurve(Focus(G), H);A: Point;B: Point;Intersection(H, G) = {A, B};YCoordinate(MidPoint(LineSegmentOf(A, B))) = 2;IsChordOf(LineSegmentOf(A, B), G)",
+      "coords": {}
+    },
+    {
+      "index": 1952,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1953,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1954,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1954.png",
+      "answer": "pm*1",
+      "fact_expr": "l: Line;C: Parabola;A: Point;B: Point;k: Number;Expression(C) = (y^2 = 4*x);Slope(l)=k;PointOnCurve(Focus(C),l);Intersection(l,C)={A,B};Abs(LineSegmentOf(A, B))=8",
+      "coords": {}
+    },
+    {
+      "index": 1955,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1956,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1956.png",
+      "answer": "pm*sqrt(2)/4",
+      "fact_expr": "l: Line;C: Parabola;E: Point;F: Point;A: Point;B: Point;k: Number;Expression(C) = (y = x^2/4);Focus(C) = F;Intersection(Directrix(C),axis)=E;PointOnCurve(F, l);Intersection(l, C) = {A, B};3*VectorOf(E, F) = VectorOf(E, A) + 2*VectorOf(E, B);Slope(l) = k",
+      "coords": {}
+    },
+    {
+      "index": 1957,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1957.png",
+      "answer": "9",
+      "fact_expr": "P: Point;G: Parabola;Expression(G) = (y^2 = 4*x);PointOnCurve(P, G);Projection(P, yAxis) = Q;Q: Point;M: Point;Coordinate(M) = (7, 8)",
+      "coords": {
+        "M": [
+          7.0,
+          8.0
+        ]
+      }
+    },
+    {
+      "index": 1958,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 2.8284271247461903,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1958.png",
+      "answer": "(x^2 - y^2/8 = 1)&(x <= -1)",
+      "fact_expr": "C: Circle;Z: Circle;A: Point;P: Point;Expression(C) = (y^2 + (x - 3)^2 = 4);Coordinate(A) = (-3, 0);Center(Z) = P;PointOnCurve(A, Z);IsOutTangent(Z, C)",
+      "coords": {
+        "A": [
+          -3.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 1959,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1959.png",
+      "answer": "11/2",
+      "fact_expr": "G: Parabola;B: Point;P: Point;F: Point;Expression(G) = (y^2 = 6*x);Focus(G) = F;PointOnCurve(P, G) = True;Coordinate(B) = (4, 3)",
+      "coords": {
+        "B": [
+          4.0,
+          3.0
+        ]
+      }
+    },
+    {
+      "index": 1960,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1960.png",
+      "answer": "2",
+      "fact_expr": "l: Line;G: Parabola;Expression(G) = (x^2 = 2*(p*y));p: Number;p>0;H: Circle;Expression(H) = (x^2 + (y + 1)^2 = 1);A: Point;B: Point;TangentPoint(l, G) = A;TangentPoint(l, H) = B;Negation(A=B);F: Point;Focus(G) = F;WhenMin(Abs(LineSegmentOf(A, F)))",
+      "coords": {}
+    },
+    {
+      "index": 1961,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1962,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_focus_mismatch",
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1962.png",
+      "answer": "y^2/8+x^2/4=1",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/b^2 + y^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;Coordinate(F1) = (0, -2);Coordinate(F2) = (0, 2);Focus(G) = {F1, F2};Length(MajorAxis(G)) = 4*sqrt(2)",
+      "coords": {
+        "F1": [
+          0.0,
+          -2.0
+        ],
+        "F2": [
+          0.0,
+          2.0
+        ]
+      },
+      "validation_reasons": [
+        "ellipse_focus_mismatch"
+      ]
+    },
+    {
+      "index": 1963,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1963.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;P: Point;O: Origin;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(P, G);Abs(LineSegmentOf(P, F)) = 5",
+      "coords": {}
+    },
+    {
+      "index": 1964,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1965,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1965.png",
+      "answer": "5/4",
+      "fact_expr": "G: Parabola;Expression(G) = (y = x^2);F: Point;Focus(G) = F;M: Point;N: Point;PointOnCurve(M, G);PointOnCurve(N, G);Abs(LineSegmentOf(M, F)) + Abs(LineSegmentOf(N, F)) = 3",
+      "coords": {}
+    },
+    {
+      "index": 1966,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.449489742783178,
+        "b": 2.449489742783178,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1966.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 6)",
+      "coords": {}
+    },
+    {
+      "index": 1967,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1967.png",
+      "answer": "{y^2=4*x,y^2=16*x}",
+      "fact_expr": "G: Parabola;p: Number;F:Point;p>0;Expression(G) = (y^2 = 2*(p*x));PointOnCurve(F, G);Distance(F,Focus(G)) = 5;Distance(F,SymmetryAxis(G)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 1968,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1968.png",
+      "answer": "y=pm*2*sqrt(2)*x",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(H) = (3, 0);OneOf(Focus(G)) = H;Coordinate(OneOf(Vertex(G)))=(1,0)",
+      "coords": {
+        "H": [
+          3.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 1969,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1969.png",
+      "answer": "-1",
+      "fact_expr": "l: Line;G: Parabola;F: Point;A: Point;B: Point;P:Point;m:Number;n:Number;Expression(G) = (y^2 = 4*x);Focus(G) = F;Intersection(Directrix(G), xAxis) = P;PointOnCurve(P, l);Intersection(l, G) = {A, B};Slope(LineSegmentOf(F, A)) = m;Slope(LineSegmentOf(F,B))=n",
+      "coords": {}
+    },
+    {
+      "index": 1970,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1971,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1971.png",
+      "answer": "y = pm*x/2",
+      "fact_expr": "C: Hyperbola;a: Number;Expression(C) = (-y^2/5 + x^2/a^2 = 1);a>0;FocalLength(C) = 10",
+      "coords": {}
+    },
+    {
+      "index": 1972,
+      "success": false,
+      "conic_type": "parabola",
+      "error": "validation: parabola_focus_mismatch",
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1972.png",
+      "answer": "y^2=2*x\n9*sqrt(2)/4",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;F: Point;Coordinate(F) = (1/2, 0);Focus(C) = F;P: Point;PointOnCurve(P, C);G: Line;Expression(G) = (x + y + 5 = 0);Q: Point;PointOnCurve(Q, G)",
+      "coords": {
+        "F": [
+          0.5,
+          0.0
+        ]
+      },
+      "validation_reasons": [
+        "parabola_focus_mismatch"
+      ]
+    },
+    {
+      "index": 1973,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1973.png",
+      "answer": "(1/4,0)",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = x)",
+      "coords": {}
+    },
+    {
+      "index": 1974,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1975,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.6457513110645907,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 7.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1975.png",
+      "answer": "4",
+      "fact_expr": "C: Ellipse;A: Point;P: Point;F: Point;Expression(C) = (x^2/9 + y^2/7 = 1);Coordinate(A) = (-sqrt(2), 1);LeftFocus(C) = F;PointOnCurve(P, C)",
+      "coords": {
+        "A": [
+          -1.4142135623730951,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 1976,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1976.png",
+      "answer": "(0,1)",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y)",
+      "coords": {}
+    },
+    {
+      "index": 1977,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1977.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;I: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(I) = (sqrt(2), sqrt(6));PointOnCurve(I, OneOf(Asymptote(G)))",
+      "coords": {
+        "I": [
+          1.4142135623730951,
+          2.449489742783178
+        ]
+      }
+    },
+    {
+      "index": 1978,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1978.png",
+      "answer": "16/5",
+      "fact_expr": "G: Hyperbola;P: Point;F2: Point;Expression(G) = (x^2/9 - y^2/16 = 1);F1: Point;Focus(G) = {F1,F2};PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F1),LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 1979,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 2.8284271247461903,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1979.png",
+      "answer": "15",
+      "fact_expr": "G: Hyperbola;H: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2 - y^2/8 = 1);Expression(H) = (x^2/16 + y^2/7 = 1);Focus(G) = {F1, F2};Focus(H) = {F1, F2};OneOf(Intersection(G, H)) = P",
+      "coords": {}
+    },
+    {
+      "index": 1980,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1980.png",
+      "answer": "sqrt(26)/4",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;P: Point;Q: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2, H);Intersection(H, RightPart(G)) = {P, Q};IsPerpendicular(LineSegmentOf(P, Q), LineSegmentOf(P, F1));Abs(LineSegmentOf(P, Q)) = (8/15)*Abs(LineSegmentOf(P, F1))",
+      "coords": {}
+    },
+    {
+      "index": 1981,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1981.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;L: LineSegment;IsChordOf(L,G) = True;PointOnCurve(F,L) = True;Intersection(L,G) = {A,B};A: Point;B: Point;L1: Line;L2: Line;PointOnCurve(A,L1) = True;IsPerpendicular(L1,yAxis) = True;PointOnCurve(B,L2) = True;IsPerpendicular(L2,yAxis) = True;FootPoint(L1,yAxis) = C;FootPoint(L2,yAxis) = D;C: Point;D: Point",
+      "coords": {}
+    },
+    {
+      "index": 1982,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_1982.png",
+      "answer": "x = -4",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;M: Point;m: Number;Coordinate(M) = (1, m);PointOnCurve(M, G);Distance(M, Focus(G)) = 5",
+      "coords": {}
+    },
+    {
+      "index": 1983,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          0.0,
+          0.0
+        ],
+        "radius": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/circle/problem_1983.png",
+      "answer": "x^2/9 + y^2/5 = 1",
+      "fact_expr": "O: Circle;G: Circle;A: Point;P: Point;Coordinate(A) = (2, 0);Expression(O) = (x^2 + y^2 = 9);IsDiameter(LineSegmentOf(A, P), G);IsInTangent(G, O)",
+      "coords": {}
+    },
+    {
+      "index": 1984,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1984.png",
+      "answer": "10",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/25 + y^2/16 = 1);PointOnCurve(P, G);Focus(G) = {F1,F2}",
+      "coords": {}
+    },
+    {
+      "index": 1985,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1985.png",
+      "answer": "8",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);P: Point;PointOnCurve(P, G);Q: Point;Coordinate(Q) = (0, -4)",
+      "coords": {
+        "Q": [
+          0.0,
+          -4.0
+        ]
+      }
+    },
+    {
+      "index": 1986,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1986.png",
+      "answer": "sqrt(5)/2",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(P) = (1, 0);Distance(P,OneOf(Asymptote(C)))=1/2",
+      "coords": {
+        "P": [
+          1.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 1987,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1987.png",
+      "answer": "2*sqrt(10)",
+      "fact_expr": "G: Hyperbola;P: Point;F2: Point;F1: Point;Expression(G) = (x^2 - y^2/9 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0",
+      "coords": {}
+    },
+    {
+      "index": 1988,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_1988.png",
+      "answer": "2;4*sqrt(2)",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*x);E: Point;a: Number;Coordinate(E) = (a, 0);l: Line;PointOnCurve(E, l);P: Point;Q: Point;x1: Number;x2: Number;y1: Number;y2: Number;Coordinate(P) = (x1, y1);Coordinate(Q) = (x2, y2);Intersection(l, C) = {P, Q};y1*y2 = -4;M: Point;N: Point;MidPoint(H) = Q;Endpoint(H) = {M, N};PointOnCurve(N, xAxis);PointOnCurve(M, C);H:LineSegment;Negation(P=Q)",
+      "coords": {}
+    },
+    {
+      "index": 1989,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1989.png",
+      "answer": "x^2-y^2/8=1",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;A: Point;F:Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (1, 2);RightFocus(C)=F;Expression(Asymptote(C))=(y=pm*(2*sqrt(2)*x));Distance(A, F) = 2*sqrt(2)",
+      "coords": {
+        "A": [
+          1.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 1990,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1991,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1991.png",
+      "answer": "8",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};l: Line;PointOnCurve(F1, l);M: Point;N: Point;Intersection(l, G) = {M, N}",
+      "coords": {}
+    },
+    {
+      "index": 1992,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          -2.0,
+          0.0
+        ],
+        "radius": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/circle/problem_1992.png",
+      "answer": "y^2=-8*x",
+      "fact_expr": "P: Circle;G: Line;C:Circle;P1:Point;Expression(C)=((x+2)^2+y^2=1);Expression(G) = (x = 1);IsOutTangent(P,C);IsTangent(P,G);Center(P)=P1",
+      "coords": {}
+    },
+    {
+      "index": 1993,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_1993.png",
+      "answer": "2",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Inclination(OneOf(Asymptote(C))) = ApplyUnit(60, degree)",
+      "coords": {}
+    },
+    {
+      "index": 1994,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1995,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1996,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1997,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1997.png",
+      "answer": "20",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/9 + x^2/a^2 = 1);a: Number;a>3;F1: Point;F2: Point;Focus(G) = {F1, F2};Abs(LineSegmentOf(F1, F2)) = 8;A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G) = True;PointOnCurve(F1, LineSegmentOf(A, B)) = True",
+      "coords": {}
+    },
+    {
+      "index": 1998,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 2.6457513110645907,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 7.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1998.png",
+      "answer": "8",
+      "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/16 + y^2/7 = 1);PointOnCurve(P, G);Distance(P, RightFocus(G)) = 2",
+      "coords": {}
+    },
+    {
+      "index": 1999,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_1999.png",
+      "answer": "[sqrt(3)/2, 1)",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;M: Point;F1: Point;F2: Point;e: Number;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(M, C);Area(TriangleOf(M, F1, F2)) = sqrt(3)*b^2;Eccentricity(C) = e",
+      "coords": {}
+    },
+    {
+      "index": 2000,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_2000.png",
+      "answer": "pm*1",
+      "fact_expr": "l: Line;G: Ellipse;M: Point;N: Point;Expression(G) = (x^2/2 + y^2 = 1);Expression(l)=(y=k*x+1);Intersection(l,G)={M,N};Abs(LineSegmentOf(M, N)) = 4*sqrt(2)/3;k:Number",
+      "coords": {}
+    },
+    {
+      "index": 2001,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 2002,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.366584062576294,
+        "b": 2.364173412322998,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.00017838670464698225,
+        "converged": true,
+        "iterations": 89
+      },
+      "output_path": "results/test/hyperbola/problem_2002.png",
+      "answer": "2*sqrt(3)",
+      "fact_expr": "G: Hyperbola;b: Number;b>0;Expression(G) = (x^2/4 - y^2/b^2 = 1);Eccentricity(G) = 2",
+      "coords": {}
+    },
+    {
+      "index": 2003,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_2003.png",
+      "answer": "2*sqrt(10)",
+      "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2 - y^2/9 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);DotProduct(VectorOf(P,F1),VectorOf(P,F2))=0",
+      "coords": {}
+    },
+    {
+      "index": 2004,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_2004.png",
+      "answer": "(1,3]",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;e: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, RightPart(G));Min(Abs(LineSegmentOf(P, F1))^2/Abs(LineSegmentOf(P, F2))) = 8*a;Eccentricity(G) = e",
+      "coords": {}
+    },
+    {
+      "index": 2005,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_2005.png",
+      "answer": "x^2/45+y^2/20=1",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;P: Point;PointOnCurve(P, G) = True;F1: Point;F2: Point;Focus(G) = {F1, F2};IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2)) = True;L1: Line;L2: Line;Directrix(G) = {L1, L2};Distance(P, L1) = 6;Distance(P, L2) = 12",
+      "coords": {}
+    },
+    {
+      "index": 2006,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_2006.png",
+      "answer": "[1/2,1)",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;P: Point;F: Point;A: Point;a>0;b>0;Expression(C) = (y^2/b + x^2/a = 1);RightFocus(C) = F;PointOnCurve(A, C);Max(Distance(A, F))= d;PointOnCurve(P, RightDirectrix(C));LineSegmentOf(P, F) = d;d:Number",
+      "coords": {}
+    },
+    {
+      "index": 2007,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_2007.png",
+      "answer": "[pi/4, pi/3] + [2*pi/3, 3*pi/4]",
+      "fact_expr": "G: Parabola;H: Circle;I: LineSegment;alpha: Number;Expression(G) = (y^2 = 4*x);Expression(H) = (x^2 + y^2 = 3/4);PointOnCurve(Focus(G), I);Inclination(I) = alpha;Negation(Length(I) > 8);IsChordOf(I, G);IsIntersect(OverlappingLine(I), H)",
+      "coords": {}
+    },
+    {
+      "index": 2008,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_2008.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;A: Point;RightVertex(C) = A;A1: Circle;Center(A1) = A;c: Number;HalfFocalLength(C) = c;Radius(A1) = c;M: Point;N: Point;Intersection(A1, OneOf(Asymptote(C))) = {M, N};Abs(LineSegmentOf(M, N)) = sqrt(3)*c",
+      "coords": {}
+    },
+    {
+      "index": 2009,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_2009.png",
+      "answer": "36",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l1: Line;l2: Line;PointOnCurve(F, l1);PointOnCurve(F, l2);IsPerpendicular(l1, l2);A: Point;B: Point;Intersection(l1, C) = {A, B};D: Point;E: Point;Intersection(l2, C) = {D, E}",
+      "coords": {}
+    },
+    {
+      "index": 2010,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_2010.png",
+      "answer": "9",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);F1: Point;F2: Point;Focus(G) = {F1,F2};P: Point;PointOnCurve(P,G) = True;IsPerpendicular(LineSegmentOf(P,F1) , LineSegmentOf(P,F2)) = True",
+      "coords": {}
+    },
+    {
+      "index": 2011,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 8.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_2011.png",
+      "answer": "4*sqrt(10)/3",
+      "fact_expr": "l: Line;G: Ellipse;M: Point;A: Point;B: Point;F: Point;Expression(G) = (x^2/8 + y^2/4 = 1);Coordinate(M) = (0, 1);PointOnCurve(M, l);Intersection(l, G) = {A, B};RightFocus(G) = F;WhenMax(Perimeter(TriangleOf(A,B,F)))",
+      "coords": {
+        "M": [
+          0.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 2012,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_2012.png",
+      "answer": "{0, 1, 4/5}",
+      "fact_expr": "l: Line;G: Curve;a: Real;Expression(G) = (y^2 = a*x);Expression(l) = (y = x*(a - 1) - 1);NumIntersection(l, G) = 1",
+      "coords": {}
+    },
+    {
+      "index": 2013,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 2014,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_2014.png",
+      "answer": "2",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;l: Line;Directrix(C) = l;M: Point;Coordinate(M) = (1, 0);G: Line;PointOnCurve(M, G);Slope(G) = sqrt(3);A: Point;Intersection(G, l) = A;B: Point;OneOf(Intersection(G, C)) = B;LineSegmentOf(A, M) = LineSegmentOf(M, B)",
+      "coords": {
+        "M": [
+          1.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 2015,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_focus_mismatch",
+      "params": {
+        "a": 1.4506710767745972,
+        "b": 2.0511727333068848,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 4.657696081267204e-06,
+        "converged": true,
+        "iterations": 116
+      },
+      "output_path": "results/test/hyperbola/problem_2015.png",
+      "answer": "10",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Expression(OneOf(Asymptote(C))) = (y = sqrt(2)*x);F: Point;Coordinate(F) = (-sqrt(3), 0);LeftFocus(C) = F;PointOnCurve(F , l) = True;Slope(l) = 2;l: Line;Intersection(l, C) = {A, B};A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), C) = True",
+      "coords": {
+        "F": [
+          -1.7320508075688772,
+          0.0
+        ]
+      },
+      "validation_reasons": [
+        "hyperbola_focus_mismatch"
+      ]
+    },
+    {
+      "index": 2016,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_2016.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;G: Line;m: Number;P: Point;Q: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y = -3*m + x);Intersection(G, C) = {P, Q};XCoordinate(MidPoint(LineSegmentOf(P, Q))) = 2*m",
+      "coords": {}
+    },
+    {
+      "index": 2017,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_2017.png",
+      "answer": "4*sqrt(3)/3",
+      "fact_expr": "l: Line;G: Parabola;O: Origin;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, l);Intersection(l,G)={A,B};VectorOf(B, A) = 4*VectorOf(B, F)",
+      "coords": {}
+    },
+    {
+      "index": 2018,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_2018.png",
+      "answer": "x=2",
+      "fact_expr": "C: Parabola;m: Number;E: Circle;Expression(C) = (y^2 = m*x);Expression(E) = (4*x + x^2 + y^2 - 1 = 0);Focus(C) = Center(E)",
+      "coords": {}
+    },
+    {
+      "index": 2019,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_2019.png",
+      "answer": "(sqrt(3)+1)/2",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, LeftPart(C)) = True;Abs(LineSegmentOf(P, F1)) = Abs(LineSegmentOf(F1, F2));AngleOf(P, F1, F2) = ApplyUnit(120, degree)",
+      "coords": {}
+    },
+    {
+      "index": 2020,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_2020.png",
+      "answer": "4",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Range(Eccentricity(G))=(0, sqrt(2)/2];A: Point;RightFocus(G) = A;B: Point;UpperVertex(G) = B;Distance(LeftFocus(G), LineOf(A, B)) = 2;A1: Point;LeftFocus(G) = A1;P: Point;PointOnCurve(P, G)",
+      "coords": {}
+    },
+    {
+      "index": 2021,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_2021.png",
+      "answer": "3*x-4*y-7=0",
+      "fact_expr": "l: Line;G: Ellipse;A: Point;B: Point;M: Point;Expression(G) = (x^2/4 + y^2/3 = 1);Coordinate(M) = (1, -1);PointOnCurve(M,l);Intersection(l, G) = {A, B};MidPoint(LineSegmentOf(A, B)) = M",
+      "coords": {
+        "M": [
+          1.0,
+          -1.0
+        ]
+      }
+    },
+    {
+      "index": 2022,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_point_off_curve",
+      "params": {
+        "a": 1.4142164151010692,
+        "b": 1.4142164151010692,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 2.2160979279094996e-06,
+        "converged": true,
+        "iterations": 95
+      },
+      "output_path": "results/test/hyperbola/problem_2022.png",
+      "answer": "x^2-y^2/3=1",
+      "fact_expr": "G: Hyperbola;SymmetryAxis(G) = axis;O: Origin;Center(G) = O;F1: Point;F2: Point;Coordinate(F1) = (-2, 0);Coordinate(F2) = (2, 0);Focus(G) = {F1, F2};P: Point;Coordinate(P) = (-2, 3);PointOnCurve(P, G)",
+      "coords": {
+        "F1": [
+          -2.0,
+          0.0
+        ],
+        "F2": [
+          2.0,
+          0.0
+        ],
+        "P": [
+          -2.0,
+          3.0
+        ]
+      },
+      "validation_reasons": [
+        "hyperbola_point_off_curve"
+      ]
+    },
+    {
+      "index": 2023,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_2023.png",
+      "answer": "3",
+      "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (x^2/4 - y^2/m = 1);Expression(Asymptote(G)) = (y = pm*(x*(sqrt(3)/2)))",
+      "coords": {}
+    },
+    {
+      "index": 2024,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_2024.png",
+      "answer": "sqrt(3)+sqrt(6)",
+      "fact_expr": "C: Hyperbola;a: Number;b: Number;F1: Point;A: Point;F2: Point;B: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(A, RightPart(C));Intersection(LineOf(A,F1), yAxis) = B;Abs(LineSegmentOf(F1, B)) = 3*Abs(LineSegmentOf(A, B));IsPerpendicular(LineSegmentOf(A, F1), LineSegmentOf(A, F2))",
+      "coords": {}
+    },
+    {
+      "index": 2025,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_2025.png",
+      "answer": "(x-2)^2+y^2=4",
+      "fact_expr": "G: Hyperbola;H: Circle;Expression(G) = (x^2 - y^2/3 = 1);RightFocus(G) = Center(H);Eccentricity(G)=Radius(H)",
+      "coords": {}
+    },
+    {
+      "index": 2026,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_2026.png",
+      "answer": "x^2/9-y^2/7=1",
+      "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/25 + y^2/9 = 1);Eccentricity(G) = 4/3;Focus(G)=Focus(H)",
+      "coords": {}
+    },
+    {
+      "index": 2027,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 2028,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_2028.png",
+      "answer": "x+4*y-5=0",
+      "fact_expr": "G: Ellipse;H: LineSegment;I: Point;Expression(G) = (x^2/16 + y^2/4 = 1);Coordinate(I) = (1, 1);MidPoint(H)=I;IsChordOf(H,G)",
+      "coords": {
+        "I": [
+          1.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 2029,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "results/test/parabola/problem_2029.png",
+      "answer": "3/2",
+      "fact_expr": "l: Line;Expression(l) = (-m + x - y = 0);m: Number;C: Parabola;Expression(C) = (y^2 = 2*p*x);p: Number;p>0;PointOnCurve(Focus(C), l) = True;Intersection(l, C) = {A, B};A: Point;B: Point;Abs(LineSegmentOf(A, B)) = 6",
+      "coords": {}
+    },
+    {
+      "index": 2030,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.125,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_2030.png",
+      "answer": "(3/4, +oo)",
+      "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;x1: Number;y1: Number;x2: Number;y2: Number;Expression(G) = (y = 2*x^2);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);PointOnCurve(A, G);PointOnCurve(B, G);PerpendicularBisector(LineSegmentOf(A, B)) = l;Slope(l) = 1/2",
+      "coords": {}
+    },
+    {
+      "index": 2031,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.8284271247461903,
+        "b": 2.0,
+        "major_axis": "y",
+        "x_coef": 4.0,
+        "y_coef": 8.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_2031.png",
+      "answer": "10",
+      "fact_expr": "C: Parabola;Expression(C) = (x^2 = -2*p*y);p: Number;p>0;F: Point;Focus(C) = F;Q: Curve;Expression(Q) = (x^2/4 + y^2/8 = 1);OneOf(Focus(Q)) = F;PointOnCurve(F, G) = True;G: Line;Intersection(G, C) = {A, B};Negation(A = B);A: Point;B: Point;Intersection(TangentOnPoint(A, C), TangentOnPoint(B, C)) = M;M: Point;XCoordinate(MidPoint(M)) = 2",
+      "coords": {}
+    },
+    {
+      "index": 2032,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_2032.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G);Length(LineSegmentOf(A, B)) = 10",
+      "coords": {}
+    },
+    {
+      "index": 2033,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_2033.png",
+      "answer": "sqrt(35)",
+      "fact_expr": "G: Ellipse;H: Line;Expression(G) = (x^2 + 4*y^2 = 16);Expression(H) = (y = x/2 + 1)",
+      "coords": {}
+    },
+    {
+      "index": 2034,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_2034.png",
+      "answer": "4*sqrt(5)",
+      "fact_expr": "G: Line;H: Curve;A: Point;B: Point;Expression(G) = (y = 2*x + 3);Expression(H) = (y = x^2);Intersection(G, H) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 2035,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 2036,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 2037,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_2037.png",
+      "answer": "3",
+      "fact_expr": "P: Point;Coordinate(P) = (2, a);a: Number;C: Parabola;Expression(C) = (y^2 = 4*x);PointOnCurve(P, C) = True",
+      "coords": {}
+    },
+    {
+      "index": 2038,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 2039,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 2040,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_focus_mismatch",
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_2040.png",
+      "answer": "(0, 2]",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;F: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (4, 0);RightFocus(G) = F;PointOnCurve(F, H);Inclination(H) = ApplyUnit(60, degree);NumIntersection(H, RightPart(G)) = 1",
+      "coords": {
+        "F": [
+          4.0,
+          0.0
+        ]
+      },
+      "validation_reasons": [
+        "hyperbola_focus_mismatch"
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+        "note": "using explicit params"
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+      "answer": "sqrt(3)",
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+        "note": "using explicit params"
+      },
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+      "fact_expr": "G: Parabola;Expression(G) = (y = 2*x^2)",
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+    },
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+        "converged": true,
+        "note": "using explicit params"
+      },
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+      "coords": {}
+    },
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+      "index": 2050,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/hyperbola/problem_2051.png",
+      "answer": "sqrt(2)",
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+      },
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+      "coords": {}
+    },
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+      "success": false,
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+        "note": "using explicit params"
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+          1.0,
+          0.0
+        ]
+      }
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+      },
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+          0.0
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+      "coords": {}
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "error": "Unsupported or unparseable expression"
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+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_2063.png",
+      "answer": "{4/7,4}",
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+      "coords": {}
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+        "converged": true,
+        "note": "using explicit params"
+      },
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+      "answer": "(sqrt(2),+oo)",
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+      "coords": {}
+    },
+    {
+      "index": 2066,
+      "success": false,
+      "conic_type": "parabola",
+      "error": "validation: parabola_point_off_curve",
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+        "iterations": 500
+      },
+      "output_path": "results/test/parabola/problem_2066.png",
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+      "coords": {
+        "H": [
+          -4.0,
+          -8.0
+        ]
+      },
+      "validation_reasons": [
+        "parabola_point_off_curve"
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+      },
+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/ellipse/problem_2067.png",
+      "answer": "2*sqrt(3)",
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+      "coords": {}
+    },
+    {
+      "index": 2068,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "results/test/parabola/problem_2068.png",
+      "answer": "32",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 8*x);F: Point;Focus(C) = F;G: Line;PointOnCurve(F, G);Inclination(G) = ApplyUnit(30, degree);A: Point;B: Point;Intersection(G, C) = {A, B}",
+      "coords": {}
+    }
+  ]
+}
\ No newline at end of file
diff --git a/samples/train/summary.json b/samples/train/summary.json
new file mode 100644
index 0000000000000000000000000000000000000000..23284d99bbfb8a4b5460ba21df105b60fdb0b0b9
--- /dev/null
+++ b/samples/train/summary.json
@@ -0,0 +1,136892 @@
+{
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+    "success": 5849,
+    "failed": 1908
+  },
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+    "hyperbola": 2025,
+    "parabola": 2000,
+    "circle": 90
+  },
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+    "hyperbola_focus_mismatch": 38,
+    "ellipse_ecc_mismatch": 19,
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+    "ellipse_point_off_curve": 39,
+    "ellipse_focus_mismatch": 25,
+    "parabola_focus_mismatch": 6,
+    "hyperbola_asymptote_mismatch": 3,
+    "circle_point_off_curve": 1
+  },
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0000.png",
+      "answer": "sqrt(3)/3",
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+      "coords": {}
+    },
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+      "answer": "sqrt(5)",
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+      "answer": "6*sqrt(2)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0032.png",
+      "answer": "2*sqrt(3)",
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+      "answer": "[\\sqrt{2}/2, 1)",
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+      "index": 35,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0035.png",
+      "answer": "34/3",
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+      "index": 36,
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+      "answer": "sqrt(5)/2",
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+    {
+      "index": 37,
+      "success": true,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0037.png",
+      "answer": "12*sqrt(2)/7",
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+    {
+      "index": 38,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 39,
+      "success": true,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0039.png",
+      "answer": "x^2=(1/4)*y",
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+      "index": 40,
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+        "direction": "right"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0040.png",
+      "answer": "(2,2)",
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+      "index": 41,
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0041.png",
+      "answer": "sqrt(5)/2",
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+    {
+      "index": 42,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 43,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.0,
+        "orientation": "horizontal"
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0043.png",
+      "answer": "{(sqrt(3),+oo),(-oo,-sqrt(3))}",
+      "fact_expr": "G: Hyperbola;H: Line;m: Real;Expression(G) = (x^2/4 - y^2 = 1);Expression(H) = (y = m + x);NumIntersection(H, G)=2",
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+    {
+      "index": 44,
+      "success": true,
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+        "direction": "right"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0044.png",
+      "answer": "-1",
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+    },
+    {
+      "index": 45,
+      "success": true,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0045.png",
+      "answer": "{ApplyUnit(60,degree),ApplyUnit(120,degree)}",
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+    {
+      "index": 46,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 47,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0047.png",
+      "answer": "1/16",
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+      "index": 48,
+      "success": true,
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+      "error": null,
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+        "b": 4.47213595499958,
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+        "x_coef": 45.0,
+        "y_coef": 20.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0048.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
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+    },
+    {
+      "index": 49,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 4.898979485566356,
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+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0049.png",
+      "answer": "x^2/56+y^2/7=1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/25 - y^2/24 = 1);H: Ellipse;Focus(G) = Focus(H);Distance(Directrix(H),Center(H)) = 8",
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+    {
+      "index": 50,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.7320508075688772,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0050.png",
+      "answer": "x^2/4+y^2/3=1",
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+      "coords": {}
+    },
+    {
+      "index": 51,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0051.png",
+      "answer": "sqrt(3)",
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+    },
+    {
+      "index": 52,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 53,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0053.png",
+      "answer": "2",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Focus(C) = {F1, F2};PointOnCurve(P, C);IsPerpendicular(VectorOf(P,F1),VectorOf(P,F2));Area(TriangleOf(P, F1, F2)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 54,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0054.png",
+      "answer": "sqrt(2)",
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+    },
+    {
+      "index": 55,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0055.png",
+      "answer": "2*x+y-2=0",
+      "fact_expr": "G: Parabola;A: Point;B: Point;O: Origin;M: Point;Expression(G) = (y^2 = 2*x);Coordinate(M) = (1, 0);PointOnCurve(M,LineOf(A,B));Intersection(LineOf(A,B), G) = {A, B};Slope(LineSegmentOf(O, A))+Slope(LineSegmentOf(O, B)) = 1",
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+        "M": [
+          1.0,
+          0.0
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+      }
+    },
+    {
+      "index": 56,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 57,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0057.png",
+      "answer": "18\n20",
+      "fact_expr": "G: Ellipse;H: Line;A: Point;B: Point;F2: Point;F1: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Focus(G) = {F1, F2};PointOnCurve(F1,H);Intersection(H,G)={A,B}",
+      "coords": {}
+    },
+    {
+      "index": 58,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_focus_mismatch",
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0058.png",
+      "answer": "1\n2",
+      "fact_expr": "C1: Hyperbola;C2:Hyperbola;b: Number;a: Number;F: Point;Expression(C1) = (-y^2/b^2 + x^2/a^2 = 1);Expression(C2) = (-y^2/16 + x^2/4 = 1);Coordinate(F) = (sqrt(5), 0);Asymptote(C1)=Asymptote(C2);RightFocus(C1) = F",
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+          0.0
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+      "validation_reasons": [
+        "hyperbola_focus_mismatch"
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+    {
+      "index": 59,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0059.png",
+      "answer": "x = -2",
+      "fact_expr": "G: Parabola;p: Number;H: Circle;p>0;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (8*y - 4*x + x^2 + y^2 + 19 = 0);PointOnCurve(Center(H), G)",
+      "coords": {}
+    },
+    {
+      "index": 60,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0060.png",
+      "answer": "20",
+      "fact_expr": "F1: Point;F2: Point;Focus(G) = {F1, F2};G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);H: Line;A: Point;B: Point;PointOnCurve(F1, H);Intersection(H, G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 61,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 62,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0062.png",
+      "answer": "x^2=4*y",
+      "fact_expr": "C: Parabola;F: Point;Coordinate(F) = (0, 1);Focus(C) = F",
+      "coords": {
+        "F": [
+          0.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 63,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
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+      "answer": "{sqrt(2),sqrt(3)}",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0068.png",
+      "answer": "(0,12)",
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+      "answer": "5",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0070.png",
+      "answer": "sqrt(5)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0071.png",
+      "answer": "[4, +oo)",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 73,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0073.png",
+      "answer": "9",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0074.png",
+      "answer": "sqrt(6)/3",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;A: Point;B: Point;M: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(B) = (0, a);Intersection(RightDirectrix(C), xAxis) = A;VectorOf(A,B)=3*VectorOf(A,M);PointOnCurve(M, C)",
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+    {
+      "index": 75,
+      "success": true,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0075.png",
+      "answer": "3, 25/3",
+      "fact_expr": "G: Ellipse;m: Real;Expression(G) = (x^2/5 + y^2/m = 1);Eccentricity(G) = sqrt(10)/5",
+      "coords": {}
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+    {
+      "index": 76,
+      "success": true,
+      "conic_type": "ellipse",
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+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0076.png",
+      "answer": "2",
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+    {
+      "index": 77,
+      "success": true,
+      "conic_type": "parabola",
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+        "direction": "up"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0077.png",
+      "answer": "27/4",
+      "fact_expr": "G: Parabola;Expression(G) = (y = x^2);A: Point;B: Point;Coordinate(A) = (1, 1);Coordinate(B) = (-2, 4);M: Point;Intersection(TangentOnPoint(A,G),TangentOnPoint(B,G))=M;PointOnCurve(A,G);PointOnCurve(B,G)",
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+        "B": [
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+    {
+      "index": 78,
+      "success": true,
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+        "y_coef": 3.0
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0078.png",
+      "answer": "3",
+      "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2/5 + y^2/m = 1);PointOnCurve(Focus(G), xAxis);e:Number;Eccentricity(G)=e;e = sqrt(10)/5",
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+    {
+      "index": 79,
+      "success": true,
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+      "error": null,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0079.png",
+      "answer": "6",
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+    {
+      "index": 80,
+      "success": true,
+      "conic_type": "ellipse",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0080.png",
+      "answer": "(\\sqrt{2} - 1, 1)",
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+    {
+      "index": 81,
+      "success": true,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0081.png",
+      "answer": "-1",
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+      "index": 82,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0082.png",
+      "answer": "3",
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+    {
+      "index": 83,
+      "success": true,
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+        "y_coef": 8.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0083.png",
+      "answer": "x+y-3=0",
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+    {
+      "index": 84,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 85,
+      "success": true,
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_0085.png",
+      "answer": "1/5",
+      "fact_expr": "G: Ellipse;k: Real;H: Circle;Expression(G) = (k*y^2 + x^2 = 1);Expression(H) = (x^2 + y^2 = 4);PointOnCurve(Focus(G), H)",
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+    {
+      "index": 86,
+      "success": true,
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+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0086.png",
+      "answer": "4",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;A: Point;F1: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (sqrt(5), 0);LeftFocus(C) = F1;RightFocus(C) = F2;RightVertex(C) = A;Eccentricity(C) = sqrt(5)/5",
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+    {
+      "index": 87,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0087.png",
+      "answer": "3*sqrt(5)/5",
+      "fact_expr": "l: Line;G: Hyperbola;b: Number;a: Number;A: Point;B: Point;C: Point;D: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(RightFocus(G),l);IsPerpendicular(l,xAxis);Intersection(l, G) = {A, B};Intersection(l,Asymptote(G))={C,D};3*Abs(LineSegmentOf(A, B)) = 2*Abs(LineSegmentOf(C, D))",
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+    {
+      "index": 88,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0088.png",
+      "answer": "(1, 2)",
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+    {
+      "index": 89,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 90,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 91,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0091.png",
+      "answer": "8/3",
+      "fact_expr": "C: Parabola;G: Line;A: Point;B: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C)=F;Inclination(G)=pi/3;PointOnCurve(F,G);Intersection(G, C) = {A, B};IsChordOf(LineSegmentOf(A, B), C)",
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+    {
+      "index": 92,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
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+      "error": "Unsupported or unparseable expression"
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+      "answer": "{-3,-25/3}",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0151.png",
+      "answer": "5/2",
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+      "answer": "(9 - sqrt(17))/8",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0155.png",
+      "answer": "4*sqrt(3)-4",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0157.png",
+      "answer": "2*sqrt(2-1)",
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+      "answer": "3*sqrt(7)",
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+      "index": 220,
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+      "index": 221,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0222.png",
+      "answer": "(\\sqrt{5} - 1)/2",
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+      "answer": "sqrt(3)-1",
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+      "answer": "x^2/15 + y^2/5 = 1",
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+      "answer": "pm*sqrt(3)",
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+      "answer": "16/5",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0300.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;M: Point;O: Origin;Expression(G) = (y^2 = 4*x);PointOnCurve(M, G);Distance(M, O) = sqrt(5)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0301.png",
+      "answer": "x^2/6 - y^2/3 = 1",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0302.png",
+      "answer": "5",
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+    {
+      "index": 303,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0303.png",
+      "answer": "1/4",
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+      "success": false,
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+      "success": true,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0305.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Ellipse;a > b;b > 0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = sqrt(6)/2",
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+    {
+      "index": 306,
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+        "y_coef": 9.0
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0306.png",
+      "answer": "8",
+      "fact_expr": "G: Ellipse;H: Line;F2: Point;A: Point;B: Point;F1: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Focus(G) = {F1, F2};PointOnCurve(F1, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(F2, A)) + Abs(LineSegmentOf(F2, B)) = 12",
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+    {
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+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 308,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0308.png",
+      "answer": "sqrt(2)/2",
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+      "answer": "1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/b^2 = 1);b: Number;b>0;Expression(Asymptote(G)) = (y = pm*1/2*x)",
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+      "error": null,
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+        "converged": true,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0310.png",
+      "answer": "18",
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+    {
+      "index": 311,
+      "success": true,
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+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0311.png",
+      "answer": "4/3",
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+    {
+      "index": 312,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0312.png",
+      "answer": "{sqrt(34)/3, sqrt(34)/5}",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/n + x^2/m = 1);m: Number;n: Number;Expression(Asymptote(G)) = (pm*3*y + 5*x = 0)",
+      "coords": {}
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+    {
+      "index": 313,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "y_coef": 16.0
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0313.png",
+      "answer": "3/2",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/25 + y^2/16 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C)",
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+    {
+      "index": 314,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0314.png",
+      "answer": "1",
+      "fact_expr": "M: Point;Coordinate(M) = (-1, 2);C: Parabola;Expression(C) = (y^2 = 4*x);G: Line;PointOnCurve(Focus(C), G);k: Number;Slope(G) = k;A: Point;B: Point;Intersection(G, C) = {A, B};AngleOf(A, M, B) = ApplyUnit(90, degree)",
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+        "M": [
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+          2.0
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+    {
+      "index": 315,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 316,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "x_coef": 12.0,
+        "y_coef": 16.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0316.png",
+      "answer": "9",
+      "fact_expr": "G: Ellipse;A: Point;B: Point;P: Point;Expression(G) = (x^2/12 + y^2/16 = 1);Coordinate(A) = (0,-2);Coordinate(B) = (0, 2);PointOnCurve(P, G);Abs(VectorOf(A, P)) - Abs(VectorOf(B, P)) = 2",
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+        "B": [
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+    {
+      "index": 317,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 318,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0318.png",
+      "answer": "2",
+      "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/25 + y^2/9 = 1);PointOnCurve(P, G);Distance(P, LeftDirectrix(G)) = 5/2",
+      "coords": {}
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+    {
+      "index": 319,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "answer": "{1/2, sqrt(3)/2}",
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+      "answer": "[\\sqrt{3}/2,1)",
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+      "index": 344,
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+      "answer": "x^2/8+y^2/2=1",
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+      "answer": "3",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0346.png",
+      "answer": "4*sqrt(7)/7",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/n^2 + x^2/m^2 = 1);m: Number;n: Number;Eccentricity(G) = 4/3;H: Hyperbola;Expression(H) = (y^2/n^2 - x^2/m^2 = 1)",
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+      "index": 348,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0349.png",
+      "answer": "5/3",
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+      "answer": "24",
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+      "answer": "pm*4*y+5*x=0",
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+    {
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+    },
+    {
+      "index": 355,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 356,
+      "success": true,
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+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0356.png",
+      "answer": "8",
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+    {
+      "index": 357,
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+    {
+      "index": 358,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0358.png",
+      "answer": "x^2+y^2=1",
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+      "coords": {}
+    },
+    {
+      "index": 359,
+      "success": true,
+      "conic_type": "parabola",
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+        "direction": "right"
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+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0359.png",
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+      "coords": {}
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+      "answer": "(4, \\pm 4\\sqrt{2})",
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+    {
+      "index": 362,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0362.png",
+      "answer": "sqrt(2)/2",
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+    {
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+      "success": true,
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+        "direction": "right"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0363.png",
+      "answer": "y = x - 1",
+      "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;P: Point;Expression(G) = (y^2 = 2*x);Coordinate(P) = (2, 1);PointOnCurve(P, H);Intersection(H, G) = {A, B};IsChordOf(LineSegmentOf(A, B), G);MidPoint(LineSegmentOf(A, B)) = P",
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+    {
+      "index": 364,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "right"
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+      "optimization": {
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0364.png",
+      "answer": "3/4",
+      "fact_expr": "G: Parabola;M: Point;Expression(G) = (y^2 = x);PointOnCurve(M, G);Distance(M, Focus(G)) = 1",
+      "coords": {}
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+    {
+      "index": 365,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0365.png",
+      "answer": "2",
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+    {
+      "index": 366,
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+      "answer": "x^2/4 - y^2/3 = 1",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0367.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "C: Ellipse;b: Number;G: Line;F: Point;A: Point;O: Origin;B: Point;b>0;c:Number;Expression(C) = (x^2/4 + y^2/b^2 = 1);Expression(G) = (7*x - sqrt(14)*y = 0);Coordinate(F) = (c, 0);RightFocus(C) = F;Intersection(G, C) = {A, B};Abs(VectorOf(A, O)) = Abs(VectorOf(A, F))",
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+      "index": 368,
+      "success": true,
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+        "b": 3.4641016151377544,
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+        "x_coef": 12.0,
+        "y_coef": 36.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0368.png",
+      "answer": "x+y-4=0",
+      "fact_expr": "G: Ellipse;B: Point;A: Point;N: Point;Expression(G) = (3*x^2 + y^2 = 36);Coordinate(N) = (1, 3);PointOnCurve(A, G);PointOnCurve(B, G);MidPoint(LineSegmentOf(A,B))=N",
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+      "index": 369,
+      "success": true,
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+      "error": null,
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+        "direction": "up"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0369.png",
+      "answer": "sqrt(m)",
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+    },
+    {
+      "index": 370,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "down"
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0370.png",
+      "answer": "7",
+      "fact_expr": "G: Parabola;M: Point;Expression(G) = (x^2 = -4*y);PointOnCurve(M, G);Distance(M, Focus(G)) = 7",
+      "coords": {}
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+    {
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+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "x_coef": 8.0,
+        "y_coef": 9.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0371.png",
+      "answer": "(0, pm*1)\n1/3",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/8 + y^2/9 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 372,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 6.0,
+        "b": 8.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0372.png",
+      "answer": "{28, 4}",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2/36 - y^2/64 = 1);P: Point;PointOnCurve(P, C);A: Point;Coordinate(A) = (-10, 0);B: Point;Coordinate(B) = (10, 0);Abs(LineSegmentOf(P, A)) = 16",
+      "coords": {
+        "A": [
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+          0.0
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+        "B": [
+          10.0,
+          0.0
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+    },
+    {
+      "index": 373,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0373.png",
+      "answer": "x^2 - y^2/2 = 1",
+      "fact_expr": "G: Hyperbola;a: Number;H: Parabola;Expression(G) = (-y^2/2 + x^2/a^2 = 1);Expression(H) = (y^2 = 4*(sqrt(3)*x));RightFocus(G) = Focus(H)",
+      "coords": {}
+    },
+    {
+      "index": 374,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0374.png",
+      "answer": "y=pm*sqrt(3)*x",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;O: Origin;F: Point;A: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;RightVertex(C) = A;MidPoint(LineSegmentOf(O,F)) = A",
+      "coords": {}
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+    {
+      "index": 375,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.75,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0375.png",
+      "answer": "[sqrt(2)/2,1]",
+      "fact_expr": "C: Parabola;M: Point;F: Point;A: Point;Expression(C) = (x^2 = 3*y);Focus(C)=F;Intersection(Directrix(C),yAxis)=A;PointOnCurve(M, C)",
+      "coords": {}
+    },
+    {
+      "index": 376,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0376.png",
+      "answer": "5/7",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;G: Line;A: Point;F: Point;B: Point;e: Number;a > b;b > 0;O: Origin;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F;PointOnCurve(O, G);Intersection(C, G) = {A, B};Abs(LineSegmentOf(A, B)) = 10;Abs(LineSegmentOf(A, F)) = 6;Cos(AngleOf(A, B, F)) = 4/5;Eccentricity(C) = e",
+      "coords": {}
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+    {
+      "index": 377,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+      "answer": "x^2/18+y^2/9^2=1",
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+      "answer": "pm*1",
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+      "answer": "16",
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+      "answer": "(0, pm*2)",
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+      "index": 403,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0403.png",
+      "answer": "(6-4*sqrt(2),6+4*sqrt(2))",
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+      "answer": "5-sqrt(2)",
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+      "answer": "(0,\\sqrt{2}/2]",
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+      "answer": "(2,0)",
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+      "answer": "(1,\\sqrt{2}]",
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+      "answer": "sqrt(2)/2",
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+      "index": 436,
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+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 438,
+      "success": true,
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+      "answer": "x^2/4+y^2/3=1",
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+      "index": 458,
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+      "answer": "{8/3, 8}",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0461.png",
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+      "index": 464,
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+      "index": 466,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0466.png",
+      "answer": "-5",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0467.png",
+      "answer": "2",
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+      "index": 468,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0468.png",
+      "answer": "3*sqrt(5)/5",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;c: Number;Coordinate(F) = (c, 0);RightFocus(C) = F;G: Circle;Center(G) = F;Radius(G) = a;A: Point;B: Point;Intersection(G, OneOf(Asymptote(C))) = {A, B};Abs(LineSegmentOf(A, B)) = (2/3)*c",
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+    {
+      "index": 469,
+      "success": true,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0469.png",
+      "answer": "3",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0470.png",
+      "answer": "5",
+      "fact_expr": "G: Hyperbola;H: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 - y^2 = 1);Expression(H) = (x^2/9 + y^2/4 = 1);Focus(H)={F1,F2};Focus(G)={F1,F2};OneOf(Intersection(H, G)) =P",
+      "coords": {}
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+    {
+      "index": 471,
+      "success": true,
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+        "direction": "up"
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+      "answer": "{pi/3,(2*pi)/3}",
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+      "answer": "y=pm*3*x",
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+      "answer": "{18, 2}",
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+      "answer": "y^2=-4*x",
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+      "answer": "1",
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+      "index": 529,
+      "success": true,
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+      "error": null,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0529.png",
+      "answer": "(2*sqrt(7))/7",
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+    {
+      "index": 530,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0530.png",
+      "answer": "2*sqrt(2)-5",
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+      "conic_type": "hyperbola",
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+        "note": "using explicit params"
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+      "answer": "5/3",
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+      "answer": "9/2",
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+      "answer": "16*sqrt(3)/3",
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+      "answer": "1",
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+      "answer": "8",
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+      "answer": "64/3",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0559.png",
+      "answer": "8*sqrt(2)",
+      "fact_expr": "l: Line;C: Parabola;O: Origin;M: Point;N: Point;Expression(C) = (y^2 = 4*x);Intersection(l, C) = {M, N};Slope(LineSegmentOf(O, M))*Slope(LineSegmentOf(O, N)) = -1/2",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0560.png",
+      "answer": "120*sqrt(3)/11",
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+      "answer": "2*sqrt(2)",
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+      "answer": "8/5",
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+      "answer": "1",
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+      "answer": "(6*sqrt(5)/5)-1",
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+      "index": 586,
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+      "answer": "sqrt(3)",
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+      "answer": "[\\sqrt{2}, +\\infty)",
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+      "index": 589,
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+      "answer": "3",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0590.png",
+      "answer": "16",
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+      "index": 591,
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+        "b": 4.0,
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+        "note": "using explicit params"
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+      "answer": "16",
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+    {
+      "index": 592,
+      "success": true,
+      "conic_type": "ellipse",
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+        "b": 2.6457513110645907,
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+        "x_coef": 9.0,
+        "y_coef": 7.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0592.png",
+      "answer": "-sqrt(7)/3",
+      "fact_expr": "l: Line;C: Ellipse;F: Point;A: Point;B: Point;O: Origin;Expression(C) = (x^2/9 + y^2/7 = 1);LeftFocus(C) = F;PointOnCurve(O, l);Intersection(l, C) = {A, B};Quadrant(A)=2;AngleOf(F, A, B) = AngleOf(B, F, O)",
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+    {
+      "index": 593,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+      "answer": "sqrt(3)",
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+      "answer": "{pi/3, ApplyUnit(60, degree)}",
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+      "answer": "2",
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+      "answer": "8",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0632.png",
+      "answer": "(-4, \\pm 4\\sqrt{2})",
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+      "answer": "2/5",
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+    {
+      "index": 634,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0634.png",
+      "answer": "6/5",
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+    {
+      "index": 635,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_0635.png",
+      "answer": "(x^2/4+y^2=1)&Negation(y=0)",
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+    {
+      "index": 636,
+      "success": true,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0636.png",
+      "answer": "y=pm*(sqrt(5)/2)*x",
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+    {
+      "index": 637,
+      "success": true,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0637.png",
+      "answer": "-5",
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+    {
+      "index": 638,
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+        "b": 1.5,
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0638.png",
+      "answer": "sqrt(3)+1",
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+      "coords": {}
+    },
+    {
+      "index": 639,
+      "success": true,
+      "conic_type": "ellipse",
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+        "b": 0.5,
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+        "y_coef": 0.25
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0639.png",
+      "answer": "y^2=pm*2*sqrt(3)*x",
+      "fact_expr": "G: Parabola;H: Ellipse;Expression(H) = (x^2 + 4*y^2 = 1);Vertex(G)=O;O:Origin;Focus(G)=OneOf(Focus(H))",
+      "coords": {}
+    },
+    {
+      "index": 640,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0640.png",
+      "answer": "3",
+      "fact_expr": "G: Parabola;l1:Line;l2:Line;P: Point;Expression(G) = (y^2 = 4*x);Expression(l1)=(4*x-3*y+11=0);Expression(l2)=(x+1=0);PointOnCurve(P, G)",
+      "coords": {}
+    },
+    {
+      "index": 641,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 642,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0642.png",
+      "answer": "{18, 2}",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/9 + x^2/a^2 = 1);a: Number;P: Point;PointOnCurve(P, G);Expression(OneOf(Asymptote(G))) = (3*x + 4*y = 0);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;Abs(LineSegmentOf(P, F1)) = 10",
+      "coords": {}
+    },
+    {
+      "index": 643,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
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+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0643.png",
+      "answer": "1/3",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;B: Point;F: Point;A: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;UpperVertex(C) = B;OneOf(Intersection(LineOf(B,F),C)) = A;LineSegmentOf(A, F)/LineSegmentOf(B, F) = 4/5",
+      "coords": {}
+    },
+    {
+      "index": 644,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0644.png",
+      "answer": "(pm*2*sqrt(2),0)\ny=pm*(sqrt(15)/3)*x",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Ellipse;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2/8 + y^2/5 = 1);Eccentricity(G) = (2*sqrt(6))/3;Vertex(G)=Focus(H)",
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+    },
+    {
+      "index": 645,
+      "success": true,
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0645.png",
+      "answer": "[\\sqrt{2}/2,1)",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;F1: Point;F2: Point;P: Point;a > b;b > 0;Focus(G)={F1,F2};PointOnCurve(P,G);Abs(LineSegmentOf(P,F1)+LineSegmentOf(P,F2))=Abs(LineSegmentOf(F1,F2))",
+      "coords": {}
+    },
+    {
+      "index": 646,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0646.png",
+      "answer": "6",
+      "fact_expr": "C: Parabola;p: Number;G: Line;A: Point;F: Point;B: Point;O: Origin;p>0;Expression(C) = (y^2 = 2*p*x);Focus(C) = F;PointOnCurve(F, G);Intersection(G, C) = {A, B};Abs(LineSegmentOf(A, F)) = 5*Abs(LineSegmentOf(B, F))",
+      "coords": {}
+    },
+    {
+      "index": 647,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0647.png",
+      "answer": "(1,\\sqrt{10}]",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = 3*x);NumIntersection(G, H)=0;Eccentricity(G)=e;e:Number",
+      "coords": {}
+    },
+    {
+      "index": 648,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0648.png",
+      "answer": "(\\pi/6, \\pi/2)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, RightPart(G));Abs(LineSegmentOf(P, F2)) = Abs(LineSegmentOf(F1, F2));Q: Point;MidPoint(LineSegmentOf(P, F1)) = Q",
+      "coords": {}
+    },
+    {
+      "index": 649,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 2.0,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0649.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 4)",
+      "coords": {}
+    },
+    {
+      "index": 650,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 651,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 652,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0652.png",
+      "answer": "7/2",
+      "fact_expr": "G: Parabola;H: Line;B: Point;A: Point;P:Point;D:Point;F: Point;Expression(G) = (x^2 = y);Coordinate(P) = (1, 0);Coordinate(D) = (0, 2);PointOnCurve(P,H);Intersection(H, G) = {A, B};Focus(G)=F;PointOnCurve(D,PerpendicularBisector(LineSegmentOf(A,B)))",
+      "coords": {
+        "P": [
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+          0.0
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+        "D": [
+          0.0,
+          2.0
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+      }
+    },
+    {
+      "index": 653,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 654,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0654.png",
+      "answer": "16",
+      "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;Expression(G) = (y^2 = 8*x);Expression(H) = (x + y - 2 = 0);Intersection(H, G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 655,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 656,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 657,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 658,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
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+      "answer": "(x-1)^2 + (y-2)^2 = 4",
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+      "answer": "4",
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+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0715.png",
+      "answer": "1/2",
+      "fact_expr": "l: Line;C: Ellipse;b: Number;a: Number;A: Point;B: Point;M: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(M) = (-2, 1);Eccentricity(C) = sqrt(3)/2;Intersection(l, C) = {A, B};MidPoint(LineSegmentOf(A,B)) = M",
+      "coords": {
+        "M": [
+          -2.0,
+          1.0
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+    },
+    {
+      "index": 716,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0716.png",
+      "answer": "12",
+      "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;x1: Number;x2: Number;y2: Number;Expression(G) = (y^2 = 8*x);Coordinate(A) = (x1, x2);Coordinate(B) = (x2, y2);PointOnCurve(Focus(G), H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = 16",
+      "coords": {}
+    },
+    {
+      "index": 717,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0717.png",
+      "answer": "(\\sqrt{5} - 1)/2",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;G: Circle;P: Point;A: Point;F: Point;O: Origin;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftVertex(C) = A;LeftFocus(C) = F;TangentPoint(LineOf(A, P), G) = P;Center(G) = O;Radius(G) = b;IsPerpendicular(LineSegmentOf(P, F), xAxis)",
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+    },
+    {
+      "index": 718,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0718.png",
+      "answer": "8",
+      "fact_expr": "l: Line;Focus(G) = F;F: Point;PointOnCurve(F,l) = True;G: Parabola;Expression(G) = (y^2 = 8*x);Intersection(l,G) = {A,B};A: Point;B: Point;Distance(MidPoint(LineSegmentOf(A,B)),yAxis) = 2",
+      "coords": {}
+    },
+    {
+      "index": 719,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0719.png",
+      "answer": "6",
+      "fact_expr": "G: Parabola;p: Number;H: Line;B: Point;C: Point;F: Point;A: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G)=F;PointOnCurve(F,H);Intersection(H,G) = {B, A};Intersection(H,Directrix(G))=C;Between(B,F,C)=True;Abs(LineSegmentOf(B, C)) = 2*Abs(LineSegmentOf(B, F));Abs(LineSegmentOf(A, F)) = 12",
+      "coords": {}
+    },
+    {
+      "index": 720,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0720.png",
+      "answer": "sqrt(6)/3",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;A: Point;P: Point;B: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Endpoint(MajorAxis(C))={A,B};PointOnCurve(P, C);AngleOf(A, P, B) = ApplyUnit(120, degree)",
+      "coords": {}
+    },
+    {
+      "index": 721,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0721.png",
+      "answer": "12",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);F: Point;Focus(G) = F;A: Point;Coordinate(A) = (5, 4);P: Point;PointOnCurve(P, G);Negation(PointOnCurve(P, LineOf(A, F)))",
+      "coords": {
+        "A": [
+          5.0,
+          4.0
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+    },
+    {
+      "index": 722,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          4.0,
+          -0.0
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+        "radius": 4.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_0722.png",
+      "answer": "sqrt(15)",
+      "fact_expr": "G: Hyperbola;a: Number;H: Circle;a>0;Expression(G) = (-y^2 + x^2/a^2 = 1);Expression(H) = (-8*x + x^2 + y^2 = 0);RightFocus(G)=Center(H)",
+      "coords": {}
+    },
+    {
+      "index": 723,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 2.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0723.png",
+      "answer": "x^2/12 - y^2/8 = 1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/4 = 1);H: Point;Coordinate(H) = (3*sqrt(2), 2);Z: Hyperbola;Focus(G) = Focus(Z);PointOnCurve(H, Z)",
+      "coords": {
+        "H": [
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+          2.0
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+    },
+    {
+      "index": 724,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0724.png",
+      "answer": "(1,3)+(3,+oo)",
+      "fact_expr": "G: Ellipse;m: Number;H: Line;Expression(G) = (y^2/3 + x^2/m = 1);Expression(H) = (y = x + 2);NumIntersection(H, G) = 2",
+      "coords": {}
+    },
+    {
+      "index": 725,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0725.png",
+      "answer": "[2, +oo)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;F: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F;PointOnCurve(F, H);Inclination(H) = (2*pi)/3;NumIntersection(H, LeftPart(G)) = 1",
+      "coords": {}
+    },
+    {
+      "index": 726,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0726.png",
+      "answer": "2",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/2 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 727,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0727.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;l: Line;A: Point;B: Point;O:Origin;c:Number;b > a;a > 0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (a, 0);Coordinate(B) = (0, b);HalfFocalLength(G)=c;Distance(O,l)=(sqrt(3)/4)*c;PointOnCurve(A,l);PointOnCurve(B,l)",
+      "coords": {}
+    },
+    {
+      "index": 728,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 4.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0728.png",
+      "answer": "10*sqrt(2)",
+      "fact_expr": "l: Line;G: Parabola;C: Point;D: Point;A: Point;B: Point;Expression(G) = (y^2 = 16*x);Expression(l) = (2*x - y + 1 = 0);Intersection(l, G) = {A, B};L1:Line;L2:Line;PointOnCurve(A,L1);PointOnCurve(B,L2);IsPerpendicular(l,L1);IsPerpendicular(l,L2);Intersection(L1, xAxis) = C ;Intersection(L2, xAxis) = D",
+      "coords": {}
+    },
+    {
+      "index": 729,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0729.png",
+      "answer": "(1/4,+\\infty)",
+      "fact_expr": "G: Ellipse;m: Number;H: Line;Expression(G) = (x^2/3 + y^2/m = 1);Expression(H) = (x + 2*y - 2 = 0);NumIntersection(G, H) = 2",
+      "coords": {}
+    },
+    {
+      "index": 730,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.125,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0730.png",
+      "answer": "1/4",
+      "fact_expr": "G: Parabola;Expression(G) = (y = 2*x^2)",
+      "coords": {}
+    },
+    {
+      "index": 731,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 732,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 733,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0733.png",
+      "answer": "4*sqrt(2)",
+      "fact_expr": "G: Parabola;P: Point;F: Point;O: Origin;Expression(G) = (x^2 = 4*y);PointOnCurve(P, G);Focus(G) = F;Abs(LineSegmentOf(P, F)) = 5",
+      "coords": {}
+    },
+    {
+      "index": 734,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0734.png",
+      "answer": "pm*sqrt(2)",
+      "fact_expr": "G: Parabola;A: Point;B: Point;O: Origin;M: Point;Expression(G) = (y^2=2*x);PointOnCurve(A,G);PointOnCurve(B,G);DotProduct(VectorOf(O,A),VectorOf(O,B))=0;Intersection(LineOf(A,B),xAxis)=M;LineSegmentOf(A,M)=2*LineSegmentOf(B,M)",
+      "coords": {}
+    },
+    {
+      "index": 735,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 736,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 737,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 738,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0738.png",
+      "answer": "(-\\infty,-4] \\cup [12,+\\infty)",
+      "fact_expr": "G: Parabola;A: Point;B: Point;C: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (4, 4);PointOnCurve(B,G);PointOnCurve(C,G);IsPerpendicular(LineSegmentOf(A, B), LineSegmentOf(B, C))",
+      "coords": {
+        "A": [
+          4.0,
+          4.0
+        ]
+      }
+    },
+    {
+      "index": 739,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0739.png",
+      "answer": "3*sqrt(3)/5",
+      "fact_expr": "F1: Point;F2: Point;Focus(G) = {F1, F2};G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);P: Point;PointOnCurve(P, G);AngleOf(P, F1, F2) = ApplyUnit(120, degree)",
+      "coords": {}
+    },
+    {
+      "index": 740,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 741,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 742,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 743,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0743.png",
+      "answer": "10/3",
+      "fact_expr": "G: Ellipse;A: Point;B: Point;F2: Point;F1: Point;x1:Number;x2:Number;y1:Number;y2:Number;Expression(G) = (x^2/25 + y^2/16 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;IsChordOf(LineSegmentOf(A,B),G);Perimeter(InscribedCircle(TriangleOf(A,B,F2)))=2*pi;Coordinate(A)=(x1,y1);Coordinate(B)=(x2,y2);PointOnCurve(F1, LineSegmentOf(A, B))",
+      "coords": {}
+    },
+    {
+      "index": 744,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0744.png",
+      "answer": "40",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 8*y);H: Line;PointOnCurve(Focus(G), H);Slope(H) = 2;A: Point;B: Point;Intersection(H, G) = {A, B};IsChordOf(LineSegmentOf(A,B),G)",
+      "coords": {}
+    },
+    {
+      "index": 745,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0745.png",
+      "answer": "[-2*sqrt(3), 2*sqrt(3)]",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);P: Point;PointOnCurve(P, C);F: Point;Focus(C) = F;A: Point;m: Real;Coordinate(A) = (3, m);Min(Abs(LineSegmentOf(P, A)) + Abs(LineSegmentOf(P, F))) = 4",
+      "coords": {}
+    },
+    {
+      "index": 746,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0746.png",
+      "answer": "17/4",
+      "fact_expr": "G: Ellipse;A: Point;M: Point;F: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Coordinate(A) = (2, 2);RightFocus(G) = F;PointOnCurve(M, G)",
+      "coords": {
+        "A": [
+          2.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 747,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
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+      "answer": "2*sqrt(3)/3",
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+      "answer": "sqrt(2)/2",
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+      "index": 777,
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+      "answer": "5/2",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0779.png",
+      "answer": "pm*1",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0787.png",
+      "answer": "y^2=20*x",
+      "fact_expr": "G: Hyperbola;H: Parabola;Expression(G) = (x^2/16 - y^2/9 = 1);Vertex(H) = Center(G);Focus(H) = RightFocus(G)",
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+      "answer": "16*sqrt(3)/5",
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+      "answer": "x^2/25+y^2/9=1",
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+      "answer": "3",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0792.png",
+      "answer": "5/4",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1);e: Number;Eccentricity(G) = e",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0794.png",
+      "answer": "5/2",
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+      "index": 795,
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+      "index": 796,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0796.png",
+      "answer": "5/4",
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+      "index": 797,
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+      "answer": "3",
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+      "index": 798,
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+      "index": 799,
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+      "index": 800,
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+      "answer": "5/3",
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+      "index": 801,
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+      "answer": "2*sqrt(3)",
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+        "y_coef": 5.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0802.png",
+      "answer": "y^2=18*x",
+      "fact_expr": "H: Ellipse;Expression(H) = (x^2/9 + y^2/5 = 1);G: Parabola;Center(H) = Vertex(G);LeftDirectrix(H) = Directrix(G)",
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+    {
+      "index": 803,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 804,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 805,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0805.png",
+      "answer": "sqrt(10)",
+      "fact_expr": "G: Parabola;H: Circle;P: Point;A: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (y^2 + (x - 4)^2 = 2);PointOnCurve(P, G);l: Line;TangentOfPoint(P, H) = l;TangentPoint(l, H) = A",
+      "coords": {}
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+    {
+      "index": 806,
+      "success": true,
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+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0806.png",
+      "answer": "2*sqrt(3)/3",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2/b^2 = 1);b: Number;Length(ImageinaryAxis(G)) = 2",
+      "coords": {}
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+    {
+      "index": 807,
+      "success": false,
+      "error": "Circle from diameter: missing point coordinates"
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+    {
+      "index": 808,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "up"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0808.png",
+      "answer": "2",
+      "fact_expr": "H: Line;Expression(H) = (m - m*y + x = 0);m: Number;G: Parabola;Expression(G) = (x^2 = 2*(p*y));p: Number;p>0;PointOnCurve(Focus(G), H)",
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+    {
+      "index": 809,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 810,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0810.png",
+      "answer": "7/2",
+      "fact_expr": "G: Parabola;P: Point;A: Point;M: Point;d: Number;Expression(G) = (y^2 = 2*x);Coordinate(A) = (7/2, 4);PointOnCurve(P, G);Distance(P, Directrix(G)) = d;Projection(P, yAxis) = M",
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+    {
+      "index": 811,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 812,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
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+      "answer": "1+sqrt(2)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0870.png",
+      "answer": "1/2",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0871.png",
+      "answer": "(1,\\sqrt{2})",
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+      "index": 873,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0873.png",
+      "answer": "{pm*1, pm*sqrt(2)}",
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+      "answer": "2*sqrt(2)/3",
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+      "index": 905,
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+      "answer": "x^2/3 - y^2/6 = 1",
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+      "index": 906,
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+        "note": "using explicit params"
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+      "answer": "sqrt(13)",
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+      "index": 907,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 908,
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+      "coords": {}
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+    {
+      "index": 909,
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+      "error": null,
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+        "direction": "up"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0909.png",
+      "answer": "24",
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+    {
+      "index": 910,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0910.png",
+      "answer": "sqrt(2)",
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+    {
+      "index": 911,
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+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 913,
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0913.png",
+      "answer": "sqrt(5)+2*sqrt(2)",
+      "fact_expr": "G: Ellipse;A: Point;P: Point;F: Point;Expression(G) = (x^2/2 + y^2 = 1);Coordinate(A) = (0, 1/2);RightFocus(G) = F;PointOnCurve(P, G)",
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+          0.5
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+    {
+      "index": 914,
+      "success": true,
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+      "error": null,
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0914.png",
+      "answer": "2",
+      "fact_expr": "C: Parabola;G: Circle;H: Line;A: Point;B: Point;P: Point;k:Number;k>0;Expression(C) = (y^2 = 8*x);Coordinate(P) = (1, 0);PointOnCurve(P, H);Slope(H) = k;Intersection(H, C) = {A, B};IsDiameter(LineSegmentOf(A,B),G);Length(InterceptChord(xAxis,G))=Length(InterceptChord(yAxis,G))",
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+          0.0
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+      "index": 915,
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+      "error": null,
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+        "b": 3.0,
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+        "x_coef": 25.0,
+        "y_coef": 9.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0915.png",
+      "answer": "20",
+      "fact_expr": "G: Ellipse;H: Line;P: Point;Q: Point;F2: Point;F1: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Focus(G) ={F1,F2};PointOnCurve(F1, H);Intersection(H, G) = {P, Q}",
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+    {
+      "index": 916,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0916.png",
+      "answer": "(2-sqrt(2))/2",
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+          1.0,
+          2.0
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+    {
+      "index": 917,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0917.png",
+      "answer": "3",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Focus(C) = {F1, F2};PointOnCurve(P, C);IsPerpendicular(VectorOf(P, F1), VectorOf(P, F2));Area(TriangleOf(P, F1, F2)) = 9",
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+    {
+      "index": 918,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0918.png",
+      "answer": "5",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);P: Point;PointOnCurve(P, G);XCoordinate(P) = 3",
+      "coords": {}
+    },
+    {
+      "index": 919,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0919.png",
+      "answer": "1/3 + 2*sqrt(21)/3",
+      "fact_expr": "G: Ellipse;C: Circle;P: Point;Q: Point;Expression(G) = (x^2 + 4*y^2 = 4);Expression(C) = (x^2 + (y - 2)^2 = 1/9);PointOnCurve(P, C);PointOnCurve(Q, G)",
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+    },
+    {
+      "index": 920,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.366584062576294,
+        "b": 2.364173412322998,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.00017838670464698225,
+        "converged": true,
+        "iterations": 89
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0920.png",
+      "answer": "-12",
+      "fact_expr": "C: Hyperbola;a: Number;G: Parabola;m: Real;a>0;Expression(C) = (-y^2/4 + x^2/a^2 = 1);Expression(G) = (y^2 = m*x);Eccentricity(C)=2*sqrt(5)/5;LeftFocus(C)=Focus(G)",
+      "coords": {}
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+    {
+      "index": 921,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 922,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0922.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);P: Point;PointOnCurve(P, G);Distance(P, Focus(G)) = 3",
+      "coords": {}
+    },
+    {
+      "index": 923,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0923.png",
+      "answer": "x-4*y+3=0",
+      "fact_expr": "G: Parabola;L: Line;A: Point;B: Point;M: Point;Expression(G) = (x^2 = 8*y);Coordinate(M) = (1, 1);Intersection(L,G)={A,B};MidPoint(LineSegmentOf(A,B))=M",
+      "coords": {
+        "M": [
+          1.0,
+          1.0
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+      }
+    },
+    {
+      "index": 924,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0924.png",
+      "answer": "12",
+      "fact_expr": "C: Parabola;Expression(C) = (x^2 = 4*y);F: Point;Focus(C) = F;PointOnCurve(F, G);G: Line;Intersection(G, C) = {A, B};A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), C) ;Distance(MidPoint(LineSegmentOf(A, B)), xAxis) = 5",
+      "coords": {}
+    },
+    {
+      "index": 925,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 926,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_point_off_curve",
+      "params": {
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+        "b": 2.450642358201144,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 1.0319236343469396e-05,
+        "converged": true,
+        "iterations": 94
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0926.png",
+      "answer": "x^2 - y^2 / 3 = 1",
+      "fact_expr": "G: Hyperbola;H: Point;Coordinate(H) = (2, 3);PointOnCurve(H, G);Expression(OneOf(Asymptote(G))) = (y = sqrt(3)*x)",
+      "coords": {
+        "H": [
+          2.0,
+          3.0
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+      },
+      "validation_reasons": [
+        "hyperbola_point_off_curve"
+      ]
+    },
+    {
+      "index": 927,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0927.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;F1: Point;F2: Point;P: Point;Expression(G) = (x^2/4 - y^2 = 1);Focus(G)={F1,F2};PointOnCurve(P, G);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0",
+      "coords": {}
+    },
+    {
+      "index": 928,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 929,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0929.png",
+      "answer": "x^2 - y^2/3 = 1",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;l: Line;OneOf(Asymptote(C)) = l;Inclination(l) = pi/3;Distance(OneOf(Focus(C)), l) = sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 930,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0930.png",
+      "answer": "(1+sqrt(5))/2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;A: Point;B: Point;F1: Point;F2: Point;a>0;b>0;Focus(G) = {F1, F2};L:Line;IsPerpendicular(L,xAxis);Intersection(L,G)={A,B};PointOnCurve(F2, L);Radius(InscribedCircle(TriangleOf(A,B,F1)))=a",
+      "coords": {}
+    },
+    {
+      "index": 931,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0931.png",
+      "answer": "12",
+      "fact_expr": "G:Parabola;Expression(G) = (y^2 = 4*x);F:Point;Focus(G)=F;l:Line;A: Point;B: Point;Intersection(l,G)={A,B};M: Point;PointOnCurve(F, l);MidPoint(LineSegmentOf(A,B))=M;Distance(M,Directrix(G))=6",
+      "coords": {}
+    },
+    {
+      "index": 932,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0932.png",
+      "answer": "2\n(2/3)*pi",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 + y^2/2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 933,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0933.png",
+      "answer": "{sqrt(6)/2,sqrt(6)/3}",
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+      "answer": "(4, 2)",
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+      "answer": "(0,sqrt(6)/3]",
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+      "answer": "8/5",
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+      "answer": "4",
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+      "index": 945,
+      "success": true,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0945.png",
+      "answer": "1/8",
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+      "answer": "16",
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+      "success": false,
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+      "answer": "x^2 - y^2/6 = 1",
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+      "answer": "1",
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "answer": "9/4",
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+      "coords": {}
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+      "index": 952,
+      "success": true,
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+      "error": null,
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+        "direction": "right"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0952.png",
+      "answer": "2",
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+      "index": 953,
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0953.png",
+      "answer": "(9/4,3)",
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+          3.0
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+      "index": 954,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0954.png",
+      "answer": "{sqrt(2), sqrt(10)/3}",
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+      "index": 955,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 956,
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+        "y_coef": 3.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_0956.png",
+      "answer": "[\\sqrt{2}/2, 1)",
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+      "index": 957,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 958,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "up"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0958.png",
+      "answer": "9",
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+          8.0
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+    {
+      "index": 959,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 960,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "p": 1.0,
+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0960.png",
+      "answer": "pi/3",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;M: Point;Intersection(Directrix(G), xAxis) = M;N: Point;PointOnCurve(N, G);Abs(LineSegmentOf(M, N)) = 2*Abs(LineSegmentOf(N, F))",
+      "coords": {}
+    },
+    {
+      "index": 961,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0961.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;F: Point;H: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F;PointOnCurve(F,H);Slope(H) = 1;NumIntersection(H,G) = 1",
+      "coords": {}
+    },
+    {
+      "index": 962,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0962.png",
+      "answer": "-sqrt(15)/sqrt(15)",
+      "fact_expr": "Slope(l) = k;k: Number;k<0;l: Line;F: Point;Coordinate(F) = (0, 1);PointOnCurve(F, l) ;H: Curve;Expression(H) = (y = x^2/4)&(x>=0);G: Line;Expression(G) = (y = -1);Intersection(l, H) = A;Intersection(l, G) = B;B: Point;A: Point;Abs(LineSegmentOf(F, B)) = 5*Abs(LineSegmentOf(F, A))",
+      "coords": {
+        "F": [
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+          1.0
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+    {
+      "index": 963,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0963.png",
+      "answer": "2*sqrt(6)",
+      "fact_expr": "G: Parabola;H: Line;A: Point;M: Point;N: Point;Expression(G) = (y^2 = 2*x);Coordinate(A) = (1, 0);PointOnCurve(A, H);Inclination(H)=pi/4;Intersection(H,G)={M,N}",
+      "coords": {
+        "A": [
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+          0.0
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+    },
+    {
+      "index": 964,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0964.png",
+      "answer": "2",
+      "fact_expr": "H: Parabola;Expression(H) = (y^2 = -8*x);G: Hyperbola;Expression(G) = (-y^2/3 + x^2/m = 1);m: Number;PointOnCurve(RightFocus(G), Directrix(H))",
+      "coords": {}
+    },
+    {
+      "index": 965,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 966,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "answer": "sqrt(30)/3",
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+      "answer": "(0,pm*1)",
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+      "index": 990,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0990.png",
+      "answer": "sqrt(2)/2",
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+      "index": 991,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_0992.png",
+      "answer": "4*sqrt(7)",
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+      "index": 993,
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+      "conic_type": "parabola",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0993.png",
+      "answer": "3/2",
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+    {
+      "index": 994,
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+        "direction": "right"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0994.png",
+      "answer": "(1, -4)",
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+      "index": 995,
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_0995.png",
+      "answer": "8*sqrt(3)",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);M: Point;PointOnCurve(M, G);F: Point;Focus(G) = F;AngleOf(M, F, O) = ApplyUnit(120, degree);N: Point;Coordinate(N) = (-2, 0);O: Origin",
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+      "index": 997,
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+      "index": 999,
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+      "index": 1000,
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+      "index": 1001,
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+      "answer": "4",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1003.png",
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+        "b": 4.0,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1004.png",
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+      "index": 1005,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1006.png",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1007.png",
+      "answer": "sqrt(41)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1009.png",
+      "answer": "5*sqrt(2)/3",
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+    {
+      "index": 1010,
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+        "x_coef": 16.0,
+        "y_coef": 12.0
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1010.png",
+      "answer": "12/5",
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+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 1012,
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+      "index": 1013,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1014.png",
+      "answer": "sqrt(5)",
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+      "index": 1015,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1015.png",
+      "answer": "[-1,4]",
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+    {
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+      "index": 1018,
+      "success": true,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1018.png",
+      "answer": "pm*sqrt(2)",
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+      "coords": {}
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+      "index": 1019,
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+        "direction": "left"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1019.png",
+      "answer": "x=1",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = -4*x)",
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+      "index": 1020,
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+          4.0
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+      "index": 1021,
+      "success": true,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1021.png",
+      "answer": "sqrt(5)",
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+    {
+      "index": 1022,
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+        "a": 1.366584062576294,
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+      "answer": "x^2 - y^2 / 3 = 1",
+      "fact_expr": "C1: Hyperbola;C2: Parabola;M: Point;F: Point;Expression(C1)=(x^2/a^2 - y^2/b^2 = 1);Eccentricity(C1)=2;RightFocus(C1)=F;Focus(C2)=F;Center(C1)=Vertex(C2);Intersection(C1,C2)=M;Abs(LineSegmentOf(M, F)) = 5;a:Number;b:Number;a>0;b>0",
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+    {
+      "index": 1023,
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+      "index": 1024,
+      "success": false,
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+      "answer": "3",
+      "fact_expr": "G: Hyperbola;P: Point;F2: Point;F1: Point;e:Number;Expression(G) = (x^2 - y^2/3 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;Eccentricity(G)=e;PointOnCurve(P,G);AngleOf(P,F2,F1)=ApplyUnit(60,degree)",
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+    {
+      "index": 1026,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 1027,
+      "success": true,
+      "conic_type": "ellipse",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1027.png",
+      "answer": "x^2/4+y^2/3=1",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Length(MajorAxis(C)) = 4;FocalLength(C) = 2",
+      "coords": {}
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+    {
+      "index": 1028,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 1029,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 3.4641016151377544,
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1029.png",
+      "answer": "7",
+      "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 - y^2/12 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 3",
+      "coords": {}
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+    {
+      "index": 1030,
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+        "x_coef": 5.0,
+        "y_coef": 4.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1030.png",
+      "answer": "5/3",
+      "fact_expr": "G: Ellipse;H: Line;O: Origin;A: Point;B: Point;Expression(G) = (x^2/5 + y^2/4 = 1);PointOnCurve(RightFocus(G), H);Slope(H) = 2;Intersection(H, G) = {A, B}",
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+    {
+      "index": 1031,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "answer": "(\\sqrt{7}-2,1)",
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+      "answer": "3*x^2/4-y^2/4=1",
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+      "answer": "7/6",
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+      "answer": "x^2/4+y^2/3=1",
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+    {
+      "index": 1174,
+      "success": false,
+      "error": "Circle from diameter: missing point coordinates"
+    },
+    {
+      "index": 1175,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1175.png",
+      "answer": "2",
+      "fact_expr": "C: Parabola;Expression(C) = (x^2 = 2*y);P: Point;Q: Point;PointOnCurve(P, C);PointOnCurve(Q, C);Abs(LineSegmentOf(P, Q)) = 5",
+      "coords": {}
+    },
+    {
+      "index": 1176,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1176.png",
+      "answer": "sqrt(10)/2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;LeftFocus(G) = F1;RightFocus(G) = F2;F1: Point;F2: Point;H: Line;PointOnCurve(F2, H) = True;Intersection(H, RightPart(G)) = {A, B};B: Point;A: Point;DotProduct(VectorOf(B, F1), VectorOf(B, F2)) = 0;Sin(AngleOf(F1, A, F2)) = 3/5",
+      "coords": {}
+    },
+    {
+      "index": 1177,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.0,
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+        "x_coef": 4.0,
+        "y_coef": 1.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1177.png",
+      "answer": "1",
+      "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 + y^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(90, degree)",
+      "coords": {}
+    },
+    {
+      "index": 1178,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1178.png",
+      "answer": "3",
+      "fact_expr": "G: Parabola;P: Point;Q: Point;F: Point;M: Point;Expression(G) = (y^2 = 4*x);Focus(G)=F;PointOnCurve(P,G);PointOnCurve(Q,G);Negation(P=Q);lambda:Number;lambda>1;VectorOf(P, F) = lambda*VectorOf(F, Q);MidPoint(LineSegmentOf(P,Q))=M;Distance(M, Directrix(G)) = 8/3",
+      "coords": {}
+    },
+    {
+      "index": 1179,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1180,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1180.png",
+      "answer": "2*sqrt(3)/3",
+      "fact_expr": "l1: Line;l2:Line;l:Line;G: Hyperbola;b: Number;a: Number;F: Point;B: Point;A: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Asymptote(G)={l1,l2};RightFocus(G)=F;PointOnCurve(F,l);IsPerpendicular(l,l1);Intersection(l1,l)=A;Intersection(l2,l)=B;VectorOf(F, B) = 2*VectorOf(A, F)",
+      "coords": {}
+    },
+    {
+      "index": 1181,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1182,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1182.png",
+      "answer": "(3/2, 2*sqrt(3))",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);F: Point;Focus(G) = F;A: Point;Coordinate(A) = (2, 2*sqrt(3));P: Point;PointOnCurve(P, G);WhenMin(Abs(LineSegmentOf(P, A)) + Abs(LineSegmentOf(P, F)))",
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+          3.4641016151377544
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+    },
+    {
+      "index": 1183,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1184,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
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+          -0.0
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+        "radius": 5.0
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+      "optimization": {
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+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_1184.png",
+      "answer": "{(y^2 = 20*x)&(x > 0), (y = 0)&(x < 0)}",
+      "fact_expr": "C: Circle;Expression(C) = (-10*x + x^2 + y^2 = 0);H: Circle;P: Point;IsTangent(H, yAxis);IsOutTangent(C, H);Center(H) = P",
+      "coords": {}
+    },
+    {
+      "index": 1185,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1186,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "vertical"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1186.png",
+      "answer": "2",
+      "fact_expr": "E: Hyperbola;Expression(E) = (-x^2/b^2 + y^2/a^2 = 1);a: Number;b: Number;a>0;b>0;C: Circle;Expression(C) = (y^2 + (x - 2)^2 = 1);IsTangent(Asymptote(E),C) = True",
+      "coords": {}
+    },
+    {
+      "index": 1187,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1187.png",
+      "answer": "2",
+      "fact_expr": "C: Parabola;Expression(C) = (x^2 = 4*y);F: Point;Focus(C) = F;M: Point;Intersection(Directrix(C), yAxis) = M;A: Point;B: Point;l: Line;PointOnCurve(F, l);Intersection(l, C) = {A, B};AngleOf(A, M, B) = ApplyUnit(90, degree)",
+      "coords": {}
+    },
+    {
+      "index": 1188,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1188.png",
+      "answer": "(x-2)^2+y^2=16",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);F: Point;Focus(G) = F;l: Line;Directrix(G) = l;H: Circle;Center(H) = F;IsTangent(l, H)",
+      "coords": {}
+    },
+    {
+      "index": 1189,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "y",
+        "x_coef": 9.0,
+        "y_coef": 25.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1189.png",
+      "answer": "25",
+      "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/9 + y^2/25 = 1);PointOnCurve(P, G);F1: Point;F2: Point;Focus(G) = {F1, F2};m: Number;Distance(P, F1)*Distance(P, F2) = m",
+      "coords": {}
+    },
+    {
+      "index": 1190,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1190.png",
+      "answer": "6*sqrt(2)",
+      "fact_expr": "G: Parabola;H: Line;O: Origin;A: Point;B: Point;F: Point;Expression(G) = (x^2 = 8*y);PointOnCurve(F, H);Focus(G) = F;Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, F)) = 6",
+      "coords": {}
+    },
+    {
+      "index": 1191,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 1192,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1192.png",
+      "answer": "16",
+      "fact_expr": "G: Hyperbola;m: Real;Expression(G) = (x^2/9 - y^2/m = 1);FocalLength(G) = 10",
+      "coords": {}
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+    {
+      "index": 1193,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_focus_mismatch",
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1193.png",
+      "answer": "3",
+      "fact_expr": "G: Hyperbola;m: Number;F1: Point;A: Point;l:Line;Expression(G) = (-y^2/16 + x^2/m = 1);Coordinate(F1) = (5, 0);OneOf(Focus(G)) = F1;OneOf(Asymptote(G))=l;PointOnCurve(F1, LineSegmentOf(F1, A));IsPerpendicular(LineSegmentOf(F1, A),l);Intersection(LineSegmentOf(F1, A), l) = A;O:Origin",
+      "coords": {
+        "F1": [
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+          0.0
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+      },
+      "validation_reasons": [
+        "hyperbola_focus_mismatch"
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+    {
+      "index": 1194,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1194.png",
+      "answer": "pm*1",
+      "fact_expr": "G: Hyperbola;a: Number;H: Ellipse;Expression(G) = (-y^2/2 + x^2/a^2 = 1);Expression(H) = (x^2/4 + y^2/a^2 = 1);Focus(G)=Focus(H)",
+      "coords": {}
+    },
+    {
+      "index": 1195,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "left"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1195.png",
+      "answer": "(-1/4,1)",
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+      "coords": {
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+          1.0
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+    {
+      "index": 1196,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1196.png",
+      "answer": "3*sqrt(15)",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/25 + y^2/9 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C);Abs(LineSegmentOf(P, F1)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 1197,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 3.872983346207417,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1197.png",
+      "answer": "13",
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+    },
+    {
+      "index": 1198,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 1199,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_point_off_curve",
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+        "b": 1.4970015287399292,
+        "orientation": "horizontal"
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+        "converged": true,
+        "iterations": 2
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1199.png",
+      "answer": "2*sqrt(3)/3",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;A: Point;F1: Point;F2: Point;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (sqrt(15)/2, 1/2);PointOnCurve(A, C);LeftFocus(C) = F1;RightFocus(C) = F2;DotProduct(VectorOf(A, F1), VectorOf(A, F2)) = 0",
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+    {
+      "index": 1200,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1201,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1202,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1203,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1203.png",
+      "answer": "1/4",
+      "fact_expr": "G: Hyperbola;H: Parabola;a: Number;A: Point;B: Point;l:Line;Expression(G) = (x^2/4 - y^2 = 1);a>0;Expression(H) = (y = a*x^2);Directrix(H)=l;L1:Line;L2:Line;Asymptote(G)={L1,L2};Intersection(l,L1)=A;Intersection(l,L2)=B;Abs(LineSegmentOf(A, B)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 1204,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1204.png",
+      "answer": "2",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b = sqrt(3)*a",
+      "coords": {}
+    },
+    {
+      "index": 1205,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1205.png",
+      "answer": "x^2/4 + y^2/2 = 1",
+      "fact_expr": "E: Ellipse;b: Number;a: Number;G: Line;P: Point;F2: Point;Q: Point;F1: Point;O:Origin;a > b;b > 0;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(E) = F1;RightFocus(E) = F2;PointOnCurve(O,G);Intersection(G, E) = {P, Q};IsPerpendicular(LineSegmentOf(P, F2), LineSegmentOf(F2, Q));Area(TriangleOf(P,F,Q2))=a^2/2;Abs(LineSegmentOf(F2, Q)) + Abs(LineSegmentOf(P, F2)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 1206,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1206.png",
+      "answer": "3",
+      "fact_expr": "E: Parabola;A: Point;F: Point;m: Number;Expression(E) = (y^2 = 4*x);Coordinate(A) = (2, m);Focus(E) = F;PointOnCurve(A, E)",
+      "coords": {}
+    },
+    {
+      "index": 1207,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1208,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "answer": "16/3",
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+      "answer": "sqrt(5)/5",
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+      "index": 1216,
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+      "index": 1219,
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+      "answer": "{(6,6*sqrt(2)),(6,-6*sqrt(2))}",
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+      "success": false,
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+      "index": 1222,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1223.png",
+      "answer": "2/3",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1224.png",
+      "answer": "4*x+9*y-13=0",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1225.png",
+      "answer": "y^2 = 4*x",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1226.png",
+      "answer": "x^2/5 - y^2/20 = 1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;H: Parabola;Expression(H) = (y^2 = 2*(p*x));p: Number;p>0;Coordinate(Intersection(OneOf(Asymptote(G)), Directrix(H))) = (-4/3, 8/3);M: Point;x0: Number;Coordinate(M) = (x0, 4);OneOf(Intersection(G, H)) = M",
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+        "note": "using explicit params"
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+      "answer": "25",
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+    {
+      "index": 1228,
+      "success": true,
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+      "error": null,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1228.png",
+      "answer": "4*sqrt(2)/3",
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+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1229.png",
+      "answer": "sqrt(2)",
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+      "coords": {}
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+    {
+      "index": 1230,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1230.png",
+      "answer": "2",
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+      "index": 1231,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1231.png",
+      "answer": "sqrt(5)",
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+      "coords": {}
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+    {
+      "index": 1232,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1232.png",
+      "answer": "1/3",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;O: Origin;P: Point;F1: Point;F2: Point;G: Point;I: Point;a > b;b > 0;lambda:Number;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);PointOnCurve(G, LineSegmentOf(O, P));Abs(LineSegmentOf(O, P)) = 4*Abs(LineSegmentOf(O, G));Incenter(TriangleOf(P,F1,F2))=I;VectorOf(I, G) = lambda*VectorOf(F1, F2)",
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+    {
+      "index": 1233,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.4142135623730951,
+        "major_axis": "y",
+        "x_coef": 2.0,
+        "y_coef": 4.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1233.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/2 + y^2/4 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 1234,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1235,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.4142135623730951,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1235.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 2)",
+      "coords": {}
+    },
+    {
+      "index": 1236,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 6.164414002968976,
+        "major_axis": "x",
+        "x_coef": 63.0,
+        "y_coef": 38.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1236.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
+      "fact_expr": "G: Ellipse;Expression(G)=(x^2/63+y^2/38=1);H: Circle;Expression(H)=(x^2+y^2-6*x-8*y=0);C:Ellipse;FocalLength(G) = FocalLength(C);PointOnCurve(Center(H),C)",
+      "coords": {}
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+    {
+      "index": 1237,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 1238,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 4.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "answer": "sqrt(3)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1254.png",
+      "answer": "3/5",
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+      "answer": "x^2/3 - y^2/9 = 1",
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+      "answer": "x^2 - y^2 / 3 = 1",
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+      "answer": "(\\sqrt{3} + \\sqrt{7})/2",
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+      "index": 1261,
+      "success": false,
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+      "error": "Unsupported or unparseable expression"
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+      "index": 1263,
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+      "answer": "3/2",
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+      "answer": "-4",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1266.png",
+      "answer": "3+2*sqrt(2)",
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+      "error": "could not convert string to float: 'a'"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1268.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "G: Ellipse;Expression(G) = (2*x^2 + y^2 = 4)",
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+      "index": 1269,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1270.png",
+      "answer": "-4/9",
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+      "index": 1271,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1271.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "E: Ellipse;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(E) = F1;RightFocus(E) = F2;O: Origin;PointOnCurve(O, G);G: Line;Intersection(G, E) = {P, Q};P: Point;Q: Point;IsPerpendicular(LineSegmentOf(P, F2), LineSegmentOf(F2, Q));Area(TriangleOf(P, F2, Q)) = a^2/2;Abs(LineSegmentOf(F2, Q)) + Abs(LineSegmentOf(P, F2)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 1272,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "vertical"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1272.png",
+      "answer": "1+sqrt(2)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;A: Point;B: Point;F1: Point;F2: Point;l:Line;a>0;b>0;Expression(G) = (-x^2/b^2 + y^2/a^2 = 1);LowerFocus(G) = F1;PointOnCurve(F1,l);IsPerpendicular(l,yAxis);Intersection(l,G)={A,B};IsDiameter(LineSegmentOf(A,B),H);PointOnCurve(F2,H);UpperFocus(G)=F2",
+      "coords": {}
+    },
+    {
+      "index": 1273,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1274,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1274.png",
+      "answer": "[-3/4,-3/8]",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2/3 = 1);A1: Point;A2: Point;Endpoint(MajorAxis(C)) = {A1, A2};P: Point;PointOnCurve(P, C) = True;Range(Slope(LineOf(P, A1))) = [1, 2]",
+      "coords": {}
+    },
+    {
+      "index": 1275,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1275.png",
+      "answer": "1",
+      "fact_expr": "C: Parabola;p: Number;A: Point;B: Point;F: Point;D: Point;d: Number;p>0;Expression(C) = (y^2 = 2*(p*x));PointOnCurve(A, C);PointOnCurve(B, C);Focus(C) = F;AngleOf(A, F, B) = ApplyUnit(60, degree);MidPoint(LineSegmentOf(A, B)) = D;Distance(D, Directrix(C)) = d",
+      "coords": {}
+    },
+    {
+      "index": 1276,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1276.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;p: Number;Expression(G) = (y^2 = 2*(p*x));P: Point;XCoordinate(P) = 4;PointOnCurve(P, G);Distance(P, Focus(G)) = 5",
+      "coords": {}
+    },
+    {
+      "index": 1277,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.7320508075688772,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1277.png",
+      "answer": "sqrt(25)-2*sqrt(3)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2 = 1);P: Point;PointOnCurve(P, RightPart(G));F: Point;RightFocus(G) = F;A: Point;Coordinate(A) = (3, 1)",
+      "coords": {
+        "A": [
+          3.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 1278,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1278.png",
+      "answer": "[1/3,1)",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C);PointOnCurve(F2, PerpendicularBisector(LineSegmentOf(P, F1))) = True",
+      "coords": {}
+    },
+    {
+      "index": 1279,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 2.23606797749979,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1279.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "G: Hyperbola;H: Line;Expression(G) = (x^2/4 - y^2/5 = 1);Expression(H) = (sqrt(5)*x - 2*y = 0)",
+      "coords": {}
+    },
+    {
+      "index": 1280,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1280.png",
+      "answer": "2*sqrt(3)/3",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;M: Point;PointOnCurve(M,Asymptote(C)) ;O: Origin;Abs(LineSegmentOf(O, F2)) = Abs(LineSegmentOf(O, M));Abs(LineSegmentOf(M, F1)) - Abs(LineSegmentOf(M, F2)) = 2*sqrt(2)*a",
+      "coords": {}
+    },
+    {
+      "index": 1281,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1281.png",
+      "answer": "x^2 - y^2 / 3 = 1",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;P: Point;F: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 8*x);OneOf(Focus(G))=F;Focus(H)=F;OneOf(Intersection(G,H))=P;Abs(LineSegmentOf(P, F)) = 5",
+      "coords": {}
+    },
+    {
+      "index": 1282,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1282.png",
+      "answer": "10",
+      "fact_expr": "E: Hyperbola;P: Point;F1: Point;F2: Point;Expression(E) = (x^2/9 - y^2/16 = 1);LeftFocus(E) = F1;RightFocus(E) = F2;PointOnCurve(P, E);Abs(LineSegmentOf(P, F1)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 1283,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1284,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1285,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1286,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1286.png",
+      "answer": "4\n-16",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);l: Line;A: Point;B: Point;F: Point;Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B};x1: Number;x2: Number;y1: Number;y2: Number",
+      "coords": {}
+    },
+    {
+      "index": 1287,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1287.png",
+      "answer": "2*sqrt(6)",
+      "fact_expr": "l: Line;G: Parabola;H: Circle;m:Real;A: Point;B: Point;Expression(G) = (y^2 = 8*x);Expression(H) = ((x - 1)^2 + (y - 1)^2 = 6);Expression(l) = (-m + m*x - y + 1 = 0);PointOnCurve(Focus(G),l);Length(InterceptChord(l,H))=LineSegmentOf(A,B)",
+      "coords": {}
+    },
+    {
+      "index": 1288,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1288.png",
+      "answer": "2*a",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>b;b>0;M: Point;PointOnCurve(M, G);Negation(M=Endpoint(MinorAxis(G)));B1: Point;B2: Point;Endpoint(MinorAxis(G)) = {B1, B2};N: Point;K: Point;Intersection(LineOf(M, B1), xAxis) = N;Intersection(LineOf(M, B2), xAxis) = K;O: Origin",
+      "coords": {}
+    },
+    {
+      "index": 1289,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1289.png",
+      "answer": "ApplyUnit(30,degree)",
+      "fact_expr": "G: Parabola;N: Point;F: Point;M: Point;Expression(G) = (x^2 = 4*y);Focus(G) = F;Intersection(Directrix(G), yAxis) = M;PointOnCurve(N, G);Abs(LineSegmentOf(N, F)) = (sqrt(3)/2)*Abs(LineSegmentOf(M, N))",
+      "coords": {}
+    },
+    {
+      "index": 1290,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1291,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1291.png",
+      "answer": "0",
+      "fact_expr": "P: Point;PointOnCurve(P,C1) = True;C1: Ellipse;Expression(C1) = (x^2/4 + y^2/3 = 1);F: Point;LeftFocus(C1) = F;C2: Circle;Expression(C2) = ((x + 2)^2 + (y - 4)^2 = 1);Q: Point;PointOnCurve(Q,C2) = True",
+      "coords": {}
+    },
+    {
+      "index": 1292,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_point_off_curve",
+      "params": {
+        "a": 2.309742261840055,
+        "b": 1.632524068785969,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 4.273270740250148e-06,
+        "converged": true,
+        "iterations": 84
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1292.png",
+      "answer": "x^2/2-y^2=1",
+      "fact_expr": "G: Hyperbola;O: Origin;P: Point;SymmetryAxis(G)=axis;Expression(Asymptote(G)) = (y =pm*(sqrt(2)/2)*x);PointOnCurve(P, G);Coordinate(P)=(2,1);Center(G)=O",
+      "coords": {
+        "P": [
+          2.0,
+          1.0
+        ]
+      },
+      "validation_reasons": [
+        "hyperbola_point_off_curve"
+      ]
+    },
+    {
+      "index": 1293,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1294,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1294.png",
+      "answer": "-3/4",
+      "fact_expr": "C: Parabola;p: Number;A: Point;F: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Coordinate(A) = (-2, 3);PointOnCurve(A, Directrix(C));Focus(C) = F",
+      "coords": {
+        "A": [
+          -2.0,
+          3.0
+        ]
+      }
+    },
+    {
+      "index": 1295,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1295.png",
+      "answer": "8/3",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 8*x);F: Point;Focus(C) = F;l: Line;Directrix(C) = l;M: Point;PointOnCurve(M, l);Q: Point;OneOf(Intersection(LineOf(M, F), C)) = Q;VectorOf(F, M) = 3*VectorOf(F, Q)",
+      "coords": {}
+    },
+    {
+      "index": 1296,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1296.png",
+      "answer": "sqrt(5)/5",
+      "fact_expr": "l: Line;C: Parabola;A: Point;F: Point;B: Point;P: Point;M: Point;N: Point;k:Number;k>0;Expression(C) = (y^2 = 4*x);Coordinate(P) = (-1, 0);Focus(C) = F;PointOnCurve(P, l);Slope(l) = k;Intersection(l, C) = {A, B};Intersection(LineOf(A,F), C) = M;Intersection(LineOf(B,F), C) = N;Abs(LineSegmentOf(A, F))/Abs(LineSegmentOf(F, M)) + Abs(LineSegmentOf(B, F))/Abs(LineSegmentOf(F, N)) = 18",
+      "coords": {
+        "P": [
+          -1.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 1297,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1298,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1299,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1300,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1300.png",
+      "answer": "3/2",
+      "fact_expr": "G: Parabola;A: Point;B: Point;Expression(G) = (y^2 = 4*x);Length(LineSegmentOf(A,B)) = 5;PointOnCurve(Endpoint(LineSegmentOf(A, B)), G)",
+      "coords": {}
+    },
+    {
+      "index": 1301,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1301.png",
+      "answer": "pm*1",
+      "fact_expr": "G: Ellipse;H: Line;m: Real;A: Point;B: Point;Expression(G) = (x^2 + 2*y^2 = 2);Expression(H) = (y = m + x);Intersection(G, H) = {A, B};Abs(LineSegmentOf(A, B)) = (4*sqrt(2))/3",
+      "coords": {}
+    },
+    {
+      "index": 1302,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.1622776601683795,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 10.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1302.png",
+      "answer": "6*sqrt(2)",
+      "fact_expr": "H: Circle;Expression(H) = (x^2 + (y - 6)^2 = 2);G: Ellipse;Expression(G) = (x^2/10 + y^2 = 1);P: Point;Q: Point;PointOnCurve(P, H);PointOnCurve(Q, G)",
+      "coords": {}
+    },
+    {
+      "index": 1303,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1303.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;c: Number;Coordinate(F) = (c, 0);Focus(G) = F;A: Point;RightVertex(G) = A;Distance(A,OneOf(Asymptote(G))) = c/2",
+      "coords": {}
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+    {
+      "index": 1304,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1304.png",
+      "answer": "4*sqrt(3)",
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+      "index": 1305,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1306.png",
+      "answer": "(1,\\sqrt{2})",
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+      "coords": {}
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+    {
+      "index": 1307,
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+        "direction": "down"
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1307.png",
+      "answer": "3",
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+      "index": 1308,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 1309,
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+        "direction": "right"
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+      "answer": "4",
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+    {
+      "index": 1310,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 1311,
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1311.png",
+      "answer": "2*sqrt(2-lambda)",
+      "fact_expr": "G: Hyperbola;Expression(Z) = (x^2/2 + y^2/lambda = 1);Z: Curve;Z = G;lambda: Number",
+      "coords": {}
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+    {
+      "index": 1312,
+      "success": true,
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1312.png",
+      "answer": "sqrt(5)/3",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;A: Point;B: Point;PointOnCurve(A, C);PointOnCurve(B, C);VectorOf(A, F1) = 2*VectorOf(F1, B);DotProduct(VectorOf(A, F2), VectorOf(A,F1)) = 0",
+      "coords": {}
+    },
+    {
+      "index": 1313,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1313.png",
+      "answer": "x^2/5 - y^2/4 = 1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;C: Circle;Expression(C) = (-6*x + x^2 + y^2 + 5 = 0);IsTangent(Asymptote(G),C) = True;Center(C) = RightFocus(G)",
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+    {
+      "index": 1314,
+      "success": true,
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+        "b": 3.0,
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+        "x_coef": 25.0,
+        "y_coef": 9.0
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1314.png",
+      "answer": "4",
+      "fact_expr": "G: Ellipse;M: Point;F1: Point;O: Origin;N: Point;Expression(G) = (x^2/25 + y^2/9 = 1);LeftFocus(G)=F1;PointOnCurve(M,G);MidPoint(LineSegmentOf(M, F1)) = N;Abs(LineSegmentOf(O, N)) = 3",
+      "coords": {}
+    },
+    {
+      "index": 1315,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1315.png",
+      "answer": "9",
+      "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 - y^2/9 = 1);PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G) = F2;Abs(LineSegmentOf(P, F1)) = 5",
+      "coords": {}
+    },
+    {
+      "index": 1316,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1316.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;P: Point;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(90, degree)",
+      "coords": {}
+    },
+    {
+      "index": 1317,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1317.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "C: Circle;Gamma:Ellipse;Expression(Gamma) = (x^2/a^2+y^2/b^2=1);Expression(C) = ((x - 1)^2 + (y - 1)^2 = 2);a:Number;b:Number;a>b;b>0;F:Point;B:Point;RightFocus(Gamma)=F;UpperVertex(Gamma)=B;PointOnCurve(F,C);PointOnCurve(B,C)",
+      "coords": {}
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+    {
+      "index": 1318,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 1319,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1320,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 3.872983346207417,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1320.png",
+      "answer": "6",
+      "fact_expr": "G: Hyperbola;H: Circle;C:Circle;P: Point;M: Point;N: Point;Expression(G) = (x^2 - y^2/15 = 1);Expression(H) = (y^2 + (x + 4)^2 = 9);Expression(C)=(y^2 + (x - 4)^2 = 1);PointOnCurve(P, RightPart(G));PointOnCurve(M, H);PointOnCurve(N, C)",
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+    },
+    {
+      "index": 1321,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1321.png",
+      "answer": "(pm*2,1)",
+      "fact_expr": "G: Parabola;P: Point;Expression(G) = (x^2 = 4*y);PointOnCurve(P, G);Distance(P, Focus(G)) = 2",
+      "coords": {}
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+    {
+      "index": 1322,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1322.png",
+      "answer": "x^2/4+y^2/3=1",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;RightFocus(C)=F;Coordinate(F)=(1,0);B: Point;UpperVertex(C)=B;Slope(LineOf(F,B)) = -sqrt(3)",
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+          0.0
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+    {
+      "index": 1323,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1323.png",
+      "answer": "x^2/9+y^2/7=1",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;Focus(C) = {F1, F2};H: Line;PointOnCurve(F1, H);A: Point;B: Point;Intersection(H, C) = {A, B};Perimeter(TriangleOf(A, B, F2)) = 12;Eccentricity(C) = sqrt(2)/3",
+      "coords": {}
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+    {
+      "index": 1324,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 0.9220545291900635,
+        "b": 2.765991687774658,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 4.6528161874448415e-06,
+        "converged": true,
+        "iterations": 186
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1324.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;Expression(OneOf(Asymptote(G))) = (y = 3*x)",
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+    {
+      "index": 1325,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1326,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1326.png",
+      "answer": "2",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);Intersection(Directrix(C),xAxis)=M;A: Point;M: Point;B: Point;F:Point;Focus(C)=F;G:Line;Intersection(G, C) = {A, B};PointOnCurve(F, G);Slope(LineOf(M,A))=k1;Slope(LineOf(M,B))=k2;k1:Number;k2:Number",
+      "coords": {}
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+    {
+      "index": 1327,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1328,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1328.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;F: Point;A: Point;B: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F;IsPerpendicular(LineSegmentOf(F, A), LineSegmentOf(F, B));L1:Line;L2:Line;Asymptote(G)={L1,L2};Intersection(RightDirectrix(G),L1)=A;Intersection(RightDirectrix(G),L2)=B",
+      "coords": {}
+    },
+    {
+      "index": 1329,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1329.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;p: Number;P: Point;K: Point;F: Point;l: Line;p>0;y0:Number;Expression(G) = (x^2 = 2*p*y);Coordinate(P) = (4,y0);Focus(G) = F;Directrix(G) = l;PointOnCurve(P, G);Intersection(l,yAxis) = K;Abs(LineSegmentOf(P, K)) = sqrt(2)*Abs(LineSegmentOf(P, F))",
+      "coords": {}
+    },
+    {
+      "index": 1330,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1330.png",
+      "answer": "2",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;A: Point;F2: Point;B: Point;F1: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, G);Intersection(G,LeftPart(C))={A,B};Sin(AngleOf(A, B, F2)/2) = 1/4;Abs(LineSegmentOf(A, B)) = Abs(LineSegmentOf(B, F2))",
+      "coords": {}
+    },
+    {
+      "index": 1331,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1332,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1332.png",
+      "answer": "7/5",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;P: Point;F2: Point;Q: Point;F1: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, G);Intersection(G, LeftPart(C)) = {P, Q};Abs(LineSegmentOf(P, F2)) = 2*c;c:Number;c=sqrt(a^2+b^2);Area(TriangleOf(P,F1,F2))/Area(TriangleOf(Q,F1,F2)) = 2/3",
+      "coords": {}
+    },
+    {
+      "index": 1333,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1334,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 3.872983346207417,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1334.png",
+      "answer": "6",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/15 = 1);P: Point;PointOnCurve(P, G);F1: Point;F2: Point;OneOf(Focus(G)) = F1;OneOf(Focus(G)) = F2;Negation(F1=F2);Distance(P, F1) = 4",
+      "coords": {}
+    },
+    {
+      "index": 1335,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.8284271247461903,
+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 8.0,
+        "y_coef": 2.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
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+      "answer": "x=-1",
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+      "answer": "sqrt(3)/2",
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+      "answer": "sqrt(2)",
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+      "answer": "1/2",
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+      "answer": "-1/4",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1414.png",
+      "answer": "pm*1/2",
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+        "direction": "down"
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+      "answer": "20",
+      "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;Expression(G) = (x^2 = -4*y);Expression(l) = (2*x - y - 1 = 0);Intersection(l, G) = {A, B}",
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+      "success": false,
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+      "success": false,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1419.png",
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+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;Coordinate(F1) = (-3, 0);Coordinate(F2) = (3, 0);LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, Asymptote(C));Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))",
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+        "direction": "right"
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+      "answer": "[4,+oo)",
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+      "answer": "11/3",
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+      "answer": "y=pm*sqrt(6)*x/2",
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+      "answer": "x^2/4 - y^2/2 = 1",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "answer": "pi/2",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1464.png",
+      "answer": "{-1, 0}",
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+      "answer": "4*sqrt(2)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1466.png",
+      "answer": "4",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1467.png",
+      "answer": "4*sqrt(41)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1468.png",
+      "answer": "(2, +oo)",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1469.png",
+      "answer": "(\\sqrt{6}/3,\\sqrt{6})",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/2 - y^2/6 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;L: Line;PointOnCurve(F2, L) = True;Intersection(L, RightPart(G)) = {A, B};A: Point;B: Point;Quadrant(A) = 1;C: Circle;IsInTangent(C, TriangleOf(A, F1, F2)) = True;Radius(C) = r;r: Number",
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+      "index": 1470,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1470.png",
+      "answer": "15/16",
+      "fact_expr": "G: Parabola;M: Point;Expression(G) = (y = 4*x^2);PointOnCurve(M, G);Distance(M, Focus(G)) = 1",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1471.png",
+      "answer": "pm*sqrt(3)",
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+      "index": 1472,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1472.png",
+      "answer": "(-(a^2-b^2)/a, (a^2-b^2)/a)",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>b;b>0;A: Point;B: Point;PointOnCurve(A, G);PointOnCurve(B, G);x0: Number;P: Point;Coordinate(P) = (x0, 0);Intersection(PerpendicularBisector(LineSegmentOf(A, B)), xAxis) = P",
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+    {
+      "index": 1473,
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+        "orientation": "vertical"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1473.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/5 + y^2/4 = 1);H: Circle;Expression(H) = (-6*x + x^2 + y^2 + 8 = 0);C: Point;Center(H) = C",
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+      "index": 1474,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 1475,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+      "optimization": {
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+        "note": "using explicit params"
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+      "answer": "2*sqrt(2)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1523.png",
+      "answer": "3/2",
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+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
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+      "answer": "[sqrt(2)/2, 1]",
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+      "answer": "{(x^2-y^2/9=1),(y^2/9-x^2=1)}",
+      "fact_expr": "G: Hyperbola;Expression(Asymptote(G)) = (y = pm*3*x);FocalLength(G)=2*sqrt(10)",
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+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "answer": "sqrt(15)",
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+        "b": 3.0,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1530.png",
+      "answer": "26",
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+    {
+      "index": 1531,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 1532,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1532.png",
+      "answer": "(2, +oo)",
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+    {
+      "index": 1533,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.0,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1533.png",
+      "answer": "2*sqrt(3)",
+      "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2 - y^2 = 1);Focus(G)={F1,F2};PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2))",
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+    {
+      "index": 1534,
+      "success": true,
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+      "error": null,
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+        "direction": "up"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1534.png",
+      "answer": "pi/4",
+      "fact_expr": "G: Parabola;M: Point;Expression(G) = (y = x^2);Coordinate(M) = (1/2, 1/4);PointOnCurve(M,G)",
+      "coords": {
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+          0.25
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+    {
+      "index": 1535,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "p": 0.5,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1535.png",
+      "answer": "-3/4",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x);H: Line;PointOnCurve(Focus(G), H)  ;Intersection(G, H) = {A, B};O: Origin;A: Point;B: Point",
+      "coords": {}
+    },
+    {
+      "index": 1536,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1536.png",
+      "answer": "2",
+      "fact_expr": "C: Hyperbola;a: Number;b: Number;G: Circle;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y^2 + (x - 2)^2 = 4);Length(InterceptChord(OneOf(Asymptote(C)),G))=2",
+      "coords": {}
+    },
+    {
+      "index": 1537,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1538,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1538.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;G: Parabola;Expression(G) = (y = x^2 + 1);IsTangent(Asymptote(C), G)",
+      "coords": {}
+    },
+    {
+      "index": 1539,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 6.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 36.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1539.png",
+      "answer": "x + 4*y - 5 = 0",
+      "fact_expr": "G: Ellipse;H: LineSegment;A: Point;Expression(G) = (x^2/36 + y^2/9 = 1);Coordinate(A) = (1, 1);IsChordOf(H, G);MidPoint(H) = A",
+      "coords": {
+        "A": [
+          1.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 1540,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.8284271247461903,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 8.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1540.png",
+      "answer": "x^2/2 - y^2/2 = 1",
+      "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/8 + y^2/4 = 1);Eccentricity(G) = sqrt(2);Focus(G) = Focus(H)",
+      "coords": {}
+    },
+    {
+      "index": 1541,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1541.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;A: Point;PointOnCurve(A, C);IsPerpendicular(LineSegmentOf(A, F2), LineSegmentOf(F1, F2));Intersection(LineOf(A, F1), C) = {A, B};B: Point;VectorOf(F1, A) = 4*VectorOf(F1, B)",
+      "coords": {}
+    },
+    {
+      "index": 1542,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1543,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1543.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);P: Point;PointOnCurve(P, G);M: Point;Projection(P, yAxis) = M;A: Point;a: Number;Coordinate(A) = (4, a);Abs(a)>4",
+      "coords": {}
+    },
+    {
+      "index": 1544,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1544.png",
+      "answer": "pm*6",
+      "fact_expr": "C: Parabola;Expression(C) = (x^2 = 2*p*y);p: Number;p>0;F: Point;Focus(C) = F;M: Point;Coordinate(M) = (x0, 3);x0: Number;PointOnCurve(M, C) = True;Distance(M, F) = 6",
+      "coords": {}
+    },
+    {
+      "index": 1545,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1546,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1546.png",
+      "answer": "10",
+      "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;P: Point;F: Point;Expression(G) = (x^2 = 4*y);Coordinate(P) = (1, 4);Focus(G) = F;PointOnCurve(P, l);Intersection(l, G) = {A,B};MidPoint(LineSegmentOf(A, B)) = P",
+      "coords": {
+        "P": [
+          1.0,
+          4.0
+        ]
+      }
+    },
+    {
+      "index": 1547,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1547.png",
+      "answer": "2*sqrt(3)",
+      "fact_expr": "G: Parabola;H: Line;F: Point;N: Point;M: Point;D: Point;l: Line;Expression(G) = (y^2 = 4*x);Focus(G) = F;Directrix(G) = l;PointOnCurve(F, H);Slope(H) = sqrt(3);Intersection(H, G) = M;IsPerpendicular(LineSegmentOf(M, N),l);FootPoint(LineSegmentOf(M, N),l)=N;Intersection(LineOf(N,F), yAxis) = D;Quadrant(M)=1",
+      "coords": {}
+    },
+    {
+      "index": 1548,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1549,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1549.png",
+      "answer": "1/2",
+      "fact_expr": "G: Parabola;M: Point;Expression(G) = (y^2 = 2*x);PointOnCurve(M, G);Distance(M, Focus(G)) = 1",
+      "coords": {}
+    },
+    {
+      "index": 1550,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1550.png",
+      "answer": "1+sqrt(2)",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;F2: Point;F1: Point;a>0;b>0;LeftFocus(C)= F1;RightFocus(C)= F2;l:Line;Slope(l)=1;PointOnCurve(F1,l);P:Point;Intersection(l,RightPart(C))=P;AngleOf(F1, F2, P)=ApplyUnit(90, degree)",
+      "coords": {}
+    },
+    {
+      "index": 1551,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1551.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);M:Point;F1:Point;F2:Point;PointOnCurve(M,G);Focus(G)={F1,F2}",
+      "coords": {}
+    },
+    {
+      "index": 1552,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1553,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1554,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1554.png",
+      "answer": "4",
+      "fact_expr": "C: Parabola;p: Number;M: Point;N: Point;F: Point;O: Origin;p>0;Expression(C) = (y^2 = 2*(p*x));PointOnCurve(M, C);Focus(C) = F;L:Line;PointOnCurve(M,L);IsPerpendicular(L,Directrix(C));FootPoint(L,Directrix(C))=N;AngleOf(M, F, O) = ApplyUnit(120, degree);Perimeter(TriangleOf(M, N, F)) = 12",
+      "coords": {}
+    },
+    {
+      "index": 1555,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1555.png",
+      "answer": "-3",
+      "fact_expr": "G: Parabola;p: Number;H: Line;A: Point;F: Point;B: Point;P: Point;l: Line;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;Directrix(G) = l;PointOnCurve(F,H);Intersection(H,G)={A,B};Quadrant(B)=1;Intersection(H, l) = P;VectorOf(A, F) = VectorOf(F, B)/2;VectorOf(A, P) = lambda*VectorOf(A, F);lambda:Number",
+      "coords": {}
+    },
+    {
+      "index": 1556,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1556.png",
+      "answer": "sqrt(5)/5",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;F2: Point;A: Point;B: Point;C: Point;F1: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) =F1;RightFocus(G)=F2;PointOnCurve(F1, H);IsPerpendicular(H,xAxis);Intersection(H, G) = {A, B};Intersection(LineOf(A,F2),G)={A,C};Area(TriangleOf(A, B, C)) = 3*Area(TriangleOf(B, C, F2))",
+      "coords": {}
+    },
+    {
+      "index": 1557,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1558,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1558.png",
+      "answer": "2*sqrt(6)",
+      "fact_expr": "G: Parabola;p: Number;H: Circle;p>0;l:Line;Expression(G) = (x^2 = 2*p*y);Expression(H) = (-6*x + x^2 + y^2 - 1 = 0);Directrix(G)=l;Length(InterceptChord(l,H))=4",
+      "coords": {}
+    },
+    {
+      "index": 1559,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 6.0,
+        "b": 4.47213595499958,
+        "major_axis": "x",
+        "x_coef": 36.0,
+        "y_coef": 20.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1559.png",
+      "answer": "[-12,84]",
+      "fact_expr": "C: Ellipse;M: Circle;A: Point;B: Point;P: Point;Expression(C) = (x^2/36 + y^2/20 = 1);Expression(M) = (y^2 + x^2 - 8*x = 0);PointOnCurve(P, C);IsDiameter(LineSegmentOf(A, B), M)",
+      "coords": {}
+    },
+    {
+      "index": 1560,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 16.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1560.png",
+      "answer": "3+2*sqrt(2)",
+      "fact_expr": "C: Parabola;G: Line;A: Point;B: Point;F: Point;Expression(C) = (y^2 = 64*x);Focus(C)=F;PointOnCurve(F,G);Intersection(G,C)={A,B};Abs(LineSegmentOf(F, A)) > Abs(LineSegmentOf(F, B));Slope(G)=1",
+      "coords": {}
+    },
+    {
+      "index": 1561,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1561.png",
+      "answer": "2*x-y-15=0",
+      "fact_expr": "P: Point;Coordinate(P) = (8, 1);G: Hyperbola;Expression(G) = (x^2 - 4*y^2 = 4);H: LineSegment;IsChordOf(H, G);MidPoint(H) = P",
+      "coords": {
+        "P": [
+          8.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 1562,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 4.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1562.png",
+      "answer": "64",
+      "fact_expr": "l: Line;C: Parabola;A: Point;B: Point;O: Origin;F:Point;Expression(C) = (y^2 = 16*x);Inclination(l)=pi/6;Focus(C)=F;PointOnCurve(F,l);Intersection(l,C) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 1563,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1563.png",
+      "answer": "y^2=sqrt(2)*x",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(C) = F;l: Line;Directrix(C) = l;A: Point;PointOnCurve(A, C);G: Circle;Center(G) = F;Radius(G)=LineSegmentOf(F,A);B: Point;D: Point;Intersection(G, l) = {B, D};AngleOf(F, B, D) = ApplyUnit(30, degree);Area(TriangleOf(A, B, D)) = sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 1564,
+      "success": false,
+      "error": "Circle from diameter: cannot parse diameter pattern"
+    },
+    {
+      "index": 1565,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1565.png",
+      "answer": "4",
+      "fact_expr": "C: Parabola;l1:Line;l2:Line;A: Point;B: Point;D: Point;E: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F,l1);PointOnCurve(F,l2);IsPerpendicular(l1,l2);Intersection(l1, C) = {A, B};Intersection(l2, C) = {D, E}",
+      "coords": {}
+    },
+    {
+      "index": 1566,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 0.7071067811865476,
+        "b": 0.7071067811865476,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1566.png",
+      "answer": "x^2/2 + y^2 = 1",
+      "fact_expr": "G: Hyperbola;H: Ellipse;O: Origin;Expression(G) = (2*x^2 - 2*y^2 = 1);Center(H) = O;Focus(H) = Focus(G);InterReciprocal(Eccentricity(G), Eccentricity(H))",
+      "coords": {}
+    },
+    {
+      "index": 1567,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1567.png",
+      "answer": "y=pm*(3/2)*x",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;C1: Circle;Expression(C1) = (x^2 + y^2 - 4*x - 6*y = 0);PointOnCurve(Center(C1), OneOf(Asymptote(C)))",
+      "coords": {}
+    },
+    {
+      "index": 1568,
+      "success": false,
+      "error": "Circle from diameter: missing point coordinates"
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+    {
+      "index": 1569,
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+      "answer": "9",
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+    {
+      "index": 1570,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1570.png",
+      "answer": "3/2",
+      "fact_expr": "G: Parabola;M: Point;N: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G)=F;PointOnCurve(M,G);PointOnCurve(N,G);Negation(M=N);Abs(LineSegmentOf(M, F)) + Abs(LineSegmentOf(N, F)) = 5",
+      "coords": {}
+    },
+    {
+      "index": 1571,
+      "success": true,
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+        "b": 2.23606797749979,
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+        "x_coef": 9.0,
+        "y_coef": 5.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1571.png",
+      "answer": "x=-2",
+      "fact_expr": "G: Parabola;p: Number;H: Ellipse;p>0;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (x^2/9 + y^2/5 = 1);PointOnCurve(OneOf(Focus(H)), Directrix(G))",
+      "coords": {}
+    },
+    {
+      "index": 1572,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 2.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1572.png",
+      "answer": "x+2*y-3=0",
+      "fact_expr": "G: Ellipse;H: LineSegment;Expression(G) = (x^2/4 + y^2/2 = 1);Coordinate(MidPoint(H)) = (1, 1);IsChordOf(H,G)",
+      "coords": {}
+    },
+    {
+      "index": 1573,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1573.png",
+      "answer": "((2*x-1)^2)/2+y^2=1",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/2 + y^2 = 1);P:Point;PointOnCurve(P,G)",
+      "coords": {}
+    },
+    {
+      "index": 1574,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1574.png",
+      "answer": "7",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2 + x^2/a^2 = 1);a: Number;a>0;Eccentricity(C) = sqrt(3)/2;F1: Point;F2: Point;Focus(C) = {F1, F2};l: Line;PointOnCurve(F1, l) = True;Intersection(l, C) = {A, B};A: Point;B: Point",
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+    {
+      "index": 1575,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1575.png",
+      "answer": "sqrt(3)/3",
+      "fact_expr": "F: Point;G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;RightFocus(G) = F;A: Point;OneOf(Endpoint(MinorAxis(G))) = A;Intersection(LineOf(A, F), G) = {A,B};B: Point;VectorOf(A, F) = 2*VectorOf(F, B)",
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+    {
+      "index": 1576,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1577,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1578,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1579,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 2.23606797749979,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1579.png",
+      "answer": "(6*sqrt(5)/5,2)",
+      "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 - y^2/5 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P,G);Quadrant(P)=1;Area(TriangleOf(P, F1, F2)) = 6",
+      "coords": {}
+    },
+    {
+      "index": 1580,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 2.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1580.png",
+      "answer": "{(-oo,-2*sqrt(3)/3),(2*sqrt(3)/3,+oo)}",
+      "fact_expr": "G: Hyperbola;H: Line;k:Number;Expression(G) = (x^2 - y^2 = 4);Expression(H) = (y=k*(x-1));NumIntersection(G,H)=0",
+      "coords": {}
+    },
+    {
+      "index": 1581,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1581.png",
+      "answer": "x^2/3+y^2/2=1",
+      "fact_expr": "l: Line;C: Ellipse;b: Number;a: Number;F1: Point;B: Point;F2: Point;A: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;Eccentricity(C)=sqrt(3)/3;PointOnCurve(F2, l);Intersection(l,C)={A,B};Perimeter(TriangleOf(A,F1,B)) = 4*sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 1582,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1582.png",
+      "answer": "(0,1/4)",
+      "fact_expr": "G: Parabola;Expression(G) = (y = x^2)",
+      "coords": {}
+    },
+    {
+      "index": 1583,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1583.png",
+      "answer": "2",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;G: Circle;Expression(G) = (x^2 + y^2 = a^2);L1: Line;L2: Line;A: Point;B: Point;TangentOfPoint(OneOf(Focus(C)), G) = {L1, L2};TangentPoint(L1, G) = A;TangentPoint(L2, G) = B;O: Origin;AngleOf(A, O, B) = ApplyUnit(120, degree)",
+      "coords": {}
+    },
+    {
+      "index": 1584,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1585,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1585.png",
+      "answer": "sqrt(5)/2",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2/4 - y^2 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 1586,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1586.png",
+      "answer": "x^2 - y^2/3 = 1",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;A: Point;O: Origin;Z: Line;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(RightVertex(C), Z);IsPerpendicular(Z, xAxis);Intersection(Z, OneOf(Asymptote(C))) = A;Center(G) = RightFocus(C);Radius(G) = 2;PointOnCurve(A, G);PointOnCurve(O, G)",
+      "coords": {}
+    },
+    {
+      "index": 1587,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1587.png",
+      "answer": "pm*sqrt(3)",
+      "fact_expr": "l: Line;G: Parabola;A: Point;F: Point;B: Point;C: Point;Expression(G) = (y^2 = 4*x);Focus(G)=F;PointOnCurve(F, l);Intersection(l,G)={A,B};Intersection(l,Directrix(G))=C;VectorOf(A, F) = (3/2)*VectorOf(B, C)",
+      "coords": {}
+    },
+    {
+      "index": 1588,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1588.png",
+      "answer": "2*sqrt(15)",
+      "fact_expr": "G: Parabola;H: Line;k: Number;A: Point;B: Point;Expression(G) = (y^2 = 8*x);Expression(H) = (y = k*x - 2);Intersection(H, G) = {A, B};XCoordinate(MidPoint(LineSegmentOf(A,B))) = 2",
+      "coords": {}
+    },
+    {
+      "index": 1589,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1589.png",
+      "answer": "2",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(C) = F;A: Point;B: Point;PointOnCurve(F,LineSegmentOf(A,B)) = True;IsChordOf(LineSegmentOf(A,B),C) = True;VectorOf(F,A) + VectorOf(F,B) = 0;G: Circle;IsDiameter(LineSegmentOf(A,B),G) = True;PointOnCurve(LeftVertex(C),G) = True",
+      "coords": {}
+    },
+    {
+      "index": 1590,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 4.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1590.png",
+      "answer": "sqrt(15)",
+      "fact_expr": "G: Hyperbola;a: Number;a>0;H:Parabola;Expression(G) = (-y^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 16*x);PointOnCurve(LeftFocus(G),Directrix(H))",
+      "coords": {}
+    },
+    {
+      "index": 1591,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1591.png",
+      "answer": "8",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;M: Point;O: Origin;Expression(G) = (x^2/25 + y^2/16 = 1);LeftFocus(G) = F1;PointOnCurve(P, G);Negation(P=Vertex(G));MidPoint(LineSegmentOf(P, F1)) = M",
+      "coords": {}
+    },
+    {
+      "index": 1592,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1593,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1593.png",
+      "answer": "6",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 8*x);F: Point;M: Point;N: Point;Focus(C) = F;PointOnCurve(M, C);Intersection(OverlappingLine(LineSegmentOf(F, M)), yAxis) = N;MidPoint(LineSegmentOf(F, N)) = M",
+      "coords": {}
+    },
+    {
+      "index": 1594,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.4641016151377544,
+        "b": 2.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1594.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/12 - y^2/4 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 1595,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.8284271247461903,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 8.0,
+        "y_coef": 4.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1595.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;O: Origin;Vertex(G) = O;H: Ellipse;Expression(H) = (x^2 + 2*y^2 = 8);Focus(G) = OneOf(Focus(H))",
+      "coords": {}
+    },
+    {
+      "index": 1596,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1596.png",
+      "answer": "4*sqrt(3)/3",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;L: Line;PointOnCurve(F, L);Inclination(L) = ApplyUnit(60, degree);Intersection(L, G) = {A, B};A: Point;B: Point;C1: Circle;IsDiameter(LineSegmentOf(A, F), C1) = True;C2: Circle;IsDiameter(LineSegmentOf(B, F), C2) = True;TangentPoint(C1, yAxis) = M;TangentPoint(C2, yAxis) = N;M: Point;N: Point",
+      "coords": {}
+    },
+    {
+      "index": 1597,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1597.png",
+      "answer": "[\\sqrt{2}-1, 1)",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;e: Number;Eccentricity(G) = e;P: Point;PointOnCurve(P, G);LineSegmentOf(P, F1)/LineSegmentOf(P, F2) = e",
+      "coords": {}
+    },
+    {
+      "index": 1598,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
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+      "answer": "24",
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+      "answer": "5*x^2-(5/4)*y^2=1",
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+      "answer": "2*sqrt(2)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1626.png",
+      "answer": "2*x-y+4=0",
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+      "answer": "14",
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+      "index": 1634,
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+      "answer": "4*x - y - 15 = 0",
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+      "answer": "y=pm*x\nsqrt(2)",
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+      "answer": "sqrt(2)",
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+      "answer": "sqrt(3)",
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+      "answer": "2",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1650.png",
+      "answer": "x^2/4-y^2=1",
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+      "answer": "(x-2)^2+y^2=10",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1652.png",
+      "answer": "{sqrt(13)/2,sqrt(13)/3}",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1653.png",
+      "answer": "sqrt(3)-1",
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+      "index": 1654,
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+      "answer": "16",
+      "fact_expr": "G: Parabola;C: Ellipse;A: Point;B: Point;O:Origin;Expression(C) = (x^2/4 + y^2/3 = 1);Vertex(G)=O;LeftDirectrix(C)=Directrix(G);Intersection(RightDirectrix(C),G)={A,B}",
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+      "index": 1655,
+      "success": false,
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+      "index": 1656,
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+      "answer": "y^2 = 4*x",
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+    {
+      "index": 1657,
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+      "answer": "x = -1/2\n2",
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+        "note": "using explicit params"
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+      "answer": "12+8*sqrt(2)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1685.png",
+      "answer": "{21/8, 51/8}",
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+      "index": 1686,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1687.png",
+      "answer": "4",
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+        "note": "using explicit params"
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+      "answer": "3/5",
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+      "success": true,
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+        "b": 1.5,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1689.png",
+      "answer": "sqrt(3)",
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+      "index": 1690,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_focus_mismatch",
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+        "b": 1.7320508075688772,
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+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1690.png",
+      "answer": "x^2/6+y^2/2=1",
+      "fact_expr": "F: Point;RightFocus(C) = F;Coordinate(F) = (2, 0);C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;l: Line;m: Number;Expression(l) = (y = m - x/3);A: Point;B: Point;Intersection(l, C) = {A, B};P: Point;MidPoint(LineSegmentOf(A, B)) = P;O: Origin;k: Number;Slope(LineOf(O, P)) = k;k = 1",
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+          0.0
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+      "index": 1691,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1692,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1693,
+      "success": true,
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+        "b": 1.7320508075688772,
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+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1693.png",
+      "answer": "7",
+      "fact_expr": "G: Parabola;H: Ellipse;m: Number;Expression(G) = (y^2 = 8*x);Expression(H) = (y^2/3 + x^2/m = 1);PointOnCurve(OneOf(Focus(H)), Directrix(G))",
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+      "index": 1694,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1695,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1696,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1696.png",
+      "answer": "(-\\infty,-1)",
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+    {
+      "index": 1697,
+      "success": true,
+      "conic_type": "parabola",
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+      "params": {
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1697.png",
+      "answer": "x=-4",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 16*x)",
+      "coords": {}
+    },
+    {
+      "index": 1698,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 1.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1698.png",
+      "answer": "4",
+      "fact_expr": "H: Parabola;Expression(H) = (y^2 = 2*(p*x));p: Real;G: Hyperbola;Expression(G) = (x^2/3 - y^2 = 1);Focus(H) = RightFocus(G)",
+      "coords": {}
+    },
+    {
+      "index": 1699,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1699.png",
+      "answer": "{0, -1, 1/2}",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);l: Line;M: Point;Coordinate(M) = (-2, 1);PointOnCurve(M, l);k: Number;Slope(l) = k;NumIntersection(l, G) = 1",
+      "coords": {
+        "M": [
+          -2.0,
+          1.0
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+      }
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+    {
+      "index": 1700,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1701,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 3.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1701.png",
+      "answer": "18",
+      "fact_expr": "F: Point;G: Parabola;Expression(G) = (y^2 = 12*x);Focus(G) = F;A: Point;B: Point;C: Point;PointOnCurve(A, G) = True;PointOnCurve(B, G) = True;PointOnCurve(C, G) = True;VectorOf(F, A) + VectorOf(F, B) + VectorOf(F, C) = 0",
+      "coords": {}
+    },
+    {
+      "index": 1702,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1703,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1703.png",
+      "answer": "2",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);G: Circle;P: Point;Q: Point;F: Point;l:Line;Focus(C) = F;Negation(P=Q);PointOnCurve(P,l);TangentPoint(l,C)=Q;IsDiameter(LineSegmentOf(P,Q),G);PointOnCurve(F,G)",
+      "coords": {}
+    },
+    {
+      "index": 1704,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
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+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1704.png",
+      "answer": "2*sqrt(5)/5",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;a > b;b > 0;Expression(G) = (x^2/b^2 + y^2/a^2 = 1);Expression(H) = (x - 2*y + 2 = 0);PointOnCurve(OneOf(Focus(G)),H);PointOnCurve(OneOf(Vertex(G)),H)",
+      "coords": {}
+    },
+    {
+      "index": 1705,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1706,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 3.0,
+        "direction": "left"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1706.png",
+      "answer": "(-4,4*\\sqrt{3})",
+      "fact_expr": "G: Parabola;M: Point;F: Point;Expression(G) = (y^2 = -12*x);Focus(G) = F;Abs(LineSegmentOf(M, F)) = 7;PointOnCurve(M,G);Quadrant(M)=2",
+      "coords": {}
+    },
+    {
+      "index": 1707,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1707.png",
+      "answer": "1/3",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, G) = {A, B};C: Point;Intersection(OverlappingLine(LineSegmentOf(F, B)), Directrix(G)) = C;Abs(LineSegmentOf(B, C)) = 2*Abs(LineSegmentOf(B, F))",
+      "coords": {}
+    },
+    {
+      "index": 1708,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1708.png",
+      "answer": "1",
+      "fact_expr": "l: Line;Expression(l) = (y = x - 1);G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;A: Point;B: Point;p>0;Intersection(l, G) = {A, B};XCoordinate(MidPoint(LineSegmentOf(A, B))) = 2;IsChordOf(LineSegmentOf(A, B), G)",
+      "coords": {}
+    },
+    {
+      "index": 1709,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1709.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(A, G);PointOnCurve(B,G);VectorOf(A, F) = 3*VectorOf(F, B);IsChordOf(LineSegmentOf(A, B), G)",
+      "coords": {}
+    },
+    {
+      "index": 1710,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1710.png",
+      "answer": "x^2/4+y^2=1",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(OneOf(Focus(C))) = (sqrt(3), 0);Eccentricity(C)=sqrt(3)/2;a: Number;b: Number",
+      "coords": {}
+    },
+    {
+      "index": 1711,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1712,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1712.png",
+      "answer": "18",
+      "fact_expr": "G: Parabola;p: Number;A: Point;B:Point;F: Point;p>0;y1:Number;y2:Number;Expression(G) = (y^2 = 2*(p*x));Coordinate(A) = (2, y1);Coordinate(B) = (1/2, y2);Focus(G) = F;PointOnCurve(A,G);PointOnCurve(B,G);Quadrant(A)=1;Quadrant(B)=4;Abs(LineSegmentOf(A, F)) = 20",
+      "coords": {}
+    },
+    {
+      "index": 1713,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1713.png",
+      "answer": "y=pm*(sqrt(2)/2)*x",
+      "fact_expr": "H: Ellipse;Expression(H) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;G: Hyperbola;Expression(G) = (-y^2/n^2 + x^2/m^2 = 1);m: Number;n: Number;m>0;n>0;F1: Point;F2: Point;Focus(H) = {F1, F2};Focus(G) = {F1, F2};P: Point;OneOf(Intersection(H, G)) = P;AngleOf(F1, P, F2) = (2/3)*pi;WhenMin(Eccentricity(H)*Eccentricity(G))",
+      "coords": {}
+    },
+    {
+      "index": 1714,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1715,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1716,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1717,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 2.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1717.png",
+      "answer": "1/2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/4 = 1);F: Point;RightFocus(G) = F;H: Line;PointOnCurve(F, H);IsParallel(H, OneOf(Asymptote(G))) ;Intersection(H, G) = P;P: Point;M: Point;PointOnCurve(M, LineOf(P, F)) ;DotProduct(VectorOf(O, M), VectorOf(P, F)) = 0;O: Origin",
+      "coords": {}
+    },
+    {
+      "index": 1718,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1718.png",
+      "answer": "y=pm*x/2",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;RightFocus(C) = F2;F2: Point;L: Line;PointOnCurve(F2, L) = True;IsPerpendicular(L, OneOf(Asymptote(C))) = True;FootPoint(L, OneOf(Asymptote(C))) = P;P: Point;Intersection(L, C) = A;A: Point;VectorOf(F2, P) = 3*VectorOf(F2, A)",
+      "coords": {}
+    },
+    {
+      "index": 1719,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1719.png",
+      "answer": "2",
+      "fact_expr": "C: Parabola;p: Number;G: Line;M: Point;A: Point;B: Point;l: Line;p>0;Expression(C) = (y^2 = 2*(p*x));Coordinate(M) = (1, 0);Directrix(C) = l;PointOnCurve(M, G);Slope(G) = sqrt(3);Intersection(G, l) = A;OneOf(Intersection(G, C)) = B;VectorOf(A, M) = VectorOf(M, B)",
+      "coords": {
+        "M": [
+          1.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 1720,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1720.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F: Point;A: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftVertex(G)=A;RightFocus(G)=F;PointOnCurve(P,RightPart(G));AngleOf(P,F,A)=2*AngleOf(P,A,F)",
+      "coords": {}
+    },
+    {
+      "index": 1721,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1721.png",
+      "answer": "[\\sqrt{2} - 1, 1)",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;c: Number;F1: Point;F2: Point;P: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);c*VectorOf(P, F2) = a*VectorOf(P, F1)",
+      "coords": {}
+    },
+    {
+      "index": 1722,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1723,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.0625,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1723.png",
+      "answer": "1/16",
+      "fact_expr": "G: Parabola;Expression(G) = (y = 4*x^2);P: Point;PointOnCurve(P, G)",
+      "coords": {}
+    },
+    {
+      "index": 1724,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1724.png",
+      "answer": "(\\sqrt{2}/2,1)",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);D:Circle;b: Number;a: Number;F: Point;a > b;b > 0;c:Number;Coordinate(F) = (-c, 0);LeftFocus(G) = F;Center(D)=F;Radius(D)=c;NumIntersection(H,LeftDirectrix(G))=2",
+      "coords": {}
+    },
+    {
+      "index": 1725,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "note": "using explicit params"
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+      "answer": "sqrt(6)/2",
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+      "index": 1726,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "answer": "(-2, -1)",
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+      "answer": "10",
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+      "answer": "y^2=2*(x-1)",
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+      "answer": "sqrt(3)+1",
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+      "answer": "(-\\infty,-4)+(-4,0)",
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+      "answer": "4",
+      "fact_expr": "G: Hyperbola;lambda: Number;Expression(G) = (x^2 - y^2/lambda = 1);lambda>0;Expression(OneOf(Asymptote(G))) = (y = 2*x)",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1737.png",
+      "answer": "sqrt(5)/2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;P: Point;A: Point;B: Point;a>0;b>0;Expression(G) = (-x^2/b^2 + y^2/a^2 = 1);PointOnCurve(P,G);PointOnCurve(P,H);IsParallel(xAxis,H);DotProduct(VectorOf(P, A), VectorOf(P, B)) = -a^2/4;L1:Line;L2:Line;Asymptote(G)={L1,L2};Intersection(H,L1)=A;Intersection(H,L2)=B",
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+        "direction": "right"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1738.png",
+      "answer": "3",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1739.png",
+      "answer": "sqrt(65)/5",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;F1: Point;A: Point;F2: Point;B: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) =F1;RightFocus(G)=F2;PointOnCurve(A, RightPart(G));Intersection(LineSegmentOf(A,F1),LeftPart(G))=B;IsPerpendicular(LineSegmentOf(A, F2), LineSegmentOf(B, F2));Abs(LineSegmentOf(B, F1)) = Abs(LineSegmentOf(A, F2))/3",
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+    {
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+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1740.png",
+      "answer": "sqrt(15)/3",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;M: Point;F2: Point;N: Point;F1: Point;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;L:Line;PointOnCurve(F1, L);IsPerpendicular(L,xAxis);Intersection(L, C) = {M, N};Cos(AngleOf(M, F2, N)) = 7/8",
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+    {
+      "index": 1741,
+      "success": true,
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+      "error": null,
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1741.png",
+      "answer": "x^2/16+y^2/4=1",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = sqrt(3)/2;Length(MinorAxis(G))=4",
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+        "y_coef": 2.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1742.png",
+      "answer": "4*sqrt(3)/3",
+      "fact_expr": "G: Line;H: Curve;Expression(G) = (y = x);Expression(H) = (2*x^2 + y^2 = 2)",
+      "coords": {}
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+      "index": 1743,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "success": true,
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+        "x_coef": 2.0,
+        "y_coef": 1.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1744.png",
+      "answer": "2*sqrt(3)",
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+        "direction": "right"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1745.png",
+      "answer": "[10,+\\infty)",
+      "fact_expr": "C: Parabola;p: Real;G: Line;A: Point;M: Point;B: Point;p>0;Expression(C) = (y^2 = 2*p*x);Expression(G) = (3*x + 4*y + 25 = 0);Focus(C) = A;Intersection(Directrix(C), xAxis) = B;PointOnCurve(M, G);AngleOf(A, M, B) = ApplyUnit(90, degree)",
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+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1746.png",
+      "answer": "(1, \\sqrt{5}]",
+      "fact_expr": "H: Line;Expression(H) = (y = 2*x);G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;NumIntersection(H, G) = 0",
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+    {
+      "index": 1747,
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+        "direction": "up"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1747.png",
+      "answer": "9",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1748.png",
+      "answer": "sqrt(5)",
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+    {
+      "index": 1749,
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1749.png",
+      "answer": "3",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 8*x);F: Point;Focus(C) = F;l: Line;Directrix(C) = l;P: Point;PointOnCurve(P, l);Q: Point;OneOf(Intersection(LineOf(P, F), C)) = Q;O: Origin;VectorOf(F, P) = 4*VectorOf(F, Q)",
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+    {
+      "index": 1750,
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1750.png",
+      "answer": "3",
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+    {
+      "index": 1751,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 1752,
+      "success": true,
+      "conic_type": "ellipse",
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+        "y_coef": 3.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1752.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "E: Ellipse;b: Number;a: Number;G: Line;A: Point;F1: Point;B: Point;F2: Point;a > b;b > 0;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(E) = F1;RightFocus(E) = F2;PointOnCurve(F1, G);Intersection(G, E) = {A, B};Abs(LineSegmentOf(A, F1)) = 3*Abs(LineSegmentOf(B, F1));Cos(AngleOf(A, F2, B)) = 3/5",
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+    {
+      "index": 1753,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 2.364173412322998,
+        "orientation": "horizontal"
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+        "iterations": 89
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1753.png",
+      "answer": "1",
+      "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (-y^2/3 + x^2/m = 1);Eccentricity(G) = 2",
+      "coords": {}
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+    {
+      "index": 1754,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1754.png",
+      "answer": "7/2",
+      "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;Expression(G) = (y^2 = 2*x);Coordinate(A) = (3, 2);Focus(G) = F;PointOnCurve(P, G)",
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+          2.0
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+    {
+      "index": 1755,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "up"
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+        "note": "using explicit params"
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+      "answer": "(0,pm*3)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1760.png",
+      "answer": "sqrt(3) + sqrt(6)",
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+      "answer": "x=-2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x)",
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+    {
+      "index": 1762,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1762.png",
+      "answer": "(\\sqrt{5}/5, 3/5)",
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+      "answer": "sqrt(10)/2",
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+      "answer": "sqrt(2)",
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+      "answer": "9",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1768.png",
+      "answer": "sqrt(5)/3",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1769.png",
+      "answer": "sqrt(3)+1",
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+      "index": 1770,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "success": true,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1771.png",
+      "answer": "y = pm*sqrt(2)/2*x",
+      "fact_expr": "C: Hyperbola;a: Number;b: Number;a > 0;b > 0;Expression(C) = (x^2/a^2 - y^2/b^2 = 1);A: Point;B: Point;PointOnCurve(A, C);PointOnCurve(B, C);VectorOf(A, O) = 1/2*VectorOf(A, B);O: Origin;M: Point;PointOnCurve(M, C);Negation(M = A);Negation(M = B);Slope(LineOf(M, A)) * Slope(LineOf(M, B)) = 1/2",
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+      "index": 1772,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 1773,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1773.png",
+      "answer": "9",
+      "fact_expr": "l: Line;G: Hyperbola;Expression(G) = (-y^2/7 + x^2/m = 1);m: Number;A: Point;B: Point;F2: Point;F1: Point;LeftFocus(G)=F1;PointOnCurve(F1,l);Intersection(l,LeftPart(G))={A,B};Abs(LineSegmentOf(A, B)) = 4;Perimeter(TriangleOf(A, B, F2)) = 20;RightFocus(G)=F2",
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+    {
+      "index": 1774,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 2.23606797749979,
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+        "note": "using explicit params"
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+      "answer": "4",
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+      "index": 1775,
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+        "y_coef": 4.0
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+        "note": "using explicit params"
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+      "answer": "x^2+3*y^2/2=1",
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+      "index": 1776,
+      "success": true,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1776.png",
+      "answer": "2",
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+      "index": 1777,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1778.png",
+      "answer": "24/7",
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+      "index": 1780,
+      "success": true,
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+        "direction": "left"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1780.png",
+      "answer": "6*sqrt(2)",
+      "fact_expr": "G: Parabola;H: Line;P: Point;d1:Number;d2:Number;Expression(G) = (y^2 = -8*x);Expression(H) = (x + y - 10 = 0);PointOnCurve(P, G);Distance(P,Directrix(G)) = d1;Distance(P,H)=d2",
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+      "index": 1781,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1781.png",
+      "answer": "(0,4)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/m = 1);PointOnCurve(Focus(G), xAxis);m: Real",
+      "coords": {}
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+      "index": 1782,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1782.png",
+      "answer": "sqrt(3)+1",
+      "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;A: Point;F: Point;B: Point;O:Origin;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;PointOnCurve(O,l);Inclination(l)=pi/3;Intersection(l,LeftPart(C))=A;Intersection(l,RightPart(C))=B;DotProduct(VectorOf(A, F), VectorOf(B, F)) = 0",
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+    {
+      "index": 1783,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1783.png",
+      "answer": "y+pm*sqrt(3)*x=0",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;LeftFocus(G) = F;A: Point;RightVertex(G) = A;H: Line;Expression(H) = (x = a);Intersection(H, OneOf(Asymptote(G))) = B;B: Point;AngleOf(B, F, A) = ApplyUnit(30, degree)",
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+    {
+      "index": 1784,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1784.png",
+      "answer": "5/3",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;Distance(RightFocus(G), Asymptote(G)) = (1/2)*Distance(RightFocus(G), LeftVertex(G));e: Number;Eccentricity(G) = e",
+      "coords": {}
+    },
+    {
+      "index": 1785,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1785.png",
+      "answer": "8/13",
+      "fact_expr": "l: Line;G: Parabola;Expression(G) = (y^2 = 4*x);Expression(l)=(5*x+12*y+3=0)",
+      "coords": {}
+    },
+    {
+      "index": 1786,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1786.png",
+      "answer": "sqrt(17)/2",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1788.png",
+      "answer": "sqrt(2)/2",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1789.png",
+      "answer": "7/6",
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+      "index": 1790,
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+      "error": "Unsupported or unparseable expression"
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+        "note": "using explicit params"
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+      "answer": "9",
+      "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (x^2/16 - y^2/m = 1);FocalLength(G) = 10",
+      "coords": {}
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+      "success": true,
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+        "direction": "up"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1792.png",
+      "answer": "1/4",
+      "fact_expr": "G: Parabola;F: Point;l: Line;Expression(G) = (x^2 = y/2);Focus(G) = F;Directrix(G) = l",
+      "coords": {}
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+      "success": true,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1793.png",
+      "answer": "(1,2]",
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+    {
+      "index": 1794,
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+        "y_coef": 1.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1794.png",
+      "answer": "4/3",
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+      "index": 1795,
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+      "error": "Unsupported or unparseable expression"
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+      "index": 1796,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 1797,
+      "success": true,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1797.png",
+      "answer": "sqrt(2)",
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+      "index": 1798,
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+      "answer": "1/8",
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+    {
+      "index": 1799,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1799.png",
+      "answer": "2*sqrt(3)/3",
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+      "coords": {}
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+    {
+      "index": 1800,
+      "success": true,
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+      "error": null,
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+        "b": 1.0,
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1800.png",
+      "answer": "x-y-3=0",
+      "fact_expr": "G: Hyperbola;H: LineSegment;P: Point;Expression(G) = (x^2 - 4*y^2 = 4);Coordinate(P) = (4, 1);IsChordOf(H,G);MidPoint(H)=P",
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+      "index": 1801,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 1802,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1802.png",
+      "answer": "3",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F2: Point;F1: Point;O:Origin;b > a;a > 0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);Distance(P,O) = b;Sin(AngleOf(P, F2, F1)) = (5/3)*Sin(AngleOf(P, F1, F2))",
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+    {
+      "index": 1803,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.0,
+        "major_axis": "y",
+        "x_coef": 1.0,
+        "y_coef": 4.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1803.png",
+      "answer": "2",
+      "fact_expr": "G: Ellipse;l1: Line;l2: Line;M: Point;N: Point;P: Point;Expression(G) = (x^2 + y^2/4 = 1);Expression(l1) = (y = 2*x);Expression(l2) = (y = -2*x);PointOnCurve(P, G);L1:Line;L2:Line;PointOnCurve(P, L1);PointOnCurve(P, L2);IsParallel(l1, L1);IsParallel(l2, L2);Intersection(L1, l2) = M;Intersection(L2, l1) = N",
+      "coords": {}
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+    {
+      "index": 1804,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
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+          0.0
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_1804.png",
+      "answer": "3",
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+      "coords": {}
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+    {
+      "index": 1805,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1805.png",
+      "answer": "sqrt(5)",
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+    {
+      "index": 1806,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 1807,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 1808,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1808.png",
+      "answer": "2",
+      "fact_expr": "l: Line;C: Parabola;p: Number;A: Point;B: Point;D: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C)=F;PointOnCurve(F,l);Slope(l)=1/2;Intersection(l,C)={A,B};IsChordOf(LineSegmentOf(A,B),C);MidPoint(LineSegmentOf(A,B))=D;F:Point;Distance(D, Directrix(C)) = 10",
+      "coords": {}
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+    {
+      "index": 1809,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1809.png",
+      "answer": "y=pm*(sqrt(3)*x/3)",
+      "fact_expr": "P: Point;x0: Number;y0: Number;Coordinate(P) = (x0, y0);E: Hyperbola;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;PointOnCurve(P, E);M: Point;N: Point;LeftVertex(E) = M;RightVertex(E) = N;Slope(LineOf(P, M))*Slope(LineOf(P, N)) = 1/3",
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+    {
+      "index": 1810,
+      "success": true,
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+      "error": null,
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+        "b": 1.4142135623730951,
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+        "y_coef": 2.0
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1810.png",
+      "answer": "x+2*y-3=0",
+      "fact_expr": "G: Ellipse;H: LineSegment;P: Point;Expression(G) = (x^2/4 + y^2/2 = 1);Coordinate(P) = (1, 1);PointOnCurve(P,H);MidPoint(H)=P;IsChordOf(H,G)",
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+          1.0
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+      "index": 1811,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 1812,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1812.png",
+      "answer": "5",
+      "fact_expr": "C: Ellipse;m: Real;m>0;Expression(C) = (y^2/3 + x^2/m = 1);PointOnCurve(Focus(C) ,xAxis);FocalLength(C) = 2*sqrt(2)",
+      "coords": {}
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+    {
+      "index": 1813,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1813.png",
+      "answer": "{pi/3,2*pi/3}",
+      "fact_expr": "l: Line;G: Parabola;p: Number;P: Point;M: Point;F: Point;N: Point;p>0;Expression(G) = (y^2 = 2*p*x);Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {M, N};Intersection(l,Directrix(G)) = P;VectorOf(P, M) = 2*VectorOf(P, F)",
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+    {
+      "index": 1814,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1814.png",
+      "answer": "sqrt(2) + 1",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;B: Point;F: Point;A: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightVertex(C) = A;RightFocus(C) = F;Intersection(LeftDirectrix(C), xAxis) = B ;MidPoint(LineSegmentOf(B, F)) = A",
+      "coords": {}
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+    {
+      "index": 1815,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1815.png",
+      "answer": "(1,3/2)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);F: Point;RightFocus(G) = F;A: Point;PointOnCurve(A, G) = True;Quadrant(A) = 1;Intersection(OverlappingLine(LineSegmentOf(A, F)), G) = B;B: Point;Intersection(OverlappingLine(LineSegmentOf(A, O)), G) = C;C: Point;O: Origin;Area(TriangleOf(A, B, C)) = 3",
+      "coords": {}
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+    {
+      "index": 1816,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1816.png",
+      "answer": "6",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/4 + x^2/a^2 = 1);a: Number;a>2;FocalLength(C) = 2*sqrt(5)",
+      "coords": {}
+    },
+    {
+      "index": 1817,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 2.23606797749979,
+        "major_axis": "x",
+        "x_coef": 7.0,
+        "y_coef": 5.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1817.png",
+      "answer": "(-\\sqrt{2},0)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/7 + y^2/5 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 1818,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
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+      "answer": "2*sqrt(3)",
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+      "answer": "{-1,-3}",
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+      "answer": "ApplyUnit(45, degree)",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1830.png",
+      "answer": "x^2 - y^2/4 = 1",
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+      "answer": "-p^2",
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+      "answer": "x=3/2",
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+      "answer": "sqrt(5)",
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+      "answer": "(1+sqrt(5))/2",
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+      "answer": "(1, 5/3]",
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+      "answer": "(\\sqrt{21}+\\sqrt{3})/3",
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+      "answer": "{x^2/25+y^2/9=1,y^2/25+x^2/9=1}",
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+      "answer": "x = 4",
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+      "answer": "y=pm*x\nx=-2",
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+      "index": 1870,
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+      "answer": "x-2*y-5=0",
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+      "answer": "4/5",
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+      "answer": "((3+sqrt(5))/2, sqrt(5)+1)",
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+      "index": 1873,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1874.png",
+      "answer": "x^2/2 + y^2 = 1",
+      "fact_expr": "C: Ellipse;a: Number;b: Number;F1: Point;F2: Point;P: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-1, 0);Coordinate(F2) = (1, 0);Coordinate(P) = (4/3, 1/3);Focus(C) = {F1, F2};PointOnCurve(P, C)",
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+      "index": 1875,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1875.png",
+      "answer": "y=pm*x",
+      "fact_expr": "l: Line;Inclination(l) = ApplyUnit(30, degree);G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, l);A: Point;B: Point;Intersection(l, G) = {A, B};PointOnCurve(F2, PerpendicularBisector(LineSegmentOf(A, B)))",
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+    {
+      "index": 1876,
+      "success": true,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1876.png",
+      "answer": "sqrt(5)/3",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;M: Point;F2: Point;O: Origin;A: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F2;PointOnCurve(M, yAxis);OneOf(Intersection(LineOf(M,F2),C))=A;Abs(LineSegmentOf(O,A))=Abs(LineSegmentOf(O,F2));Abs(LineSegmentOf(O,F2))=2*Abs(LineSegmentOf(O,M))",
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+    {
+      "index": 1877,
+      "success": true,
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+        "direction": "up"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1877.png",
+      "answer": "3",
+      "fact_expr": "C: Parabola;Expression(C) = (x^2 = 4*y);F: Point;Focus(C) = F;M: Point;PointOnCurve(M, C);N: Point;Intersection(OverlappingLine(LineSegmentOf(F, M)), xAxis) = N;MidPoint(LineSegmentOf(F, N)) = M",
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+    {
+      "index": 1878,
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+        "y_coef": 3.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1878.png",
+      "answer": "sqrt(2)",
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+    {
+      "index": 1879,
+      "success": true,
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+      "answer": "3*sqrt(2)/4",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 1888,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "error": "Unsupported or unparseable expression"
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+      "answer": "sqrt(3)",
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+    {
+      "index": 1894,
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+    {
+      "index": 1895,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 1896,
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+      "answer": "[4*sqrt(2)-sqrt(17),4*sqrt(2)+sqrt(17)]",
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+      "index": 1897,
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+      "answer": "{sqrt(6)-sqrt(3), sqrt(9-6*sqrt(2))}",
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+      "index": 1898,
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+      "error": "Unsupported or unparseable expression"
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+      "index": 1899,
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+      "answer": "pm*3*y+4*x=0",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1900.png",
+      "answer": "sqrt(2)*2/3",
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+    {
+      "index": 1901,
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+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1901.png",
+      "answer": "sqrt(2) + 1",
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+    {
+      "index": 1902,
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+      "answer": "2",
+      "fact_expr": "G: Hyperbola;b: Number;b>0;Expression(G) = (x^2 - y^2/b^2 = 1);Expression(OneOf(Asymptote(G)))=(y=2*x)",
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+      "index": 1903,
+      "success": true,
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+        "y_coef": 3.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1903.png",
+      "answer": "sqrt(5)/3",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;A: Point;B: Point;O: Origin;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F;IsChordOf(LineSegmentOf(A,B),C);PointOnCurve(F,LineSegmentOf(A,B));Intersection(PerpendicularBisector(LineSegmentOf(A, B)), xAxis) = P;VectorOf(A, F) = 2*VectorOf(F, B);MidPoint(LineSegmentOf(O, F)) = P;F:Point;P:Point",
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+    {
+      "index": 1904,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1905,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1905.png",
+      "answer": "sqrt(3)",
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+    },
+    {
+      "index": 1906,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.0,
+        "orientation": "horizontal"
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1906.png",
+      "answer": "2*x-y-15=0",
+      "fact_expr": "P: Point;Coordinate(P) = (8, 1);H: Line;PointOnCurve(P, H);G: Hyperbola;Expression(G) = (x^2 - 4*y^2 = 4);A: Point;B: Point;Intersection(H, G) = {A, B};MidPoint(LineSegmentOf(A, B)) = P",
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+          1.0
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+    {
+      "index": 1907,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_point_off_curve",
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+        "b": 2.2412689930799665,
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+        "iterations": 95
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1907.png",
+      "answer": "16/3",
+      "fact_expr": "G: Hyperbola;C: Circle;H: Point;Coordinate(H) = (4, (4*sqrt(7))/3);PointOnCurve(H, G);Expression(Asymptote(G)) = (y = pm*(4/3)*x);PointOnCurve(OneOf(Vertex(G)),C);PointOnCurve(OneOf(Focus(G)),C);PointOnCurve(Center(C),G)",
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+          3.5276684147527875
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+        "hyperbola_point_off_curve"
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+    {
+      "index": 1908,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "p": 0.25,
+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1908.png",
+      "answer": "sqrt(3)/2+1",
+      "fact_expr": "G: Parabola;H: Line;A: Point;F: Point;B: Point;Expression(G) = (y^2 = x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B}",
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+    {
+      "index": 1909,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1909.png",
+      "answer": "4",
+      "fact_expr": "C: Parabola;p: Number;O: Origin;F: Point;A: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;OneOf(Intersection(PerpendicularBisector(LineSegmentOf(O,F)), C)) = A;Abs(LineSegmentOf(O, A)) = 3",
+      "coords": {}
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+    {
+      "index": 1910,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 2.364173412322998,
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+        "iterations": 89
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1910.png",
+      "answer": "y=pm*sqrt(3)*x",
+      "fact_expr": "G: Hyperbola;a: Number;a>0;Expression(G) = (-y^2/12 + x^2/a^2 = 1);Eccentricity(G) = 2",
+      "coords": {}
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+    {
+      "index": 1911,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1911.png",
+      "answer": "4*sqrt(2)",
+      "fact_expr": "C: Hyperbola;b: Number;A: Point;P: Point;B: Point;b>0;Expression(C) = (x^2/4 - y^2/b^2 = 1);LeftVertex(C) = A;RightVertex(C) = B;PointOnCurve(P, C);Slope(LineOf(P, A)) * Slope(LineOf(P, B)) = 1",
+      "coords": {}
+    },
+    {
+      "index": 1912,
+      "success": true,
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+      "answer": "8/3",
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+      "answer": "12",
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+        "note": "using explicit params"
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+      "answer": "6",
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+      "index": 1919,
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+      "answer": "4/3",
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+      "index": 1920,
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+      "error": "Unsupported or unparseable expression"
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+      "index": 1921,
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+      "index": 1922,
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+      "answer": "20",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1923.png",
+      "answer": "sqrt(13)/3",
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+      "index": 1924,
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+      "answer": "(2, 4)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1925.png",
+      "answer": "32",
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+      "answer": "2",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1927.png",
+      "answer": "y=pm*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/4 = 1)",
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+      "answer": "2",
+      "fact_expr": "G: Ellipse;H: Circle;I: Line;n: Number;m: Number;P:Point;Expression(G) = (x^2/7 + y^2/3 = 1);Expression(H) = (x^2 + y^2 = 3);Expression(I) = (m*x + n*y - 3 = 0);Coordinate(P) = (m, n);NumIntersection(I,H)=0;L:Line;PointOnCurve(P,L)",
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+    {
+      "index": 1929,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1929.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;F2: Point;P: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, RightDirectrix(G));c: Number;HalfFocalLength(G) = c;YCoordinate(P) = sqrt(3)*c;LineSegmentOf(F1, F2) = LineSegmentOf(F2, P)",
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+    {
+      "index": 1930,
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+        "y_coef": 3.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1930.png",
+      "answer": "sqrt(57)/4",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);H: Circle;PointOnCurve(RightVertex(G), H) ;PointOnCurve(RightFocus(G), H);PointOnCurve(Center(H), G)",
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+      "index": 1931,
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+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "answer": "2",
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+      "index": 1932,
+      "success": true,
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+        "direction": "right"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1932.png",
+      "answer": "3/2",
+      "fact_expr": "G: Parabola;H: Line;A: Point;F: Point;B: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, F)) = 3",
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+    {
+      "index": 1933,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 1934,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 1935,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 1936,
+      "success": true,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1936.png",
+      "answer": "3/5",
+      "fact_expr": "G: Line;Expression(G) = (5*x + 4*y - 1 = 0);C: Ellipse;Expression(C) = (x^2/b^2 + y^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Intersection(G, C) = {M, N};M: Point;N: Point;MidPoint(LineSegmentOf(M, N)) = P;P: Point;O: Origin;Slope(LineOf(O, P)) = 5/4",
+      "coords": {}
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+    {
+      "index": 1937,
+      "success": false,
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+      "error": "validation: ellipse_point_off_curve",
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+        "iterations": 93
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1937.png",
+      "answer": "{x^2/9 + y^2 = 1, x^2/9 + y^2/81 = 1}",
+      "fact_expr": "G: Ellipse;A: Point;Coordinate(A) = (3, 0);Length(MajorAxis(G)) = 3 * Length(MinorAxis(G));OverlappingLine(MajorAxis(G), axis);OverlappingLine(MinorAxis(G), axis);PointOnCurve(A, G)",
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+    {
+      "index": 1938,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 1939,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 3.0,
+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1939.png",
+      "answer": "6",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/9 = 1)",
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+    {
+      "index": 1940,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 1941,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 2.0,
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+        "y_coef": 4.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1941.png",
+      "answer": "5",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 + y^2/4 = 1);PointOnCurve(P, G);Focus(G) = {F1, F2}",
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+    {
+      "index": 1942,
+      "success": true,
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+      "error": null,
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+        "y_coef": 3.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1942.png",
+      "answer": "4",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 + y^2/3 = 1);PointOnCurve(P, G);Focus(G)={F1,F2};Abs(LineSegmentOf(P,F1))*Abs(LineSegmentOf(P,F2))=k;k:Number",
+      "coords": {}
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+    {
+      "index": 1943,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1943.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 + (x - 2)^2 = 2);IsTangent(H, Asymptote(G))",
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+    {
+      "index": 1944,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1944.png",
+      "answer": "1/16",
+      "fact_expr": "A: Point;B: Point;C: Point;D: Point;G: Parabola;Expression(G) = (y^2 = 8*x);F: Point;Focus(G) = F;PointOnCurve(F,LineSegmentOf(A,B));IsChordOf(LineSegmentOf(A,B), G) ;PointOnCurve(F, LineSegmentOf(C,D));IsChordOf(LineSegmentOf(C,D),G) ;IsPerpendicular(LineSegmentOf(A,B), LineSegmentOf(C,D))",
+      "coords": {}
+    },
+    {
+      "index": 1945,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1945.png",
+      "answer": "(4*sqrt(5)/5)-1",
+      "fact_expr": "G: Parabola;H: Line;P: Point;Q: Point;M: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (2*x - y + 2 = 0);PointOnCurve(P,G);L:Line;PointOnCurve(P,L);IsPerpendicular(L,yAxis);FootPoint(L,yAxis)=Q;PointOnCurve(M, H)",
+      "coords": {}
+    },
+    {
+      "index": 1946,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1946.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
+      "fact_expr": "H: Ellipse;O: Origin;Center(H) = O;G: Parabola;Expression(G) = (y^2 = 8*x);OneOf(Focus(H)) = Focus(G);Coordinate(OneOf(Vertex(H))) = (0, 2)",
+      "coords": {}
+    },
+    {
+      "index": 1947,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1947.png",
+      "answer": "pm*4",
+      "fact_expr": "G: Parabola;P: Point;O: Origin;F:Point;Focus(G)=F;Coordinate(P) = (m,-2);Vertex(G) = O;PointOnCurve(Focus(G),yAxis);PointOnCurve(P,G);Distance(P,F)=4;m:Number",
+      "coords": {}
+    },
+    {
+      "index": 1948,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1948.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
+      "fact_expr": "G: Hyperbola;F1: Point;M: Point;F2: Point;Expression(G) = (x^2/4 - y^2/9 = 1);Focus(G) = {F1, F2};PointOnCurve(M, G);AngleOf(F1, M, F2) = ApplyUnit(120, degree)",
+      "coords": {}
+    },
+    {
+      "index": 1949,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1949.png",
+      "answer": "[1/3,1)",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);e: Number;Abs(LineSegmentOf(P,F1))=(3/2)*e*Abs(LineSegmentOf(P,F2));Eccentricity(G) = e",
+      "coords": {}
+    },
+    {
+      "index": 1950,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1950.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
+      "fact_expr": "P: Point;F: Point;C: Parabola;p: Number;Q: Point;M: Point;l: Line;p>0;Expression(C) = (y^2 = 2*(p*x));Coordinate(P) = (0, 2);Focus(C) = F;Intersection(LineSegmentOf(P, F), C) = M;PointOnCurve(M, l);IsPerpendicular(l, Directrix(C));FootPoint(l, Directrix(C)) = Q;AngleOf(P, Q, F) = ApplyUnit(90, degree)",
+      "coords": {
+        "P": [
+          0.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 1951,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1951.png",
+      "answer": "[1/2, sqrt(2)/2]",
+      "fact_expr": "l: Line;Expression(l) = (-2*k + k*x - y + 1 = 0);C1: Ellipse;Expression(C1) = (y^2/b^2 + x^2/a^2 = 1);Intersection(l, C1) = {A, B};A: Point;B: Point;C2: Circle;Expression(C2) = ((x - 2)^2 + (y - 1)^2 = 1);Intersection(l, C2) = {C, D};C: Point;D: Point;k: Number;In(k, [-3/2, -1]);VectorOf(A, C) = VectorOf(D, B);b:Number;a>b;b>0",
+      "coords": {}
+    },
+    {
+      "index": 1952,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1952.png",
+      "answer": "y=pm*sqrt(3)*x",
+      "fact_expr": "C: Hyperbola;a: Number;b: Number;G: Circle;F1: Point;F2: Point;P: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;IsDiameter(LineSegmentOf(F1,F2),G);Intersection(G,Asymptote(C))=P;Abs(LineSegmentOf(P, F1)) = 2*b;Quadrant(P)=1",
+      "coords": {}
+    },
+    {
+      "index": 1953,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          5.0,
+          0.0
+        ],
+        "radius": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_1953.png",
+      "answer": "(x^2/9-y^2/16=1)&(x<=-3)",
+      "fact_expr": "G1: Circle;G2: Circle;Expression(G2) = (y^2 + (x + 5)^2 = 1);G3: Circle;Expression(G3) = (y^2 + (x - 5)^2 = 49);IsOutTangent(G1, G2) = True;IsOutTangent(G1, G3) = True",
+      "coords": {}
+    },
+    {
+      "index": 1954,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1954.png",
+      "answer": "x^2/8 + y^2/4 = 1",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;G: Circle;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2 + y^2 = 4);PointOnCurve(Focus(C), G);PointOnCurve(Endpoint(MinorAxis(C)), G)",
+      "coords": {}
+    },
+    {
+      "index": 1955,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1955.png",
+      "answer": "4*sqrt(3)",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p>0;F: Point;Focus(C) = F;l: Line;Directrix(C) = l;Expression(l) = (x=-1);M: Point;PointOnCurve(M, C) = True;A: Point;Projection(M,l) = A;Slope(LineOf(A,F)) = -sqrt(3);p: Number",
+      "coords": {}
+    },
+    {
+      "index": 1956,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1956.png",
+      "answer": "3*sqrt(5)/5",
+      "fact_expr": "G: Hyperbola;b: Number;P: Point;O: Origin;F: Point;H: Line;b>0;Expression(G) = (x^2/5 - y^2/b^2 = 1);RightFocus(G) = F;PointOnCurve(F, H);IsPerpendicular(Asymptote(G), H);FootPoint(Asymptote(G), H) = P;Area(TriangleOf(P, O, F)) = sqrt(5)",
+      "coords": {}
+    },
+    {
+      "index": 1957,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1957.png",
+      "answer": "9",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2/4 - y^2 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;G: Line;PointOnCurve(F1, G) = True;Intersection(G, LeftPart(C)) = {A, B};A: Point;B: Point",
+      "coords": {}
+    },
+    {
+      "index": 1958,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_focus_mismatch",
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1958.png",
+      "answer": "1\n2",
+      "fact_expr": "C1: Hyperbola;C2: Hyperbola;F: Point;a: Number;b: Number;a > 0;b > 0;Expression(C1) = (x^2/a^2 - y^2/b^2 = 1);Expression(C2) = (x^2/4 - y^2/16 = 1);Asymptote(C1) = Asymptote(C2);Coordinate(F) = (sqrt(5), 0);RightFocus(C1) = F",
+      "coords": {
+        "F": [
+          2.23606797749979,
+          0.0
+        ]
+      },
+      "validation_reasons": [
+        "hyperbola_focus_mismatch"
+      ]
+    },
+    {
+      "index": 1959,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1959.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;p: Number;M: Point;O: Origin;F: Point;p>0;Expression(G) = (y^2 = 2*p*x);Coordinate(M) = (4, y1);Focus(G)=F;Distance(M,F)=5;y1:Number;PointOnCurve(M,G)",
+      "coords": {}
+    },
+    {
+      "index": 1960,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1960.png",
+      "answer": "{(4,-4),(16,8)}",
+      "fact_expr": "G: Parabola;H: Line;P: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (y = x);PointOnCurve(P, G);Distance(P, H) = 4*sqrt(2)",
+      "coords": {}
+    },
+    {
+      "index": 1961,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1962,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1962.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Parabola;H: Point;P: Point;Expression(G) = (y^2 = 4*x);Coordinate(H) = (0, -1);PointOnCurve(P, G)",
+      "coords": {
+        "H": [
+          0.0,
+          -1.0
+        ]
+      }
+    },
+    {
+      "index": 1963,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1963.png",
+      "answer": "sqrt(5)/3",
+      "fact_expr": "E: Ellipse;b: Number;a: Number;G: Line;A: Point;F1: Point;F2: Point;B: Point;a > b;b > 0;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(E) = F1;RightFocus(E) = F2;PointOnCurve(F2, G);Intersection(G, E) = {A, B};DotProduct(VectorOf(A, F1), VectorOf(A, F2)) = 0;VectorOf(A, F2) = 2*VectorOf(F2, B)",
+      "coords": {}
+    },
+    {
+      "index": 1964,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1965,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1965.png",
+      "answer": "sqrt(3)/2",
+      "fact_expr": "G: Hyperbola;H: Parabola;I: Ellipse;O: Origin;Expression(G) = (x^2/16 - y^2 = 1);Expression(H) = (x^2 = 8*y);Center(I) = O;SymmetryAxis(I) = axis;PointOnCurve(Focus(H), I);PointOnCurve(Vertex(G), I)",
+      "coords": {}
+    },
+    {
+      "index": 1966,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1966.png",
+      "answer": "3",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l: Line;Directrix(C) = l;P: Point;PointOnCurve(P, l);Q: Point;OneOf(Intersection(LineOf(P, F), C)) = Q;VectorOf(Q, P) = 3*VectorOf(Q, F)",
+      "coords": {}
+    },
+    {
+      "index": 1967,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1967.png",
+      "answer": "25/4",
+      "fact_expr": "l: Line;G: Parabola;Expression(G) = (y^2 = 8*x);A: Point;B: Point;Intersection(l, G) = {A, B};F: Point;Focus(G) = F;PointOnCurve(F, l);Coordinate(A) = (8, 8)",
+      "coords": {
+        "A": [
+          8.0,
+          8.0
+        ]
+      }
+    },
+    {
+      "index": 1968,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1968.png",
+      "answer": "2*sqrt(2)/3",
+      "fact_expr": "l: Line;G: Parabola;A: Point;F: Point;B: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B};Abs(LineSegmentOf(A, F)) = 3;O:Origin",
+      "coords": {}
+    },
+    {
+      "index": 1969,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1970,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1970.png",
+      "answer": "4*sqrt(3)/3",
+      "fact_expr": "G: Hyperbola;b2: Number;a2: Number;H: Ellipse;a1: Number;b1: Number;F1: Point;P: Point;F2: Point;a2>0;b2>0;Expression(G) = (-y^2/b2^2 + x^2/a2^2 = 1);a1 > b1;b1 > 0;Expression(H) = (y^2/b1^2 + x^2/a1^2 = 1);OneOf(Intersection(H, G)) = P;Focus(G) = {F1, F2};Focus(H) = {F1, F2};e1: Number;e2: Number;Eccentricity(H) = e1;Eccentricity(G) = e2;AngleOf(F1, P, F2) = pi/3",
+      "coords": {}
+    },
+    {
+      "index": 1971,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1972,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 1973,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
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+      "answer": "-1/2",
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+      "answer": "sqrt(21)",
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+      "answer": "1/2",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1978.png",
+      "answer": "(x-1/2)^2+y^2=1",
+      "fact_expr": "G: Parabola;H: Circle;Expression(G) = (y^2 = 2*x);Focus(G)=Center(H);IsTangent(Directrix(G),H)",
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+      "success": true,
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+      "answer": "26",
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+    {
+      "index": 1980,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1980.png",
+      "answer": "2*sqrt(5)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2/2 = 1)",
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+    },
+    {
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 1982,
+      "success": true,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1982.png",
+      "answer": "2*pi/3",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 + y^2/2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 4",
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+    {
+      "index": 1983,
+      "success": true,
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+      "answer": "4",
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+    {
+      "index": 1984,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 1985,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1985.png",
+      "answer": "-1/2",
+      "fact_expr": "m: Line;G: Ellipse;Expression(G) = (x^2/2 + y^2 = 1);P1: Point;P2: Point;Intersection(m, G) = {P1, P2};P: Point;MidPoint(LineSegmentOf(P1, P2)) = P;k1: Number;Negation(k1=0);Slope(m) = k1;O: Origin;k2: Number;Slope(LineOf(O, P)) = k2",
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+    {
+      "index": 1986,
+      "success": true,
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+      "answer": "(x-5)^2+y^2=4^2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);H: Circle;Center(H) = RightFocus(G);IsTangent(Asymptote(G), H) = True",
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+    {
+      "index": 1987,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 1988,
+      "success": true,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1988.png",
+      "answer": "3",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);P: Point;PointOnCurve(P, G);Q: Point;H: Circle;PointOnCurve(Q, H);C: Point;Coordinate(C) = (2, 0);Center(H) = C;Length(Radius(H)) = 1",
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+    {
+      "index": 1989,
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+        "note": "using explicit params"
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+      "answer": "2",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;M: Point;l1: Line;l2: Line;F1: Point;F2: Point;G: Circle;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;IsDiameter(LineSegmentOf(F1, F2), G);OneOf(Asymptote(C)) = l1;OneOf(Asymptote(C)) = l2;Negation(l1 = l2);Intersection(G, l1) = M;Abs(LineSegmentOf(M, F1)) > Abs(LineSegmentOf(M, F2));PointOnCurve(MidPoint(LineSegmentOf(M, F1)), l2)",
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+    {
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1990.png",
+      "answer": "(8/3,+oo)",
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+    {
+      "index": 1991,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 1992,
+      "success": true,
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+      "answer": "17",
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+    },
+    {
+      "index": 1993,
+      "success": true,
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+        "direction": "up"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1993.png",
+      "answer": "6",
+      "fact_expr": "G: Parabola;p: Number;H: Line;p>0;Expression(G) = (x^2=2*p*y);Expression(H) = (2*x-y+3=0);PointOnCurve(Focus(G), H)",
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+    },
+    {
+      "index": 1994,
+      "success": true,
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1994.png",
+      "answer": "21/10",
+      "fact_expr": "l: Line;G: Parabola;p: Number;H: Line;B: Point;A: Point;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (x = -1/2);alpha:Number;Coordinate(F) = (-Sin(alpha)*Cos(alpha), 0);Tan(alpha) = -2;Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B};Abs(LineSegmentOf(A, B)) = 4",
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+    {
+      "index": 1995,
+      "success": true,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1995.png",
+      "answer": "6*sqrt(7)/7",
+      "fact_expr": "G: Hyperbola;L1: Line;L2: Line;Expression(G) = (x^2/3 - y^2/4 = 1);Directrix(G) = {L1, L2}",
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+    },
+    {
+      "index": 1996,
+      "success": true,
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+        "direction": "left"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_1996.png",
+      "answer": "5",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = -4*x);F: Point;Focus(G) = F;A: Point;Coordinate(A) = (3, -3);P: Point;PointOnCurve(P, G)",
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+        "A": [
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+          -3.0
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+      "index": 1997,
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+        "y_coef": 3.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_1997.png",
+      "answer": "(1,0)",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;e:Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Focus(G)={F1,F2};Eccentricity(G)=e;e=sqrt(2)/2;UpperVertex(G)=P;Area(TriangleOf(P, F1, F2)) = 1;RightFocus(G)=F2",
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+    {
+      "index": 1998,
+      "success": true,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_1998.png",
+      "answer": "8",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/5 + x^2/m = 1);m: Number;m>0;Expression(OneOf(Asymptote(G))) = (sqrt(5)*x + 2*y = 0);F: Point;LeftFocus(G) = F;P: Point;PointOnCurve(P, RightPart(G));H: Circle;Expression(H) = (x^2 + (y - 4)^2 = 1);Q: Point;PointOnCurve(Q, H)",
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+    {
+      "index": 1999,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 2000,
+      "success": true,
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+        "x_coef": 9.0,
+        "y_coef": 5.0
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2000.png",
+      "answer": "5*sqrt(3)/3",
+      "fact_expr": "G: Ellipse;P: Point;F2: Point;F1: Point;Expression(G) = (x^2/9 + y^2/5 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(60, degree)",
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+    },
+    {
+      "index": 2001,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 2.30381441116333,
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+      "optimization": {
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+        "iterations": 122
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2001.png",
+      "answer": "x^2/4 - y^2/12 = 1",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 16*x);Expression(OneOf(Asymptote(G)))=(y=sqrt(3)*x);OneOf(Focus(G))=Focus(H)",
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+    },
+    {
+      "index": 2002,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2002.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;H: Parabola;p: Number;Expression(G) = (x^2 - y^2/3 = 1);Expression(H) = (y^2 = 2*(p*x));Focus(H) = RightFocus(G)",
+      "coords": {}
+    },
+    {
+      "index": 2003,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2003.png",
+      "answer": "sqrt(5)/2",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(C) = F;L: Line;PointOnCurve(F,L) = True;IsPerpendicular(L,OneOf(Asymptote(C))) = True;FootPoint(L,OneOf(Asymptote(C))) = A;O: Origin;Abs(LineSegmentOf(O, A)) = 2*Abs(LineSegmentOf(A, F));A: Point",
+      "coords": {}
+    },
+    {
+      "index": 2004,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "answer": "sqrt(10)",
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+      "answer": "(0,1/8)",
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+      "answer": "[2,+oo)",
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+      "answer": "y=pm*(1/2)*x",
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+      "answer": "(2*sqrt(5))/5",
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+      "answer": "[-1, 3]",
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+      "success": false,
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+      "answer": "pm*2*sqrt(6)",
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+      "answer": "sqrt(2)",
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+    {
+      "index": 2055,
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+      "answer": "24",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 2057,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 2058,
+      "success": true,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2058.png",
+      "answer": "8*sqrt(3)/3",
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+    {
+      "index": 2059,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2059.png",
+      "answer": "x^2/16 - y^2/9 = 1",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Slope(OneOf(Asymptote(C))) = 3/4;FocalLength(C) = 10",
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+    {
+      "index": 2060,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 2061,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "up"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2061.png",
+      "answer": "sqrt(2)+1",
+      "fact_expr": "G: Hyperbola;H: Parabola;P: Point;F: Point;A: Point;Expression(H) = (y = x^2/4);Intersection(SymmetryAxis(H),Directrix(H))=A;Focus(H) = F;PointOnCurve(P, H);Abs(LineSegmentOf(P, F)) = m*Abs(LineSegmentOf(P, A));WhenMin(m);m:Number;Focus(G)={A,F};PointOnCurve(P, G)",
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+    {
+      "index": 2062,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2062.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;p: Number;H: Circle;p>0;Expression(G) = (y^2 = 2*p*x);Expression(H) = (y^2 + (x - 3)^2 = 16);IsTangent(Directrix(G),H)",
+      "coords": {}
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+    {
+      "index": 2063,
+      "success": true,
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+      "error": null,
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2063.png",
+      "answer": "pi/3",
+      "fact_expr": "G: Parabola;p: Number;M: Point;N: Point;F: Point;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;Intersection(Directrix(G), xAxis) = M;PointOnCurve(N, G);Abs(LineSegmentOf(M, N)) = 2*Abs(LineSegmentOf(N, F))",
+      "coords": {}
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+    {
+      "index": 2064,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 3.0,
+        "major_axis": "y",
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+        "y_coef": 25.0
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+      "answer": "20",
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+      "index": 2065,
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+      "answer": "12",
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+      "answer": "5",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2067.png",
+      "answer": "4*sqrt(2)",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2068.png",
+      "answer": "9",
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+      "answer": "5",
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+      "answer": "x^2/8+y^2/4=1",
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+      "index": 2072,
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+        "note": "using explicit params"
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+      "answer": "2",
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+      "index": 2074,
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+      "error": "Unsupported or unparseable expression"
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+      "answer": "1",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x)",
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+      "answer": "8",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2077.png",
+      "answer": "1/4",
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+    {
+      "index": 2078,
+      "success": true,
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+        "b": 4.242640687119285,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2078.png",
+      "answer": "9",
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+    {
+      "index": 2079,
+      "success": true,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2079.png",
+      "answer": "10",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1)",
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+    {
+      "index": 2080,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2080.png",
+      "answer": "pm*2*sqrt(5)/5",
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+    {
+      "index": 2081,
+      "success": true,
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2081.png",
+      "answer": "sqrt(26)-1",
+      "fact_expr": "G: Parabola;H: Circle;I: Line;P: Point;Q: Point;Expression(G) = (y^2 = 4*x);Expression(H) = ((x - 2)^2 + (y - 5)^2 = 1);Expression(I) = (x = -1);PointOnCurve(P, H);PointOnCurve(Q, G)",
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+    {
+      "index": 2082,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 2083,
+      "success": true,
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+        "iterations": 81
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2083.png",
+      "answer": "-4",
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+          2.0
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+        "B": [
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+          -2.0
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+      "index": 2084,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2084.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "E: Ellipse;A: Point;B: Point;M: Point;a:Number;b:Number;a>b;b>0;Coordinate(M) = (2, 1);Expression(E) = (y^2/b^2 + x^2/a^2 = 1);IsChordOf(LineSegmentOf(A,B),E);Inclination(LineSegmentOf(A,B))=ApplyUnit(135,degree);MidPoint(LineSegmentOf(A, B)) = M",
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+        "M": [
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+          1.0
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+    {
+      "index": 2085,
+      "success": true,
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+      "error": null,
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+        "b": 5.0,
+        "orientation": "horizontal"
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2085.png",
+      "answer": "-1",
+      "fact_expr": "G: Hyperbola;H: Circle;P: Point;F1: Point;M: Point;N: Point;O: Origin;Expression(G) = (x^2/16 - y^2/25 = 1);Expression(H) = (x^2 + y^2 = 16);LeftFocus(G) = F1;PointOnCurve(P, RightPart(G));TangentPoint(LineSegmentOf(P,F1),H)=N;MidPoint(LineSegmentOf(P,F1))=M",
+      "coords": {}
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+    {
+      "index": 2086,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 2087,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "left"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2087.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = -8*x)",
+      "coords": {}
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+    {
+      "index": 2088,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 2089,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2089.png",
+      "answer": "sqrt(3)",
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+      "coords": {}
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+    {
+      "index": 2090,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2090.png",
+      "answer": "[\\sqrt{2}, +\\infty)",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(C) = F;G: Line;PointOnCurve(F, G);Inclination(G) = ApplyUnit(45, degree);NumIntersection(G, RightPart(C)) = 1",
+      "coords": {}
+    },
+    {
+      "index": 2091,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 2092,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2092.png",
+      "answer": "4/3",
+      "fact_expr": "G: Parabola;p: Number;A: Point;B: Point;F: Point;p>0;Expression(G) = (y^2 = 2*p*x);Focus(G) = F;PointOnCurve(A, G);PointOnCurve(B, G);Quadrant(A) = 1;Quadrant(B) = 4;VectorOf(A, F) = 4*VectorOf(F, B)",
+      "coords": {}
+    },
+    {
+      "index": 2093,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.5,
+        "direction": "left"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2093.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
+      "fact_expr": "G: Hyperbola;a: Number;H: Parabola;a>0;Expression(G) = (-y^2 + x^2/a^2 = 1);Expression(H) = (y^2 = -6*x);OneOf(Directrix(G)) = Directrix(H)",
+      "coords": {}
+    },
+    {
+      "index": 2094,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2094.png",
+      "answer": "(1, \\sqrt{5})",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;H: Line;PointOnCurve(RightVertex(G), H);Slope(H) = 2;NumIntersection(H, RightPart(G)) = 2",
+      "coords": {}
+    },
+    {
+      "index": 2095,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2095.png",
+      "answer": "2",
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+      "index": 2105,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2105.png",
+      "answer": "2*x+y+1=0",
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+      "answer": "12",
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+      "index": 2107,
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+      "answer": "5*x^2/32-5*y^2/18=1",
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+      "answer": "sqrt(5)/2",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2109.png",
+      "answer": "pm*4",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2110.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
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+    {
+      "index": 2111,
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+      "answer": "2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1);P: Point;PointOnCurve(P, G);M: Point;Coordinate(M) = (3, 0)",
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+          0.0
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+      "index": 2112,
+      "success": true,
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+      "error": null,
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2112.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "C: Ellipse;F1: Point;P: Point;F2: Point;Expression(C) = (x^2/4 + y^2/3 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);AngleOf(F1, P, F2) = pi/3",
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+    {
+      "index": 2113,
+      "success": true,
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+      "error": null,
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2113.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "M: Ellipse;b: Number;a: Number;N: Circle;Expression(M) = (y^2/b^2 + x^2/a^2 = 1);Expression(N) = (x^2 + y^2 = b^2);PointOnCurve(Focus(M), N)",
+      "coords": {}
+    },
+    {
+      "index": 2114,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 8.0,
+        "b": 4.0,
+        "major_axis": "y",
+        "x_coef": 16.0,
+        "y_coef": 64.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2114.png",
+      "answer": "y^2/36 - x^2/12 = 1",
+      "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (4*x^2 + y^2 = 64);Focus(G) = Focus(H);InterReciprocal(Eccentricity(G), Eccentricity(H))",
+      "coords": {}
+    },
+    {
+      "index": 2115,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2115.png",
+      "answer": "-1",
+      "fact_expr": "G: Parabola;C: Circle;a: Real;l: Line;Expression(G) = (y^2 = 4*x);Expression(C) = (-2*sqrt(3)*y + 2*a*x + x^2 + y^2 = 0);IsTangent(l,C);Directrix(G)=l",
+      "coords": {}
+    },
+    {
+      "index": 2116,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2116.png",
+      "answer": "2*sqrt(7)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/9 + x^2/a^2 = 1);a: Number;H: Parabola;Expression(H) = (y^2 = 16*x);Focus(H)=OneOf(Focus(G))",
+      "coords": {}
+    },
+    {
+      "index": 2117,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2117.png",
+      "answer": "16*sqrt(3)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1) ;LeftFocus(G) = F1;RightFocus(G) = F2;F1: Point;F2: Point;P: Point;PointOnCurve(P, G) = True;Abs(LineSegmentOf(P, F1))*Abs(LineSegmentOf(P, F2)) = 64;Area(TriangleOf(P, F1, F2)) = S;S: Number",
+      "coords": {}
+    },
+    {
+      "index": 2118,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2118.png",
+      "answer": "y = pm*sqrt(3)*x",
+      "fact_expr": "G: Hyperbola;a: Number;H: Parabola;a>0;Expression(G) = (-y^2/12 + x^2/a^2 = 1);Expression(H) = (y^2 = 16*x);Focus(H) = OneOf(Focus(G))",
+      "coords": {}
+    },
+    {
+      "index": 2119,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2119.png",
+      "answer": "x+4*y-5=0",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/4 = 1);A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G);M: Point;Coordinate(M) = (1, 1);MidPoint(LineSegmentOf(A, B)) = M",
+      "coords": {
+        "M": [
+          1.0,
+          1.0
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+      }
+    },
+    {
+      "index": 2120,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2120.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "l: Line;C: Parabola;A: Point;B: Point;Q: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;Slope(l)>0;PointOnCurve(F,l);Intersection(l,C)={A,B};Intersection(l,Directrix(C))=Q;VectorOf(A, B) = VectorOf(B, Q)",
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+    },
+    {
+      "index": 2121,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 2122,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "major_axis": "x",
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+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2122.png",
+      "answer": "sqrt(10)/4",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;O: Origin;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0;Abs(VectorOf(P, F1)) = 3*Abs(VectorOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 2123,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2123.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
+      "fact_expr": "G: Ellipse;a: Number;a>0;Expression(G) = (y^2/3 + x^2/a^2 = 1);Expression(OneOf(Directrix(G))) = (x = 4)",
+      "coords": {}
+    },
+    {
+      "index": 2124,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2124.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;H: Circle;L: Line;M: Point;N: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(RightFocus(G),L);IsPerpendicular(L,xAxis);Intersection(L,G) = {M, N};IsDiameter(LineSegmentOf(M,N),H);PointOnCurve(LeftFocus(G),H)",
+      "coords": {}
+    },
+    {
+      "index": 2125,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2125.png",
+      "answer": "x + 2*y - 4 = 0",
+      "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/4 + y^2/3 = 1);Coordinate(P) = (1, 3/2);PointOnCurve(P, G)",
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+        "P": [
+          1.0,
+          1.5
+        ]
+      }
+    },
+    {
+      "index": 2126,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 2127,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2127.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "M: Point;N: Point;G: Line;a: Number;H: Curve;C:Curve;Expression(G) = (y = a);Expression(H) = (y^2 = x);Expression(C) = (y = e^x);Intersection(G, H) = M;Intersection(G, C) = N;WhenMin(LineSegmentOf(M, N))",
+      "coords": {}
+    },
+    {
+      "index": 2128,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.366584062576294,
+        "b": 2.364173412322998,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.00017838670464698225,
+        "converged": true,
+        "iterations": 89
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2128.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "G: Hyperbola;a: Number;a>0;Expression(G) = (-y^2/9 + x^2/a^2 = 1);Eccentricity(G) = 2",
+      "coords": {}
+    },
+    {
+      "index": 2129,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 2130,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2130.png",
+      "answer": "4",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2 + 2*y^2 = 4)",
+      "coords": {}
+    },
+    {
+      "index": 2131,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 2132,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 2133,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2133.png",
+      "answer": "32/2",
+      "fact_expr": "l: Line;C: Parabola;A: Point;M: Point;B: Point;F: Point;Expression(C) = (y^2 = 8*x);Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};Intersection(l, Directrix(C)) = M;MidPoint(LineSegmentOf(A, M)) = F",
+      "coords": {}
+    },
+    {
+      "index": 2134,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2134.png",
+      "answer": "pm*sqrt(2)",
+      "fact_expr": "F1: Point;LeftFocus(C) = F1;C: Ellipse;Expression(C) = (x^2/4 + y^2 = 1);l: Line;PointOnCurve(F1, l);A: Point;B: Point;Intersection(l, C) = {A, B};VectorOf(B, F1) = 3*VectorOf(F1, A)",
+      "coords": {}
+    },
+    {
+      "index": 2135,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2135.png",
+      "answer": "[13/12, +oo)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;A: Point;B: Point;C: Point;D: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(RightFocus(G), H);IsPerpendicular(H,xAxis);Intersection(H, G) = {A, B};Intersection(H,Asymptote(G))={C,D};Abs(LineSegmentOf(A, B)) >= (5/13)*Abs(LineSegmentOf(C, D))",
+      "coords": {}
+    },
+    {
+      "index": 2136,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2136.png",
+      "answer": "sqrt(13)/2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Ellipse;P: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2/4 + y^2/3 = 1);Intersection(H,OneOf(Asymptote(G)))=P;XCoordinate(P) = 1",
+      "coords": {}
+    },
+    {
+      "index": 2137,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2137.png",
+      "answer": "sqrt(30)/3",
+      "fact_expr": "G: Ellipse;H: Point;Expression(G) = (x^2 + 2*y^2 = 4);Coordinate(H) = (1, 1);L: LineSegment;IsChordOf(L, G);MidPoint(L) = H",
+      "coords": {
+        "H": [
+          1.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 2138,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2138.png",
+      "answer": "4*sqrt(3)",
+      "fact_expr": "G: Parabola;M: Point;F: Point;l: Line;K: Point;Z: Line;Expression(G) = (y^2 = 4*x);PointOnCurve(M, G);Focus(G) = F;LineSegmentOf(M, F) = 4;PointOnCurve(M, Z);Directrix(G) = l;IsPerpendicular(Z, l);FootPoint(Z, l) = K",
+      "coords": {}
+    },
+    {
+      "index": 2139,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 2140,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2140.png",
+      "answer": "sqrt(19)/2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;A: Point;O: Origin;C: Point;B: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = 2*b);Intersection(H,LeftPart(G))=B;Intersection(H,RightPart(G))=C;RightVertex(G)=A;AngleOf(A, O, C) = AngleOf(B, O, C)",
+      "coords": {}
+    },
+    {
+      "index": 2141,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2141.png",
+      "answer": "{5, 3}",
+      "fact_expr": "E: Ellipse;m: Number;Expression(E) = (y^2/4 + x^2/m = 1);FocalLength(E) = 2",
+      "coords": {}
+    },
+    {
+      "index": 2142,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.0625,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2142.png",
+      "answer": "17/8",
+      "fact_expr": "G: Parabola;Expression(G) = (y = 4*x^2);l: Line;PointOnCurve(Focus(G), l);A: Point;B: Point;x1: Number;x2: Number;y1: Number;y2: Number;Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);Intersection(l, G) = {A, B};y1 + y2 = 2",
+      "coords": {}
+    },
+    {
+      "index": 2143,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2143.png",
+      "answer": "x^2/10 - y^2/5 = 1",
+      "fact_expr": "G: Hyperbola;C: Hyperbola;H: Point;Expression(G) = (x^2/2 - y^2 = 1);Coordinate(H) = (4, sqrt(3));PointOnCurve(H, C);Asymptote(C)=Asymptote(G)",
+      "coords": {
+        "H": [
+          4.0,
+          1.7320508075688772
+        ]
+      }
+    },
+    {
+      "index": 2144,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2144.png",
+      "answer": "1",
+      "fact_expr": "C: Parabola;Q: Point;P: Point;Expression(C) = (y^2 = 8*x);Coordinate(Q) = (1, 0);PointOnCurve(P, C)",
+      "coords": {
+        "Q": [
+          1.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 2145,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.2631799883090584,
+        "b": 2.5275303393962245,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 4.481942266905428e-06,
+        "converged": true,
+        "iterations": 95
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2145.png",
+      "answer": "3",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/(a + 3)^2 + x^2/a^2 = 1);a: Number;a>0;Expression(Asymptote(G)) = (y = pm*2*x)",
+      "coords": {}
+    },
+    {
+      "index": 2146,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 2.23606797749979,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2146.png",
+      "answer": "10*sqrt(2)",
+      "fact_expr": "G: Hyperbola;M: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 - y^2/5 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(M, G);Perimeter(TriangleOf(M, F1, F2)) = 20",
+      "coords": {}
+    },
+    {
+      "index": 2147,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2147.png",
+      "answer": "sqrt(2)-1",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/2 + y^2 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;M: Point;Negation(M = Endpoint(MajorAxis(G)));PointOnCurve(M, G);I: Point;Incenter(TriangleOf(M, F1, F2)) = I",
+      "coords": {}
+    },
+    {
+      "index": 2148,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2148.png",
+      "answer": "(2,2)",
+      "fact_expr": "C: Parabola;p: Number;A: Point;p>0;Expression(C) = (y^2 = 2*p*x);Expression(Directrix(C)) = (x = -1/2);PointOnCurve(A,C);Quadrant(A)=1;Distance(A, Focus(C)) = 5/2",
+      "coords": {}
+    },
+    {
+      "index": 2149,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2149.png",
+      "answer": "-3/4",
+      "fact_expr": "G: Parabola;H: Line;O: Origin;A:Point;B: Point;Expression(G) = (y^2 = 2*x);Intersection(G, H) = {A,B};PointOnCurve(Focus(G), H)",
+      "coords": {}
+    },
+    {
+      "index": 2150,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2150.png",
+      "answer": "5",
+      "fact_expr": "C: Parabola;G: Line;M: Point;A: Point;B: Point;Expression(C) = (y^2 = 4*x);Coordinate(M) = (-1, 1);PointOnCurve(Focus(C), G);Intersection(G, C) = {A, B};AngleOf(A, M, B) = ApplyUnit(90, degree);IsChordOf(LineSegmentOf(A,B),C)",
+      "coords": {
+        "M": [
+          -1.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 2151,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 2152,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          -2.0,
+          0.0
+        ],
+        "radius": 1.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_2152.png",
+      "answer": "y = pm*(sqrt(3)/3)*x",
+      "fact_expr": "C: Hyperbola;a: Number;G: Circle;a>0;Expression(C) = (-y^2 + x^2/a^2 = 1);Expression(G) = (y^2 + (x - 2)^2 = 1);IsTangent(G, Asymptote(C))",
+      "coords": {}
+    },
+    {
+      "index": 2153,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 2154,
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+      "answer": "2*sqrt(3)",
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+      "success": false,
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+      "error": "validation: parabola_focus_mismatch",
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2179.png",
+      "answer": "5/2",
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+        "A": [
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+      "index": 2180,
+      "success": true,
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+        "b": 3.4641016151377544,
+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2180.png",
+      "answer": "24",
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+    {
+      "index": 2181,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2181.png",
+      "answer": "4*sqrt(7)/7",
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+      "answer": "y^2=20*x",
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+      "answer": "{(x^2/2 - y^2/14 = 1) & (x >= sqrt(2))}",
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+      "answer": "(1/2,0)",
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+      "answer": "sqrt(3)/2",
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+      "answer": "{0}",
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+      "index": 2301,
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+      "answer": "4",
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+      "coords": {}
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+    {
+      "index": 2302,
+      "success": true,
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+      "error": null,
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "answer": "8",
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+      "index": 2303,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2304.png",
+      "answer": "1\n2",
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+        "direction": "right"
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+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2305.png",
+      "answer": "10",
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+      "coords": {}
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+    {
+      "index": 2306,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 2.23606797749979,
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+        "x_coef": 9.0,
+        "y_coef": 5.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2306.png",
+      "answer": "20/3",
+      "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/9 + y^2/5 = 1);PointOnCurve(P, G);Focus(G)={F1,F2};AngleOf(F1, P, F2) = ApplyUnit(60, degree)",
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+    {
+      "index": 2307,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 2308,
+      "success": true,
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+        "y_coef": 9.0
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2308.png",
+      "answer": "81*pi/16",
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+    {
+      "index": 2309,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2309.png",
+      "answer": "2*sqrt(3)/3",
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+    {
+      "index": 2310,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2310.png",
+      "answer": "2*sqrt(3)/3",
+      "fact_expr": "E: Parabola;p: Number;p>0;Expression(E) = (y^2 = 2*(p*x));F: Point;Focus(E) = F;M: Point;PointOnCurve(M, E);O: Origin",
+      "coords": {}
+    },
+    {
+      "index": 2311,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2311.png",
+      "answer": "-4",
+      "fact_expr": "G: Parabola;p: Number;H: Circle;p<0;Expression(G) = (y^2 = p*x);Expression(H) = (y^2 + (x + 1)^2 = 1);Center(H)=Focus(G)",
+      "coords": {}
+    },
+    {
+      "index": 2312,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2312.png",
+      "answer": "pi/6",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;M: Point;Intersection(Directrix(G), xAxis) = M;N: Point;PointOnCurve(N, G);Abs(LineSegmentOf(N, F)) = (sqrt(3)/2)*Abs(LineSegmentOf(M, N))",
+      "coords": {}
+    },
+    {
+      "index": 2313,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 2314,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 36.0,
+        "y_coef": 9.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2314.png",
+      "answer": "x + 2*y - 8 = 0",
+      "fact_expr": "l: Line;G: Ellipse;A: Point;B: Point;P: Point;Expression(G) = (x^2/36 + y^2/9 = 1);Coordinate(P) = (4, 2);PointOnCurve(P, l);Intersection(l, G) = {A, B};MidPoint(LineSegmentOf(A, B)) = P",
+      "coords": {
+        "P": [
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+          2.0
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+    },
+    {
+      "index": 2315,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2315.png",
+      "answer": "3",
+      "fact_expr": "G: Parabola;C: Circle;A: Point;B: Point;P: Point;Expression(C) = (y^2 + (x-2)^2 = 1);Expression(G) = (y^2 = 4*x);PointOnCurve(A, C);PointOnCurve(B, C);PointOnCurve(P, G);Abs(LineSegmentOf(A, B)) = 2",
+      "coords": {}
+    },
+    {
+      "index": 2316,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2316.png",
+      "answer": "sqrt(3)/2",
+      "fact_expr": "l: Line;O: Origin;PointOnCurve(O, l);Negation(l = axis);C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;P: Point;Q: Point;Intersection(l, C) = {P, Q};B: Point;PointOnCurve(B, C);Negation(B = P);Negation(B = Q);Slope(LineOf(B, P)) * Slope(LineOf(B, Q)) = -1/4",
+      "coords": {}
+    },
+    {
+      "index": 2317,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 2318,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 2319,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2319.png",
+      "answer": "y=pm*sqrt(3)*x",
+      "fact_expr": "G: Hyperbola;a: Number;H: Parabola;a>0;Expression(G) = (-y^2/3 + x^2/a^2 = 1);Expression(H) = (y^2 = 8*x);Focus(H) = RightFocus(G)",
+      "coords": {}
+    },
+    {
+      "index": 2320,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2320.png",
+      "answer": "8",
+      "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;x1:Number;y1:Number;x2:Number;y2:Number;Expression(G) = (y^2 = 4*x);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);Focus(G)=F;PointOnCurve(F,H);Intersection(H,G) = {B, A};x1+x2=6;F:Point",
+      "coords": {}
+    },
+    {
+      "index": 2321,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2321.png",
+      "answer": "sqrt(5)/2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;m: Number;P: Point;A: Point;B: Point;a>0;b>0;Negation(m=0);l1:Line;l2:Line;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (m + x - 3*y = 0);Coordinate(P) = (m, 0);Asymptote(G)={l1,l2};Intersection(H,l1)=A;Intersection(H,l2)=B;Abs(LineSegmentOf(P, A)) = Abs(LineSegmentOf(P, B))",
+      "coords": {}
+    },
+    {
+      "index": 2322,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 2323,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 2.23606797749979,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 5.0
+      },
+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2323.png",
+      "answer": "5*x+9*y-14=0",
+      "fact_expr": "G: Ellipse;H: LineSegment;I: Point;Expression(G) = (x^2/9 + y^2/5 = 1);Coordinate(I) = (1, 1);IsChordOf(H,G);MidPoint(H)=I",
+      "coords": {
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+          1.0
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+    },
+    {
+      "index": 2324,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 2325,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "left"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2325.png",
+      "answer": "(x+2)^2+y^2=16",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = -8*x);H:Circle;Center(H)=Focus(G);IsTangent(Directrix(G),H)",
+      "coords": {}
+    },
+    {
+      "index": 2326,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 2327,
+      "success": false,
+      "error": "Circle from diameter: missing point coordinates"
+    },
+    {
+      "index": 2328,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2328.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;Coordinate(P) = (x0, y0);x0: Number;y0: Number;0 < x0^2/4 + y0^2/3;x0^2/4 + y0^2/3 < 1",
+      "coords": {}
+    },
+    {
+      "index": 2329,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2329.png",
+      "answer": "4*x^2-4*y^2=1",
+      "fact_expr": "P: Point;PointOnCurve(P, G);G: Hyperbola;Expression(G) = (x^2 - y^2 = 1);O: Origin;M: Point;MidPoint(LineSegmentOf(P, O)) = M",
+      "coords": {}
+    },
+    {
+      "index": 2330,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2330.png",
+      "answer": "(1,\\sqrt{2}]",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;y0: Number;x0: Number;H: Line;P: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = ((x - x0)^2 + (y - y0)^2 = 2);Expression(H) = (2*a - a*y + b*x = 0);Coordinate(P) = (x0, y0);PointOnCurve(P, H);NumIntersection(G, RightPart(C))=0",
+      "coords": {}
+    },
+    {
+      "index": 2331,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 2332,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2332.png",
+      "answer": "sqrt(5)/3",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C) = True;G: Circle;Expression(G) = (x^2 + y^2 = b^2);TangentPoint(LineSegmentOf(P, F2), G) = Q;Q: Point;MidPoint(LineSegmentOf(P, F2)) = Q;e: Number;Eccentricity(C) = e",
+      "coords": {}
+    },
+    {
+      "index": 2333,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 2334,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2356.png",
+      "answer": "8*sqrt(3)",
+      "fact_expr": "F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;G: Hyperbola;Expression(G) = (x^2/4 - y^2/8 = 1);P: Point;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))",
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+      "answer": "y=pm*sqrt(3)*x",
+      "fact_expr": "G: Hyperbola;a: Number;a>0;Expression(G) = (-y^2/3 + x^2/a^2 = 1);Eccentricity(G) = 2*a",
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+      "answer": "15/8",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = y/2);M: Point;PointOnCurve(M, G);Distance(M, Focus(G)) = 2",
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+      "success": false,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2360.png",
+      "answer": "(0,4-2*sqrt(3)]+[4+2*sqrt(3),+oo)",
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+      "index": 2361,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2361.png",
+      "answer": "y^2=4*x",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2362.png",
+      "answer": "3*x+4*y-5=0",
+      "fact_expr": "G: Hyperbola;L: LineSegment;M: Point;Expression(G) = (x^2/4 - y^2 = 1);Coordinate(M) = (3, -1);IsChordOf(L,G);MidPoint(L)=M;PointOnCurve(M,L)",
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+      "index": 2363,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2363.png",
+      "answer": "(x^2/4 + y^2 = 1)&Negation(x = pm*2)",
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+      "answer": "sqrt(3)/2",
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+      "answer": "8/3",
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+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
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+      "index": 2386,
+      "success": true,
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2387.png",
+      "answer": "5*sqrt(5)/3",
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+    {
+      "index": 2388,
+      "success": true,
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+      "error": null,
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+        "y_coef": 1.0
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2388.png",
+      "answer": "x^2+y^2=1",
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+    {
+      "index": 2389,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 3.0,
+        "major_axis": "y",
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+        "y_coef": 25.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2389.png",
+      "answer": "{(3,0),(-3,0)}",
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+    {
+      "index": 2390,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2390.png",
+      "answer": "4/3",
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+    {
+      "index": 2391,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 2392,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2392.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
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+    {
+      "index": 2393,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 3.0,
+        "y_coef": 2.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2393.png",
+      "answer": "6*sqrt(13)/13",
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+      "index": 2394,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+      "answer": "4",
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+      "index": 2411,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 2412,
+      "success": true,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2412.png",
+      "answer": "4",
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+      "index": 2413,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2413.png",
+      "answer": "-1",
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+      "index": 2414,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 2415,
+      "success": true,
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+        "direction": "right"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2415.png",
+      "answer": "5",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;M: Point;PointOnCurve(M, C);N: Point;Intersection(OverlappingLine(LineSegmentOf(F, M)), yAxis) = N;LineSegmentOf(F, M) = (1/2)*LineSegmentOf(M, N)",
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+      "answer": "2",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "success": true,
+      "conic_type": "hyperbola",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2418.png",
+      "answer": "sqrt(5)",
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+    {
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+        "note": "using explicit params"
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+      "answer": "y=pm*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;FocalLength(G) = 2*c;c: Number;A: Point;RightVertex(G) = A;H: Parabola;Expression(H) = (x^2 = 2*p*y);p: Number;p>0;F: Point;Focus(H) = F;Length(InterceptChord(Directrix(H), G)) = 2*c;Abs(LineSegmentOf(P, A)) = c;P: Point",
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+    {
+      "index": 2420,
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+      "answer": "((x-2*m)^2+y^2=m^2)&Negation(m=0)",
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+    {
+      "index": 2421,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 2422,
+      "success": true,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2422.png",
+      "answer": "Negation(k=pm*(2*sqrt(3)/3))",
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+    {
+      "index": 2423,
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+        "y_coef": 3.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2423.png",
+      "answer": "(2*sqrt(5))/5",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;A: Point;RightVertex(C) = A;P: Point;PointOnCurve(P, C);O: Origin;AngleOf(P, O, A) = ApplyUnit(60, degree);IsPerpendicular(LineSegmentOf(O, P), LineSegmentOf(A, P))",
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+    {
+      "index": 2424,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.0,
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+        "x_coef": 4.0,
+        "y_coef": 1.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2424.png",
+      "answer": "3",
+      "fact_expr": "P: Point;G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);PointOnCurve(P, G);l: Line;PointOnCurve(P, l);H: Circle;Expression(H) = (x^2 + y^2 = 4);A: Point;B: Point;Intersection(l, H) = {A, B}",
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+    {
+      "index": 2425,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_point_off_curve",
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+      "answer": "(0, 4)",
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+      "error": "Unsupported or unparseable expression"
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+      "answer": "x^2/15+y^2/10=1",
+      "fact_expr": "G: Ellipse;H: Point;C:Ellipse;Expression(G) = (x^2/9 + y^2/4 = 1);Coordinate(H) = (3, 2);PointOnCurve(H, C);Focus(G)=Focus(C)",
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+      "answer": "y^2=12*x",
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+      "answer": "5/4",
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+      "answer": "ApplyUnit(90,degree)",
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+      "answer": "6",
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+      "answer": "5*sqrt(2)",
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+      "answer": "y^2 = 12*x",
+      "fact_expr": "G: Hyperbola;H: Parabola;a: Number;Expression(G) = (x^2/7 - y^2/2 = 1);a>0;Expression(H) = (y^2 = a*x);Focus(H) = OneOf(Focus(G))",
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+        "note": "using explicit params"
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+      "answer": "y=pm*sqrt(3)*x",
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+    {
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+        "b": 1.0,
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2523.png",
+      "answer": "x - y - 3 = 0",
+      "fact_expr": "l: Line;G: Hyperbola;M: Point;Z: LineSegment;Coordinate(M) = (4, 1);Expression(G) = (x^2 - 4*y^2 = 4);IsChordOf(Z, G);MidPoint(Z) = M;OverlappingLine(Z, l)",
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+          1.0
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+        "direction": "up"
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2524.png",
+      "answer": "ApplyUnit(30, degree)",
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+      "success": false,
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2526.png",
+      "answer": "2*sqrt(2)",
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+      "index": 2527,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 2528,
+      "success": true,
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2528.png",
+      "answer": "6",
+      "fact_expr": "G: Parabola;A: Point;B: Point;Expression(G) = (y^2 = 4*x);IsChordOf(LineSegmentOf(A,B),G);XCoordinate(MidPoint(LineSegmentOf(A,B)))=2",
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+    {
+      "index": 2529,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2529.png",
+      "answer": "{y^2=8*x,y^2=-16*x}",
+      "fact_expr": "G: Parabola;m: Number;H: Line;Expression(G) = (y^2 = m*x);Expression(H) = (x = 1);Distance(Directrix(G),H) = 3",
+      "coords": {}
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+    {
+      "index": 2530,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2530.png",
+      "answer": "y = pm*sqrt(6)*x",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G) = F2;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);2*Abs(LineSegmentOf(P, F1)) = 3*Abs(LineSegmentOf(P, F2));AngleOf(F1, P, F2) = ApplyUnit(60, degree)",
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+    {
+      "index": 2531,
+      "success": true,
+      "conic_type": "parabola",
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2531.png",
+      "answer": "(4, 4), (4, -4)",
+      "fact_expr": "G: Parabola;P: Point;F: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(P, G);Abs(LineSegmentOf(P, F)) = 5;Focus(G) = F",
+      "coords": {}
+    },
+    {
+      "index": 2532,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2532.png",
+      "answer": "48",
+      "fact_expr": "G: Hyperbola;P: Point;F2: Point;F1: Point;Expression(G) = (x^2/9 - y^2/16 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(P,RightPart(G));Abs(LineSegmentOf(P,F2))=Abs(LineSegmentOf(F1,F2))",
+      "coords": {}
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+    {
+      "index": 2533,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2533.png",
+      "answer": "9/4",
+      "fact_expr": "C: Parabola;G: Line;A: Point;B: Point;O: Origin;F: Point;Expression(C) = (y^2 = 3*x);Focus(C) = F;PointOnCurve(F, G);Inclination(G) = ApplyUnit(30, degree);Intersection(G, C) = {A, B}",
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+    {
+      "index": 2534,
+      "success": true,
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+        "y_coef": 3.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2534.png",
+      "answer": "x^2/36+y^2/9=1",
+      "fact_expr": "G: Ellipse;a: Number;b: Number;H: Line;P: Point;Q: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x + 2*y + 8 = 0);Eccentricity(G) = sqrt(3)/2;Intersection(G, H) = {P, Q};Abs(LineSegmentOf(P, Q)) = sqrt(10)",
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+    {
+      "index": 2535,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+      "optimization": {
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2535.png",
+      "answer": "13",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 16*x);P: Point;PointOnCurve(P, G);Distance(P, xAxis) = 12;F: Point;Focus(G) = F;Distance(P, F) = Abs(LineSegmentOf(P, F))",
+      "coords": {}
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+    {
+      "index": 2536,
+      "success": true,
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+      "params": {
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2536.png",
+      "answer": "9/2",
+      "fact_expr": "G: Parabola;A: Point;M: Point;F: Point;Expression(G) = (y^2 = 2*x);Coordinate(A) = (4, 2);Focus(G) = F;PointOnCurve(M, G);WhenMin(Abs(LineSegmentOf(M,A))+Abs(LineSegmentOf(M,F)))",
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+          2.0
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+    {
+      "index": 2537,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2537.png",
+      "answer": "sqrt(2)+1",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;F1: Point;P: Point;F2: Point;H:Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(F1,H);IsPerpendicular(H,xAxis);Intersection(H,G)=P;AngleOf(F1, P, F2) = ApplyUnit(45, degree)",
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+      "index": 2538,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 2539,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2539.png",
+      "answer": "{x+y+1=0,x-y+1=0}",
+      "fact_expr": "C: Parabola;A: Point;P: Point;F: Point;Expression(C) = (y^2 = 4*x);Coordinate(A) = (-1, 0);Focus(C) = F;PointOnCurve(P, C);WhenMin(Abs(LineSegmentOf(P,F))/Abs(LineSegmentOf(P,A)))",
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+          0.0
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+      "index": 2540,
+      "success": false,
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+      "index": 2541,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+        "x_coef": 25.0,
+        "y_coef": 9.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2542.png",
+      "answer": "6",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);D: Point;PointOnCurve(D, G) = True;Distance(D, LeftDirectrix(G)) = 5",
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+      "index": 2543,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 2544,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 2545,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2545.png",
+      "answer": "sqrt(3)+1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;Coordinate(F) = (-c, 0);LeftFocus(G) = F;c: Number;c>0;PointOnCurve(F, LineOf(F, E));E: Point;Inclination(LineOf(F, E)) = pi/6;P: Point;Intersection(LineOf(F, E), RightPart(G)) = P;O: Origin;VectorOf(O, E) = (1/2)*(VectorOf(O, F) + VectorOf(O, P));DotProduct(VectorOf(O, E), VectorOf(E, F)) = 0",
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+    {
+      "index": 2546,
+      "success": true,
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2546.png",
+      "answer": "sqrt(10) - 1",
+      "fact_expr": "G: Parabola;P: Point;A: Point;M: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (2, 3);PointOnCurve(P, G);Projection(P, yAxis) = M",
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+        "y_coef": 2.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2547.png",
+      "answer": "ApplyUnit(120,degree)",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 + y^2/2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 4",
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+    {
+      "index": 2548,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 2549,
+      "success": true,
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+      "error": null,
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+        "b": 2.449489742783178,
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+        "x_coef": 9.0,
+        "y_coef": 6.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2549.png",
+      "answer": "y=pm*(sqrt(2)/2)*x",
+      "fact_expr": "G: Hyperbola;a: Number;b:Number;H: Ellipse;a>0;b>0;Expression(G) = (-y^2 + x^2/a^2 = 1);Expression(H) = (x^2/9 + y^2/6 = 1);Focus(G) = Focus(H)",
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+    {
+      "index": 2550,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "major_axis": "x",
+        "x_coef": 18.0,
+        "y_coef": 9.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2550.png",
+      "answer": "12",
+      "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/18 + y^2/9 = 1);PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G) = F2;Area(TriangleOf(F1, P, F2)) = 3*sqrt(3)",
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+    {
+      "index": 2551,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 2552,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 2553,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2553.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;H: Parabola;p: Number;Expression(G) = (x^2 - y^2/3 = 1);p>0;Expression(H) = (y^2 = 2*(p*x));Focus(H) = OneOf(Focus(G))",
+      "coords": {}
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+    {
+      "index": 2554,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2554.png",
+      "answer": "2*sqrt(3)/3",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;F: Point;Focus(G) = F;l: Line;Directrix(G) = l;H: Line;PointOnCurve(F,H);A: Point;B: Point;Intersection(H, G) = {A, B};L1: Line;L2: Line;PointOnCurve(A,L1);PointOnCurve(B,L2);IsPerpendicular(l,L1);IsPerpendicular(l,L2);C: Point;D: Point;FootPoint(l,L1) = C;FootPoint(l,L2) = D;Abs(LineSegmentOf(A, F)) = 2*Abs(LineSegmentOf(B, F));Area(TriangleOf(C, D, F)) = sqrt(2)",
+      "coords": {}
+    },
+    {
+      "index": 2555,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2555.png",
+      "answer": "2*sqrt(3)/3",
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+      "answer": "4*sqrt(7)/7",
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+      "answer": "1",
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+      "answer": "y^2=12*x",
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+      "answer": "{3,16/3}",
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+      "answer": "4",
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+      "answer": "2*x-y-2=0",
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+      "answer": "2*sqrt(2)",
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+        "note": "using explicit params"
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+      "answer": "2*sqrt(5)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (2*x^2 - 3*y^2 = 6)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2577.png",
+      "answer": "3",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2578.png",
+      "answer": "15/8",
+      "fact_expr": "Length(LineSegmentOf(A, B)) = 4;A: Point;B: Point;Endpoint(LineSegmentOf(A, B)) = {A, B};G: Parabola;Expression(G) = (y = 2*x^2);PointOnCurve(A, G);PointOnCurve(B, G);M: Point;MidPoint(LineSegmentOf(A, B)) = M",
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+      "index": 2579,
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+      "conic_type": "ellipse",
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+        "y_coef": 1.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2579.png",
+      "answer": "3",
+      "fact_expr": "C: Ellipse;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/4 + y^2 = 1);PointOnCurve(P, C);LeftFocus(C) = F1;RightFocus(C) = F2;Abs(LineSegmentOf(P, F1)) = 3*Abs(LineSegmentOf(P, F2))",
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+    {
+      "index": 2580,
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+      "params": {
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2580.png",
+      "answer": "sqrt(5)",
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+      "index": 2581,
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+      "conic_type": "ellipse",
+      "error": "validation: ellipse_point_off_curve",
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+        "b": 4.898979485566356,
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+        "x_coef": 49.0,
+        "y_coef": 24.0
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+        "note": "using explicit params"
+      },
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+      "answer": "24",
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+    {
+      "index": 2608,
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+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 2.0,
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2608.png",
+      "answer": "{pm*1, pm*2*sqrt(3)/3}",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 4);l: Line;Expression(l) = (y = k*(x - 1));k: Number;NumIntersection(l, G) = 1",
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+      "index": 2609,
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+        "b": 4.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2609.png",
+      "answer": "5/3",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1)",
+      "coords": {}
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+    {
+      "index": 2610,
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+        "x_coef": 4.0,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2610.png",
+      "answer": "Sin(alpha)/(Sin(gamma)+Sin(beta))",
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+      "coords": {}
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+      "index": 2611,
+      "success": true,
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+      "error": null,
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+        "direction": "up"
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2611.png",
+      "answer": "{x-y-1=0, x+y+1=0}",
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+      "coords": {}
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+    {
+      "index": 2612,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+        "final_loss": 0.0,
+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2612.png",
+      "answer": "3",
+      "fact_expr": "l: Line;G: Parabola;p: Number;H: Line;B: Point;A: Point;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (x = -3);Focus(G) = F;PointOnCurve(F, l);Intersection(l,G)={A,B};Abs(LineSegmentOf(A, B)) = 7;Distance(MidPoint(LineSegmentOf(A,B)), H) = 5",
+      "coords": {}
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+    {
+      "index": 2613,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "a": 1.0,
+        "b": 2.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2613.png",
+      "answer": "2*sqrt(5)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/4 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 2614,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2614.png",
+      "answer": "pi/6",
+      "fact_expr": "G: Parabola;N: Point;F: Point;M: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;Intersection(Directrix(G), xAxis) = M;PointOnCurve(N, G);Abs(LineSegmentOf(N, F)) = (sqrt(3)/2)*Abs(LineSegmentOf(M, N))",
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+    {
+      "index": 2615,
+      "success": true,
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+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2615.png",
+      "answer": "-5",
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+      "coords": {}
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+    {
+      "index": 2616,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "major_axis": "y",
+        "x_coef": 2.0,
+        "y_coef": 3.0
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+        "final_loss": 0.0,
+        "converged": true,
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+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
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+    {
+      "index": 2617,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2617.png",
+      "answer": "1",
+      "fact_expr": "G: Hyperbola;b: Number;b>0;Expression(G) = (x^2/4 - y^2/b^2 = 1);Expression(Asymptote(G)) = (y = pm*(x/2))",
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+    {
+      "index": 2618,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 2619,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "p": 0.5,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2619.png",
+      "answer": "5/2",
+      "fact_expr": "M: Point;Coordinate(M) = (-3, 2);G: Parabola;Expression(G) = (y^2 = 2*x);F: Point;Focus(G) = F;Q: Point;PointOnCurve(Q, G)",
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+        "M": [
+          -3.0,
+          2.0
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+    {
+      "index": 2620,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2620.png",
+      "answer": "sqrt(2)+1",
+      "fact_expr": "H: Parabola;Expression(H) = (x^2 = 4*y);A: Point;Intersection(SymmetryAxis(H), Directrix(H)) = A;B: Point;Focus(H) = B;P: Point;PointOnCurve(P, H);IsTangent(LineSegmentOf(P, A), H);G: Hyperbola;PointOnCurve(P, G);Focus(G) = {A, B}",
+      "coords": {}
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+    {
+      "index": 2621,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.366584062576294,
+        "b": 2.364173412322998,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.00017838670464698225,
+        "converged": true,
+        "iterations": 89
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2621.png",
+      "answer": "2*sqrt(3)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = 2;PointOnCurve(RightFocus(G),H);IsPerpendicular(H,xAxis);Length(InterceptChord(H,G))=m;m:Number",
+      "coords": {}
+    },
+    {
+      "index": 2622,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 2623,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 2624,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2624.png",
+      "answer": "8",
+      "fact_expr": "G: Ellipse;C: Point;F1: Point;F2: Point;Expression(G) = (x^2/25 + y^2/16 = 1);PointOnCurve(C, G);LeftFocus(G) = F1;RightFocus(G) = F2;Abs(LineSegmentOf(C, F1)) = 2",
+      "coords": {}
+    },
+    {
+      "index": 2625,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_point_off_curve",
+      "params": {
+        "a": 3.0,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2625.png",
+      "answer": "x^2/15 + y^2/10 = 1",
+      "fact_expr": "G: Ellipse;H: Point;Coordinate(H) = (3,-2);C:Curve;Expression(C)=(x^2/9+y^2/4=1);Focus(C)=Focus(G);PointOnCurve(H,G)",
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+        "H": [
+          3.0,
+          -2.0
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+      "validation_reasons": [
+        "ellipse_point_off_curve"
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+    {
+      "index": 2626,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2626.png",
+      "answer": "34/3",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);P: Point;PointOnCurve(P, G);Distance(P, LeftDirectrix(G)) = 16/5",
+      "coords": {}
+    },
+    {
+      "index": 2627,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          0.0,
+          0.0
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+        "radius": 2.8284271247461903
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+        "final_loss": 0.0,
+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_2627.png",
+      "answer": "x^2/8+y^2/2=1",
+      "fact_expr": "G: Circle;Q: Point;P: Point;M:Point;Expression(G) = (x^2 + y^2 = 8);PointOnCurve(P,G);L:Line;PointOnCurve(P,L);IsPerpendicular(L,xAxis);FootPoint(L,xAxis)=Q;MidPoint(LineSegmentOf(P,Q))=M",
+      "coords": {}
+    },
+    {
+      "index": 2628,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2628.png",
+      "answer": "(-5, 0)",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;H: Circle;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (-6*x + x^2 + y^2 + 8 = 0);Center(H) = OneOf(Focus(G));Length(MinorAxis(G)) = 8",
+      "coords": {}
+    },
+    {
+      "index": 2629,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2629.png",
+      "answer": "y=pm*sqrt(3)*x",
+      "fact_expr": "H: Parabola;Expression(H) = (y^2 = 2*x);G: Hyperbola;Expression(G) = (-y^2/3 + x^2/m = 1);m: Number;Directrix(H) = OneOf(Directrix(G))",
+      "coords": {}
+    },
+    {
+      "index": 2630,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2630.png",
+      "answer": "x^2/4 + y^2/2 = 1",
+      "fact_expr": "G: Hyperbola;H: Parabola;C:Ellipse;b: Number;a: Number;Expression(G) = (x^2 - y^2 = 1);Expression(H) = (y^2 = 8*x);Expression(C) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(Focus(H),C);Focus(G)=Focus(C)",
+      "coords": {}
+    },
+    {
+      "index": 2631,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2631.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "C: Ellipse;a: Number;G: Point;Expression(C) = (y^2/4 + x^2/a^2 = 1);Coordinate(G) = (2, 0);OneOf(Focus(C)) = G",
+      "coords": {
+        "G": [
+          2.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 2632,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2632.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;P: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(P) = (m, 2*sqrt(3));PointOnCurve(P, G);Focus(G) = F;m:Number",
+      "coords": {}
+    },
+    {
+      "index": 2633,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "y",
+        "x_coef": 1.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2633.png",
+      "answer": "x = 1-\\sqrt{2}",
+      "fact_expr": "G: Parabola;H: Ellipse;b: Number;P: Point;F1: Point;F2: Point;O:Origin;0 < b;b < 1;Expression(H) = (x^2 + y^2/b^2 = 1);LeftFocus(H) = F1;RightFocus(H)=F2;Vertex(G) = O;Focus(G)=F2;Intersection(G, H) = P;AngleOf(P, F1, F2) = ApplyUnit(45, degree)",
+      "coords": {}
+    },
+    {
+      "index": 2634,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 8.0,
+        "b": 6.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2634.png",
+      "answer": "33",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/64 - y^2/36 = 1);P: Point;PointOnCurve(P, G);F1: Point;F2: Point;Focus(G) = {F1, F2};Abs(LineSegmentOf(P, F1)) = 17",
+      "coords": {}
+    },
+    {
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+      "answer": "19/20",
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+      "answer": "(3, \\sqrt{10}]",
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+      "answer": "sqrt(15)/2",
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+      "success": false,
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+      "success": false,
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+      "answer": "4*sqrt(3)",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2682.png",
+      "answer": "sqrt(2)",
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+      "answer": "sqrt(2)/2",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2684.png",
+      "answer": "sqrt(2)",
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+    {
+      "index": 2685,
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+      "answer": "sqrt(3)",
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+      "index": 2686,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 2687,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2687.png",
+      "answer": "pm*(sqrt(3)/4)",
+      "fact_expr": "G: Ellipse;F1: Point;P: Point;M: Point;Expression(G) = (x^2/12 + y^2/3 = 1);OneOf(Focus(G)) = F1;PointOnCurve(P, G);MidPoint(LineSegmentOf(P,F1))=M;PointOnCurve(M, yAxis)",
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+    {
+      "index": 2688,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2688.png",
+      "answer": "-3/13",
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+    {
+      "index": 2689,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.7320508075688772,
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2689.png",
+      "answer": "2*sqrt(5)/5",
+      "fact_expr": "alpha: Number;Inclination(H) = alpha;H: Line;G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1);Slope(H) = Eccentricity(G)",
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+      "index": 2690,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2690.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C)=F;O: Origin;P: Point;PointOnCurve(P,C);IsPerpendicular(LineSegmentOf(P,F),LineSegmentOf(O,F))",
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+      "index": 2691,
+      "success": true,
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+      "error": null,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2691.png",
+      "answer": "x^2/9 - y^2/7 = 1",
+      "fact_expr": "H: Ellipse;Expression(H) = (x^2/25 + y^2/9 = 1);Focus(H) = Focus(G);G: Hyperbola;e: Number;Eccentricity(G) = e;e = 4/3",
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+      "index": 2692,
+      "success": true,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2714.png",
+      "answer": "1/2",
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+      "answer": "y=pm*(sqrt(3)/3)*x",
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+      "answer": "(x+1)^2/4+y^2/3=1",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2721.png",
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+        "note": "using explicit params"
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+      "answer": "3*sqrt(3)",
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+    {
+      "index": 2723,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2723.png",
+      "answer": "2*sqrt(3)-3",
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+      "index": 2724,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 2725,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "success": true,
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+      "answer": "-1/2",
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+      "answer": "4*sqrt(2)",
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+      "answer": "y = -1/8",
+      "fact_expr": "G: Parabola;Expression(G) = (y = 2*x^2)",
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+      "answer": "4",
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+      "answer": "-50/17",
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+      "answer": "-1/2",
+      "fact_expr": "M: Point;Coordinate(M) = (0, 1);l: Line;PointOnCurve(M, l);C: Ellipse;Expression(C) = (x^2/9 + y^2/5 = 1);A: Point;B: Point;Intersection(l, C) = {A, B};F1: Point;LeftFocus(C) = F1;WhenMax(Perimeter(TriangleOf(A, B, F1)))",
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+      "answer": "3",
+      "fact_expr": "l: Line;C: Hyperbola;k:Number;B: Point;A: Point;M: Point;Expression(C) = (x^2/2 - y^2/3 = 1);Coordinate(M) = (2, 1);Slope(l)=k;Intersection(l,C) = {A, B};MidPoint(LineSegmentOf(A,B))=M",
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+      "answer": "2+sqrt(3)",
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+      "answer": "x^2/9+y^2/4=1",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2740.png",
+      "answer": "[2,4)",
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+      "answer": "3",
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+        "y_coef": 4.0
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2742.png",
+      "answer": "8",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2 + 4*y^2 = 16)",
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+    {
+      "index": 2743,
+      "success": true,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2743.png",
+      "answer": "b^3/(4*a)",
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+      "index": 2744,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2744.png",
+      "answer": "y^2=4*x-4",
+      "fact_expr": "G: Parabola;F: Point;P: Point;Expression(G) = (y^2 = 8*x);Focus(G)=F;PointOnCurve(P, G)",
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+      "index": 2745,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2745.png",
+      "answer": "1",
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+      "index": 2746,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 2747,
+      "success": true,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2747.png",
+      "answer": "sqrt(2)",
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2748.png",
+      "answer": "4",
+      "fact_expr": "M: Point;Coordinate(M) = (-2, 8);G: Parabola;Expression(G) = (y^2 = 2*p*x);PointOnCurve(M, Directrix(G));p: Real",
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2749.png",
+      "answer": "2/3",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;I: Point;Coordinate(I) = (2, 0);H: Line;PointOnCurve(I, H) = True;Intersection(H, G) = {A, B};A: Point;B: Point",
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+      "index": 2750,
+      "success": true,
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+      "error": null,
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+        "y_coef": 3.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2750.png",
+      "answer": "4*sqrt(41)",
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+    {
+      "index": 2751,
+      "success": true,
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+      "error": null,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2751.png",
+      "answer": "y=pm*(sqrt(3)/3)*x",
+      "fact_expr": "G: Hyperbola;a: Number;b: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Distance(OneOf(Focus(G)),OneOf(Asymptote(G)))=(1/4)*FocalLength(G)",
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+      "index": 2752,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 2753,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 2754,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2754.png",
+      "answer": "2",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;Eccentricity(G) = sqrt(2)/2;A: Point;B: Point;C: Point;PointOnCurve(Vertex(TriangleOf(A, B, C)), G);D: Point;E: Point;F: Point;MidPoint(LineSegmentOf(A, B)) = D;MidPoint(LineSegmentOf(B, C)) = E;MidPoint(LineSegmentOf(A, C)) = F;k1: Number;k2: Number;k3: Number;Slope(OverlappingLine(LineSegmentOf(A, B))) = k1;Slope(OverlappingLine(LineSegmentOf(B, C))) = k2;Slope(OverlappingLine(LineSegmentOf(A, C))) = k3;Negation(k1*k2*k3=0);O: Origin;Slope(LineOf(O, D)) + Slope(LineOf(O, E)) + Slope(LineOf(O, F)) = -1",
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+    {
+      "index": 2755,
+      "success": true,
+      "conic_type": "ellipse",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2755.png",
+      "answer": "pm*sqrt(5)/4",
+      "fact_expr": "G: Ellipse;H: Line;k: Number;Expression(G) = (x^2/16 + y^2/4 = 1);Expression(H) = (k*x - y + 3 = 0);IsTangent(H,G)",
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+    {
+      "index": 2756,
+      "success": true,
+      "conic_type": "parabola",
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+      "answer": "x^2/3 - y^2/3 = 1",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2783.png",
+      "answer": "y=pm*2*sqrt(2)*x",
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+      "answer": "5",
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+      "answer": "8+sqrt(2)",
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+      "answer": "sqrt(7)/2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/3 = 1)",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2791.png",
+      "answer": "4",
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+      "index": 2792,
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+      "answer": "9",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2794.png",
+      "answer": "x=-1",
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+      "answer": "20/3",
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2796.png",
+      "answer": "pm*4",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2797.png",
+      "answer": "x^2/20+y^2/16=1",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2798.png",
+      "answer": "(0,pm*2*sqrt(2))",
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+        "y_coef": 0.25
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2799.png",
+      "answer": "sqrt(3)/2",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2 + 4*y^2 = 1)",
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+    {
+      "index": 2800,
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+        "note": "using explicit params"
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+      "answer": "12/5",
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+      "index": 2801,
+      "success": true,
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+        "y_coef": 2.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2801.png",
+      "answer": "3",
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+      "success": true,
+      "conic_type": "parabola",
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2802.png",
+      "answer": "4",
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+    {
+      "index": 2803,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2803.png",
+      "answer": "(-sqrt(3), -3/2) + (3/2, sqrt(3))",
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+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 1.0,
+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2804.png",
+      "answer": "sqrt(5)/2\ny=pm*(1/2)*x",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2/4 - y^2 = 1)",
+      "coords": {}
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+      "index": 2805,
+      "success": true,
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+        "b": 4.0,
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+        "y_coef": 16.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2805.png",
+      "answer": "5",
+      "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/25 + y^2/16 = 1);Coordinate(P) = (x1, y1);PointOnCurve(P, G);x1:Number;y1:Number",
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+    {
+      "index": 2806,
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+      "error": null,
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2806.png",
+      "answer": "2*sqrt(5)/5",
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+      "index": 2807,
+      "success": true,
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+        "direction": "up"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2807.png",
+      "answer": "(0,1/4)",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = y)",
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+      "index": 2808,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2808.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "l: Line;G: Ellipse;b: Number;B: Point;F2: Point;A: Point;F1: Point;0 < b;b < 2;Expression(G) = (x^2/4 + y^2/b^2 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(F1, l);Intersection(l, G) = {A, B};Max(Abs(LineSegmentOf(A, F2)) + Abs(LineSegmentOf(B, F2))) = 5",
+      "coords": {}
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+    {
+      "index": 2809,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 2.0,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2809.png",
+      "answer": "6",
+      "fact_expr": "G: Hyperbola;H: Parabola;p: Number;Expression(G) = (x^2/5 - y^2/4 = 1);p>0;Expression(H) = (y^2 = 2*(p*x));Focus(H) = RightFocus(G)",
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+    {
+      "index": 2810,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2810.png",
+      "answer": "-9",
+      "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (x^2/16 + y^2/m = 1);Eccentricity(G) = 5/4",
+      "coords": {}
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+    {
+      "index": 2811,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 4.898979485566356,
+        "major_axis": "x",
+        "x_coef": 49.0,
+        "y_coef": 24.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2811.png",
+      "answer": "10",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/49 + y^2/24 = 1)",
+      "coords": {}
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+    {
+      "index": 2812,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 2813,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 2814,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "vertical"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2814.png",
+      "answer": "4",
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+        "note": "using explicit params"
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+      "answer": "y^2 = 16*x",
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+        "note": "using explicit params"
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+      "answer": "[3, 4]",
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "answer": "sqrt(2)",
+      "fact_expr": "G: Hyperbola;m: Number;m>0;Expression(G) = (x^2/2 - y^2/m = 1);Distance(OneOf(Vertex(G)), Asymptote(G)) = 1",
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+      "index": 2818,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+        "converged": true
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+      "answer": "5",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2825.png",
+      "answer": "sqrt(10)",
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+      "index": 2827,
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2827.png",
+      "answer": "pm*sqrt(2)/2",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2828.png",
+      "answer": "sqrt(3)/2",
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+        "note": "using explicit params"
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+      "answer": "sqrt(3)/2",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+        "y_coef": 3.0
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+        "note": "using explicit params"
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+      "answer": "2",
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+        "y_coef": 36.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2832.png",
+      "answer": "34/5",
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2833.png",
+      "answer": "{(6,6*sqrt(2)),(6,-6*sqrt(2))}",
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2834.png",
+      "answer": "3*sqrt(2)/4",
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+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2835.png",
+      "answer": "5/3",
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+      "index": 2836,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 2837,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_point_off_curve",
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+        "iterations": 2
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+      "answer": "8",
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2838.png",
+      "answer": "{0,pm*1}",
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+        "b": 4.0,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2839.png",
+      "answer": "(pm*5,0)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1)",
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+      "index": 2840,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 2841,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 2842,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2842.png",
+      "answer": "3",
+      "fact_expr": "H: Ellipse;Expression(H) = (x^2/12 + y^2/m = 1);m: Number;G: Hyperbola;Expression(G) = (x^2 - 8*y^2 = 8);Focus(H) =Focus(G)",
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+      "index": 2843,
+      "success": true,
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2843.png",
+      "answer": "2",
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+      "index": 2844,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2844.png",
+      "answer": "1, 9",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/8 = 1);P: Point;PointOnCurve(P, G);F1: Point;F2: Point;OneOf(Focus(G)) = F1;OneOf(Focus(G)) = F2;Negation(F1=F2);Distance(P, F1) = 10;N: Point;MidPoint(LineSegmentOf(P, F1)) = N;O: Origin",
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+    {
+      "index": 2845,
+      "success": true,
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2845.png",
+      "answer": "6",
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+    {
+      "index": 2846,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2846.png",
+      "answer": "2*sqrt(3)/3",
+      "fact_expr": "C: Parabola;O: Origin;P: Point;F: Point;Expression(C) = (y^2 = 4*x);PointOnCurve(P, C);Focus(C) = F",
+      "coords": {}
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+    {
+      "index": 2847,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2847.png",
+      "answer": "2",
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+    {
+      "index": 2848,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2848.png",
+      "answer": "16/3",
+      "fact_expr": "l: Line;G: Parabola;P: Point;A: Point;B: Point;Expression(G) = (y^2 = 4*x);Coordinate(P) = (1, 0);Inclination(l) = ApplyUnit(60, degree);Intersection(l, G) = {A, B};IsChordOf(LineSegmentOf(A, B), G);PointOnCurve(P, l)",
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+    {
+      "index": 2849,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.5,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2849.png",
+      "answer": "1",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 6*x);H: Circle;P: Point;Coordinate(P) = (-2, 1);Center(H) = P;Length(InterceptChord(Directrix(G), H)) = sqrt(3)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2881.png",
+      "answer": "2",
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+      "answer": "12/5",
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+      "index": 2883,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "answer": "(\\sqrt{5}, +\\infty)",
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+      "index": 2885,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 2886,
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+      },
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+      "answer": "(3,0)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - 2*y^2 = 6)",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2887.png",
+      "answer": "3/5",
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+    {
+      "index": 2888,
+      "success": true,
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+      "error": null,
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+        "direction": "right"
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2888.png",
+      "answer": "6",
+      "fact_expr": "C: Parabola;E: Ellipse;A: Point;B: Point;O:Origin;Expression(C) = (y^2 = 8*x);Center(E)=O;Eccentricity(E)=1/2;RightFocus(E) = Focus(C);Intersection(Directrix(C),E)={A,B}",
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+    {
+      "index": 2889,
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+      "error": null,
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+        "b": 3.0,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2889.png",
+      "answer": "5/4",
+      "fact_expr": "G: Hyperbola;Expression(G)=(x^2/16 - y^2/9 = 1)",
+      "coords": {}
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+      "index": 2890,
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+      "error": null,
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+        "direction": "up"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2890.png",
+      "answer": "8",
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+    {
+      "index": 2891,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.7320508075688772,
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2891.png",
+      "answer": "3/4",
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+    {
+      "index": 2892,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 2893,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 2894,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2894.png",
+      "answer": "sqrt(10)",
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+          -3.0
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+      "index": 2895,
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+        "y_coef": 4.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2895.png",
+      "answer": "[1,2]",
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+      "coords": {}
+    },
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+      "index": 2896,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 2897,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_focus_mismatch",
+      "params": {
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+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2897.png",
+      "answer": "1",
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+          0.0
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+          2.0
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+      "index": 2898,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 2899,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2899.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
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+          1.0
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+    {
+      "index": 2900,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2900.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;F2: Point;F1: Point;N: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, LeftPart(C));LeftFocus(C) = F1;RightFocus(C) = F2;DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0;MidPoint(LineSegmentOf(P, F2)) = N;Quadrant(N) = 1;PointOnCurve(N, OneOf(Asymptote(C)))",
+      "coords": {}
+    },
+    {
+      "index": 2901,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.6457513110645907,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2901.png",
+      "answer": "pm*sqrt(15)/7",
+      "fact_expr": "G: Hyperbola;H: Circle;F1: Point;P: Point;M: Point;Expression(G) = (x^2/9 - y^2/7 = 1);Expression(H) = (x^2 + y^2 = 16);LeftFocus(G) = F1;PointOnCurve(P,RightPart(G));Intersection(LineSegmentOf(P,F1),H) = M;VectorOf(F1, M) = VectorOf(M, P)",
+      "coords": {}
+    },
+    {
+      "index": 2902,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2902.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;E: Circle;c: Number;P: Point;F: Point;M: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(E) = (y^2 + (-c/3 + x)^2 = b^2/9);Coordinate(F) = (c, 0);RightFocus(C) = F;PointOnCurve(P, LeftPart(C));TangentPoint(LineOf(F, P), E) = M;VectorOf(P, M) = 2*VectorOf(M, F)",
+      "coords": {}
+    },
+    {
+      "index": 2903,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2903.png",
+      "answer": "10",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);H: Line;PointOnCurve(Focus(G), H);A: Point;B: Point;x1: Number;x2: Number;y1: Number;y2: Number;Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);Intersection(H, G) = {A,B};x1 + x2 =6",
+      "coords": {}
+    },
+    {
+      "index": 2904,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2904.png",
+      "answer": "x^2/20+y^2/4=1",
+      "fact_expr": "G: Ellipse;H: Point;C:Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);Coordinate(H) = (sqrt(5), sqrt(3));Focus(G)=Focus(C);PointOnCurve(H,C)",
+      "coords": {
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+          1.7320508075688772
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+      }
+    },
+    {
+      "index": 2905,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2905.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;F: Point;Focus(G) = F;H: Line;PointOnCurve(F,H) = True;Intersection(H,G) = {M,N};M: Point;N: Point;Quadrant(M) = 1;Slope(LineOf(M,N)) = sqrt(3);XCoordinate(M) = 6",
+      "coords": {}
+    },
+    {
+      "index": 2906,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2906.png",
+      "answer": "3",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Focus(C)={F1,F2};PointOnCurve(P, C);IsPerpendicular(VectorOf(P, F2),VectorOf(P, F1));Area(TriangleOf(P, F1, F2)) = 9",
+      "coords": {}
+    },
+    {
+      "index": 2907,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2907.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;a: Number;H: Line;A: Point;B: Point;x1: Number;x2: Number;y1: Number;y2: Number;Expression(G) = (y^2 = a*x);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);PointOnCurve(Focus(G), H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = 10;x1 + x2 = 8",
+      "coords": {}
+    },
+    {
+      "index": 2908,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 2909,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2909.png",
+      "answer": "-1",
+      "fact_expr": "G: Parabola;H: Line;a: Real;Expression(G) = (y^2 = 4*x);Expression(H) = (a*x - y + 1 = 0);PointOnCurve(Focus(G), H)",
+      "coords": {}
+    },
+    {
+      "index": 2910,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2910.png",
+      "answer": "sqrt(3)-1",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G) = True;P: Point;AngleOf(P, F1, F2) = ApplyUnit(30, degree);AngleOf(P, F2, F1) = ApplyUnit(60, degree);e: Number;Eccentricity(G) = e",
+      "coords": {}
+    },
+    {
+      "index": 2911,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.7320508075688772,
+        "b": 1.0,
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+      "answer": "x^2/16 + y^2/11 = 1",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2924.png",
+      "answer": "1/2",
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+      "answer": "a^2",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2926.png",
+      "answer": "(\\sqrt{2}-1,1)",
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+      "answer": "1+2*sqrt(3)",
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+      "answer": "-3",
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+        "y_coef": 3.0
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+        "note": "using explicit params"
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+      "answer": "1/2",
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+      "index": 2930,
+      "success": true,
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+        "direction": "up"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2930.png",
+      "answer": "{(-oo, -3), (0, +oo)}",
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+      "index": 2932,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2935.png",
+      "answer": "3/4",
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+        "direction": "up"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2936.png",
+      "answer": "sqrt(5)-1",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2937.png",
+      "answer": "3*sqrt(5)/5",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2938.png",
+      "answer": "x^2/9 - y^2/16 = 1",
+      "fact_expr": "G: Hyperbola;a: Number;b: Number;H: Circle;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;IsDiameter(LineSegmentOf(F1, F2), H);Coordinate(OneOf(Intersection(H, Asymptote(G)))) = (3, 4)",
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+    {
+      "index": 2939,
+      "success": true,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2939.png",
+      "answer": "5/3",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;A:Point;F1: Point;P: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, RightPart(G));LeftFocus(G) = F1;RightFocus(G) = F2;Center(H)=O;O:Origin;Radius(H)=a;TangentPoint(LineOf(P,F1),H)=A;PointOnCurve(F2,PerpendicularBisector(LineSegmentOf(P,F1)))",
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+    {
+      "index": 2940,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 2941,
+      "success": true,
+      "conic_type": "hyperbola",
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+      "answer": "{18, 2}",
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+      "success": false,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2968.png",
+      "answer": "sqrt(14)/2",
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+      "success": true,
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2969.png",
+      "answer": "4",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/3 + y^2/4 = 1)",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2970.png",
+      "answer": "2*sqrt(5)/5",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1)",
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+      "answer": "2-sqrt(2)",
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+        "b": 2.23606797749979,
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+      "answer": "7",
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+      "answer": "2",
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+      "index": 2993,
+      "success": true,
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+      "answer": "sqrt(3)",
+      "fact_expr": "G: Hyperbola;a: Number;a>0;Expression(G) = (-y^2/9 + x^2/a^2 = 1);Eccentricity(G) = 2",
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+      "index": 2994,
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2994.png",
+      "answer": "2",
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+    {
+      "index": 2995,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 2996,
+      "success": true,
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+        "b": 3.0,
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2996.png",
+      "answer": "6",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1)",
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+      "index": 2997,
+      "success": true,
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+      "error": null,
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_2997.png",
+      "answer": "a",
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+    {
+      "index": 2998,
+      "success": true,
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+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_2998.png",
+      "answer": "sqrt(3)+1",
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+      "index": 2999,
+      "success": true,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_2999.png",
+      "answer": "pi/3",
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+      "coords": {}
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+    {
+      "index": 3000,
+      "success": true,
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+      "error": null,
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+        "b": 1.5,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3000.png",
+      "answer": "sqrt(34)/5",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;c:Number;c>0;Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);Focus(C)={F1,F2};M: Point;Coordinate(M) = (3*c, 0);l1: Line;PointOnCurve(M, l1);IsPerpendicular(l1,Asymptote(C));P: Point;FootPoint(l1,Asymptote(C))=P;O: Origin;Abs(LineSegmentOf(O, P)) = 5*b",
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+      "index": 3001,
+      "success": true,
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+      "error": null,
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+        "note": "using explicit params"
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+      "answer": "3/2",
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+      "index": 3002,
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+        "note": "using explicit params"
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+      "answer": "6*sqrt(2)",
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+      "index": 3003,
+      "success": true,
+      "conic_type": "parabola",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3003.png",
+      "answer": "2",
+      "fact_expr": "H: Line;Expression(H) = (y = k*x - 2);k: Number;G: Parabola;Expression(G) = (y^2 = 8*x);A: Point;B: Point;Negation(A=B);Intersection(H, G) = {A, B};XCoordinate(MidPoint(LineSegmentOf(A, B))) = 2",
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+      "answer": "8",
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+      "answer": "pm*y+2*x=0;sqrt(5)",
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+      "index": 3007,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3008.png",
+      "answer": "[\\sqrt{2}/2,1)",
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+    {
+      "index": 3009,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3009.png",
+      "answer": "3*sqrt(5)/5",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;C: Circle;Expression(C) = (y^2 + (x - 3)^2 = 4);l1:Line;l2:Line;Asymptote(G)={l1,l2};IsTangent(l1,C);IsTangent(l2,C)",
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+    {
+      "index": 3010,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3010.png",
+      "answer": "6;2",
+      "fact_expr": "G: Hyperbola;a: Number;H: Parabola;F: Point;P: Point;F2: Point;a>0;Expression(G) = (-y^2/5 + x^2/a^2 = 1);Expression(H) = (y^2 = 12*x);OneOf(Focus(G)) = F;OneOf(Focus(G)) = F2;Negation(F = F2);Focus(H) = F;PointOnCurve(P,G);Distance(P,F) = 2",
+      "coords": {}
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+    {
+      "index": 3011,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3011.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/2 + y^2 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 3012,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3012.png",
+      "answer": "0",
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+      "index": 3013,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 3014,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3014.png",
+      "answer": "(0, \\sqrt{2}/2)",
+      "fact_expr": "G: Ellipse;m: Number;n: Number;H: Curve;e: Number;m > n;n > 0;Expression(G) = (y^2/n + x^2/m = 1);Expression(H) = (x^2 + y^2 = m - n);NumIntersection(G, H)=0;Eccentricity(G) = e",
+      "coords": {}
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+    {
+      "index": 3015,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3016,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3016.png",
+      "answer": "(3-sqrt(3))/3",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;A: Point;P: Point;F: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;LeftVertex(C) = A;PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(P,F),xAxis);Slope(LineOf(A,P)) = sqrt(3)/3",
+      "coords": {}
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+    {
+      "index": 3017,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3017.png",
+      "answer": "2*sqrt(5)",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, C) = {A, B};Abs(LineSegmentOf(A, B)) = 6;G: Circle;IsDiameter(LineSegmentOf(A, B), G)",
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+    {
+      "index": 3018,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3018.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;H: Line;PointOnCurve(F, H);Intersection(H, G) = {A, B};A: Point;B: Point;Abs(LineSegmentOf(F, A))*Abs(LineSegmentOf(F, B)) = 6;Abs(LineSegmentOf(A, B)) = 6",
+      "coords": {}
+    },
+    {
+      "index": 3019,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3019.png",
+      "answer": "y^2=8*x",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;O: Origin;M: Point;PointOnCurve(M, G);Abs(LineSegmentOf(M, F)) = 4*Abs(LineSegmentOf(O, F));Area(TriangleOf(M, F, O)) = 4*sqrt(3)",
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+    {
+      "index": 3020,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 3021,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3021.png",
+      "answer": "sqrt(6)/3",
+      "fact_expr": "l: Line;C: Ellipse;b: Number;a: Number;A: Point;B: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Slope(l) = -1/3;Intersection(l, C) = {A, B};Coordinate(MidPoint(LineSegmentOf(A,B))) = (1, 1)",
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+    {
+      "index": 3022,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3023,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3023.png",
+      "answer": "5/2",
+      "fact_expr": "C: Parabola;F: Point;Expression(C) = (y^2 = 2*x);Focus(C) = F;P: Point;PointOnCurve(P, C);a: Number;Abs(LineSegmentOf(P, F)) = a;a > 1/2;l: Line;Directrix(C) = l;M: Point;PointOnCurve(P,LineSegmentOf(P,M));IsPerpendicular(LineSegmentOf(P,M),l);FootPoint(LineSegmentOf(P,M),l) = M;Area(InscribedCircle(TriangleOf(M,P,F))) = (3-sqrt(5))*a*pi/4",
+      "coords": {}
+    },
+    {
+      "index": 3024,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
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+          0.0
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+        "radius": 1.4142135623730951
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+      "optimization": {
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+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_3024.png",
+      "answer": "0",
+      "fact_expr": "G: Circle;H: Line;O: Origin;A: Point;B: Point;Expression(G) = (x^2 + y^2 = 2);Expression(H) = (x + 2*y + sqrt(5) = 0);Intersection(H, G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 3025,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3025.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;HalfFocalLength(G) = c;c: Number;H: Line;Expression(H) = (a*y + b*x = a*b);O: Origin;Distance(O, H) = c/4 + 1",
+      "coords": {}
+    },
+    {
+      "index": 3026,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 36.0,
+        "y_coef": 9.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3026.png",
+      "answer": "-1/2",
+      "fact_expr": "G: Ellipse;H: LineSegment;I: Point;Expression(G) = (x^2/36 + y^2/9 = 1);Coordinate(I) = (4, 2);IsChordOf(H, G);MidPoint(H) = I",
+      "coords": {
+        "I": [
+          4.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 3027,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3027.png",
+      "answer": "(1, 2)",
+      "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (3, 2);Focus(G) = F;PointOnCurve(P, G);WhenMin(Abs(LineSegmentOf(P, A))+Abs(LineSegmentOf(P, F)))",
+      "coords": {
+        "A": [
+          3.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 3028,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3028.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Point;F: Point;A: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F;PointOnCurve(F,LineSegmentOf(F,H));IsPerpendicular(LineSegmentOf(F,H),Asymptote(G));FootPoint(LineSegmentOf(F,H),Asymptote(G))=H;Intersection(LineSegmentOf(F,H),RightPart(C))=A;MidPoint(LineSegmentOf(F,H))=A",
+      "coords": {}
+    },
+    {
+      "index": 3029,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3030,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3030.png",
+      "answer": "5/3",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;I: Line;A: Point;F1: Point;F2: Point;b > a;a > 0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(RightFocus(G), I);IsParallel(I, OneOf(Asymptote(G)));Intersection(I, G) = A;Radius(InscribedCircle(TriangleOf(A, F1, F2))) = b/4",
+      "coords": {}
+    },
+    {
+      "index": 3031,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.25,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3031.png",
+      "answer": "3*sqrt(3)/2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 9*x);F: Point;Focus(G) = F;Intersection(Directrix(G), xAxis) = M;M: Point;PointOnCurve(N, G) = True;N: Point;sqrt(6)*Abs(LineSegmentOf(N, F)) = 2*Abs(LineSegmentOf(M, N))",
+      "coords": {}
+    },
+    {
+      "index": 3032,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3032.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 + y^2/2 = 1);Focus(G)={F1,F2};PointOnCurve(P,G);Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 3033,
+      "success": true,
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+        "note": "using explicit params"
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+      "answer": "5",
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+      "answer": "2*sqrt(2)",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3035.png",
+      "answer": "2*sqrt(10)/5",
+      "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 + y^2/9 = 1);PointOnCurve(P, G);Focus(G) = {F1,F2};Cos(AngleOf(F1, P, F2)) = 1/3",
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+    {
+      "index": 3036,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3036.png",
+      "answer": "5*sqrt(3)/3",
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+    {
+      "index": 3037,
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+        "note": "using explicit params"
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+      "answer": "sqrt(3)-1",
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+      "answer": "2",
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+      "answer": "12",
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+      "answer": "3+2*sqrt(2)",
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+      "answer": "2*sqrt(7)",
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+      "answer": "sqrt(2)",
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+      "answer": "8",
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+      "answer": "16/3",
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+      "success": false,
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+      "answer": "(-12,0)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3048.png",
+      "answer": "sqrt(15)/15",
+      "fact_expr": "F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;C: Hyperbola;Expression(C) = (x^2/4 - y^2 = 1);a: Number;b: Number;a>0;b>0;P: Point;PointOnCurve(P, C) = True;AngleOf(F1, P, F2) = ApplyUnit(120, degree)",
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+      "index": 3049,
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+      "answer": "x^2/9-y^2=1",
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+      "answer": "12",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 3052,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+        "iterations": 2
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3053.png",
+      "answer": "3/2",
+      "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2/2 + y^2/m = 1);PointOnCurve(Focus(G), xAxis);Eccentricity(G) = 1/2",
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+      "success": false,
+      "error": "Circle from diameter: missing point coordinates"
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 3056,
+      "success": true,
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+      "error": null,
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3056.png",
+      "answer": "sqrt(15)",
+      "fact_expr": "G: Parabola;H: Line;Expression(G) = (y^2 = 12*x);Expression(H) = (y = 2*x + 1)",
+      "coords": {}
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3057.png",
+      "answer": "2",
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3058.png",
+      "answer": "y=pm*sqrt(2)*x",
+      "fact_expr": "G: Hyperbola;a: Number;Expression(G) = (-y^2/2 + x^2/a = 1);Coordinate(OneOf(Focus(G))) = (-sqrt(3), 0)",
+      "coords": {}
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+      "index": 3059,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3059.png",
+      "answer": "x^2/2 - y^2 = 1",
+      "fact_expr": "G: Hyperbola;H: Ellipse;P: Point;Expression(H) = (x^2/4 + y^2 = 1);Coordinate(P) = (2, 1);Focus(H) = Focus(G);PointOnCurve(P, G)",
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+          1.0
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+      "index": 3060,
+      "success": true,
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+        "b": 1.4142135623730951,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3060.png",
+      "answer": "2*sqrt(3)",
+      "fact_expr": "C: Hyperbola;P: Point;F: Point;O: Origin;Expression(C) = (x^2/4 - y^2/2 = 1);RightFocus(C) = F;PointOnCurve(P, OneOf(Asymptote(C)));Abs(LineSegmentOf(O, P)) = 2*Abs(LineSegmentOf(O, F))",
+      "coords": {}
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+    {
+      "index": 3061,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3061.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
+      "fact_expr": "F1: Point;F2: Point;C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Focus(C) = {F1, F2};P: Point;PointOnCurve(P, C) = True;IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2)) = True;AngleOf(P, F1, F2) = ApplyUnit(30, degree)",
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+    {
+      "index": 3062,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3062.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;p: Number;H: Line;p>0;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (y = x + 1);PointOnCurve(OneOf(Intersection(H,axis)),Directrix(G))",
+      "coords": {}
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+    {
+      "index": 3063,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 20.0,
+        "y_coef": 16.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3063.png",
+      "answer": "20",
+      "fact_expr": "C: Ellipse;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/20 + y^2/16 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);m:Number;Abs(VectorOf(P,F1))*Abs(VectorOf(P,F2))=m",
+      "coords": {}
+    },
+    {
+      "index": 3064,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3064.png",
+      "answer": "8",
+      "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);PointOnCurve(Focus(G), H);Intersection(H, G) = {A,B};x1+x2=6;x1:Number;x2:Number;y1:Number;y2:Number",
+      "coords": {}
+    },
+    {
+      "index": 3065,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3065.png",
+      "answer": "5+sqrt(17)",
+      "fact_expr": "C: Parabola;M: Point;P: Point;F: Point;Expression(C) = (x^2 = 8*y);Coordinate(M) = (-4, 3);Focus(C) = F;PointOnCurve(P, C)",
+      "coords": {
+        "M": [
+          -4.0,
+          3.0
+        ]
+      }
+    },
+    {
+      "index": 3066,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3066.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "l: Line;E: Parabola;A: Point;C: Point;B: Point;Expression(E) = (y^2 = 4*x);PointOnCurve(Focus(E), l);Intersection(l, E) = {A, B};Intersection(l,Directrix(E))=C;Quadrant(A)=1;MidPoint(LineSegmentOf(A, C)) = B",
+      "coords": {}
+    },
+    {
+      "index": 3067,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 3.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3067.png",
+      "answer": "16.5",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1);P: Point;PointOnCurve(P, G);A: Point;Coordinate(A) = (5, 0);Distance(P, A) = 8.5;B: Point;Coordinate(B) = (-5, 0)",
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+        "A": [
+          5.0,
+          0.0
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+        "B": [
+          -5.0,
+          0.0
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+    },
+    {
+      "index": 3068,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 4.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3068.png",
+      "answer": "4",
+      "fact_expr": "C: Ellipse;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/16 + y^2/4 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 3069,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "vertical"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3069.png",
+      "answer": "sqrt(13)/3",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-x^2/b^2 + y^2/a^2 = 1);Expression(OneOf(Asymptote(C))) = (3*x - 2*y = 0)",
+      "coords": {}
+    },
+    {
+      "index": 3070,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3070.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, L) = True;Intersection(L, LeftPart(G)) = M;Intersection(L, RightPart(G)) = N;L: Line;M: Point;N: Point;IsDiameter(LineSegmentOf(M, N), H) = True;H: Circle;PointOnCurve(F2, H) = True;Abs(LineSegmentOf(M, F2)) = Abs(LineSegmentOf(N, F2))",
+      "coords": {}
+    },
+    {
+      "index": 3071,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3071.png",
+      "answer": "6",
+      "fact_expr": "l: Line;C: Parabola;p: Number;p>0;Expression(C) = (y^2 = 2*(p*x));Expression(l)=(x + 2*y - 3 = 0);PointOnCurve(Focus(C),l)",
+      "coords": {}
+    },
+    {
+      "index": 3072,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3073,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "down"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3073.png",
+      "answer": "(0,-2)",
+      "fact_expr": "G: Parabola;Expression(G) = (y = -x^2/8)",
+      "coords": {}
+    },
+    {
+      "index": 3074,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3074.png",
+      "answer": "sqrt(3)/2",
+      "fact_expr": "E: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(E) = F1;RightFocus(E) = F2;PointOnCurve(P, E);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0;Area(TriangleOf(P, F1, F2)) = (a*b)/2",
+      "coords": {}
+    },
+    {
+      "index": 3075,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_focus_mismatch",
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3075.png",
+      "answer": "y=pm*sqrt(3)*x",
+      "fact_expr": "C: Hyperbola;a: Number;F: Point;a>0;Expression(C) = (-y^2/3 + x^2/a^2 = 1);Coordinate(F) = (2, 0);OneOf(Focus(C)) = F",
+      "coords": {
+        "F": [
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+          0.0
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+      "validation_reasons": [
+        "hyperbola_focus_mismatch"
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+    {
+      "index": 3076,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 5.0,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3076.png",
+      "answer": "x^2=20*y",
+      "fact_expr": "G: Parabola;O: Origin;Vertex(G) = O;F: Point;Coordinate(F) = (0, 5);Focus(G) = F",
+      "coords": {
+        "F": [
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+          5.0
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+    },
+    {
+      "index": 3077,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 2.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3077.png",
+      "answer": "24",
+      "fact_expr": "l: Line;PointOnCurve(RightFocus(G), l);G: Hyperbola;Expression(G) = (x^2/16 - y^2/4 = 1);A: Point;B: Point;Intersection(l, RightPart(G)) = {A, B};Abs(LineSegmentOf(A, B)) = 4;F: Point;LeftFocus(G) = F",
+      "coords": {}
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+    {
+      "index": 3078,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 3079,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 3.0,
+        "major_axis": "x",
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+        "y_coef": 9.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3079.png",
+      "answer": "-9/16",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/9 = 1);A: Point;B: Point;LeftVertex(G) = A;RightVertex(G) = B;P: Point;PointOnCurve(P, G)",
+      "coords": {}
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+    {
+      "index": 3080,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3080.png",
+      "answer": "4*sqrt(3)",
+      "fact_expr": "C: Parabola;P: Point;F: Point;O: Origin;Expression(C) = (y^2 = 8*x);Focus(C) = F;PointOnCurve(P, C);Abs(LineSegmentOf(P, F)) = 6",
+      "coords": {}
+    },
+    {
+      "index": 3081,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3081.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;A: Point;P: Point;B: Point;Q: Point;F: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F;LeftVertex(G)=A;RightVertex(G)=B;PointOnCurve(F,H);IsPerpendicular(H,xAxis);Intersection(H,G)={P,Q};IsPerpendicular(LineSegmentOf(A, P), LineSegmentOf(B, Q))",
+      "coords": {}
+    },
+    {
+      "index": 3082,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 3.0,
+        "y_coef": 1.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3082.png",
+      "answer": "[2*sqrt(2),2*sqrt(3))\n0",
+      "fact_expr": "C: Ellipse;G: Line;y0: Number;x0: Number;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/3 + y^2 = 1);Expression(G) = (y*y0 + (x*x0)/3 = 1);Coordinate(P) = (x0, y0);Focus(C) = {F1,F2};0<x0^2/3+y0^2;x0^2/3+y0^2<1",
+      "coords": {}
+    },
+    {
+      "index": 3083,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 6.0,
+        "b": 5.196152422706632,
+        "major_axis": "x",
+        "x_coef": 36.0,
+        "y_coef": 27.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3083.png",
+      "answer": "90",
+      "fact_expr": "G: Ellipse;C1:Circle;C2:Circle;P: Point;M: Point;N: Point;L1:Line;L2:Line;Expression(G) = (x^2/36 + y^2/27 = 1);Expression(C1)=((x+3)^2+y^2=4);Expression(C2)=((x-3)^2+y^2=1);PointOnCurve(P,G);TangentOfPoint(P,C1)=L1;TangentOfPoint(P,C2)=L2;TangentPoint(L1,C1)=M;TangentPoint(L2,C2)=N",
+      "coords": {}
+    },
+    {
+      "index": 3084,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3084.png",
+      "answer": "pi/6",
+      "fact_expr": "G: Parabola;N: Point;F: Point;M: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;Intersection(Directrix(G), xAxis) = M;PointOnCurve(N, G);Abs(LineSegmentOf(N, F)) = (sqrt(3)/2)*Abs(LineSegmentOf(M, N))",
+      "coords": {}
+    },
+    {
+      "index": 3085,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3085.png",
+      "answer": "3/5",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;M: Circle;Expression(M) = (y^2 + (x - 3)^2 = 1);OneOf(Focus(C)) = Center(M);Length(MajorAxis(C)) = 10",
+      "coords": {}
+    },
+    {
+      "index": 3086,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3087,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_focus_mismatch",
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3087.png",
+      "answer": "x^2/2-y^2=1",
+      "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;F: Point;A: Point;B: Point;O: Origin;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (-sqrt(3), 0);LeftFocus(C) = F;Intersection(l,LeftPart(C)) = A;Intersection(l,RightPart(C)) = B;Abs(LineSegmentOf(F, A)) + Abs(LineSegmentOf(F, B)) = 4;DotProduct(VectorOf(F, A), VectorOf(F,B)) = 0;PointOnCurve(O,l)",
+      "coords": {
+        "F": [
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+          0.0
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+      },
+      "validation_reasons": [
+        "hyperbola_focus_mismatch"
+      ]
+    },
+    {
+      "index": 3088,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3088.png",
+      "answer": "(x-1)^2+y^2=1",
+      "fact_expr": "H: Circle;Expression(H) = ((-a + x)^2 + (-b + y)^2 = r^2);b: Number;a: Number;r: Number;G: Parabola;Expression(G) = (y^2 = 4*x);Center(H) = Focus(G);L: Line;Expression(L) = (3*x + 4*y + 2 = 0);IsTangent(H, L)",
+      "coords": {}
+    },
+    {
+      "index": 3089,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3089.png",
+      "answer": "sqrt(34) - 1",
+      "fact_expr": "G: Parabola;P: Point;Q: Point;M: Point;Expression(G) = (y^2 = 4*x);Coordinate(M) = (4, 5);PointOnCurve(P, G);Projection(P, yAxis) = Q",
+      "coords": {
+        "M": [
+          4.0,
+          5.0
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+      }
+    },
+    {
+      "index": 3090,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3091,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3092,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.7320508075688772,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3092.png",
+      "answer": "y^2 = pm*6*x",
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+      "answer": "x \\pm \\sqrt{3} y - 1 = 0",
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+      "answer": "x^2/3 - y^2/6 = 1",
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+      "answer": "(1, 2]",
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+      "answer": "x^2/4-y^2=1",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3115.png",
+      "answer": "(-oo, sqrt(2)]",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3116.png",
+      "answer": "sqrt(5)/2",
+      "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (y^2/4 - x^2/m = 1);Expression(OneOf(Asymptote(G))) = (y = 2*x)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3117.png",
+      "answer": "(x-2)^2+(y-2*sqrt(2))^2=9",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);C: Circle;Expression(C) = ((-a + x)^2 + (-b + y)^2 = r^2);r: Number;b: Number;a: Number;A: Point;b>0;PointOnCurve(A,C);Coordinate(A) = (3, 0);PointOnCurve(Center(C),G);IsTangent(C,Directrix(G))",
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+      "index": 3118,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3118.png",
+      "answer": "A straight line does not exist",
+      "fact_expr": "l: Line;G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1);A: Point;B: Point;Intersection(l, G) = {A, B};P: Point;Coordinate(P) = (6, 3);MidPoint(LineSegmentOf(A, B)) = P",
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+      "index": 3119,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3119.png",
+      "answer": "sqrt(5)/3",
+      "fact_expr": "P: Point;PointOnCurve(P,G) = True;G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Focus(G) = {F1,F2};F1: Point;F2: Point;DotProduct(VectorOf(P,F1),VectorOf(P,F2))=0;Tan(AngleOf(P,F1,F2)) = 1/2",
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+      "answer": "-4",
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+      "answer": "sqrt(5)/2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1)",
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+    {
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+      "answer": "x^2/16 - y^2/2 = 1",
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+      "index": 3132,
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+      "answer": "(4,\\pm 4)",
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+        "b": 1.5,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3134.png",
+      "answer": "sqrt(5)",
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+    {
+      "index": 3135,
+      "success": true,
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+      "answer": "{ApplyUnit(60, degree), ApplyUnit(120, degree)}",
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+      "index": 3136,
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+      "answer": "8",
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+      "index": 3139,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3139.png",
+      "answer": "5*sqrt(2)",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/2 + y^2 = 1);F: Point;LeftFocus(C) = F;P: Point;PointOnCurve(P, C);Q: Point;Coordinate(Q) = (4, 3)",
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+      "index": 3140,
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+        "direction": "right"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3140.png",
+      "answer": "(0, 2)",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3141.png",
+      "answer": "sqrt(5)",
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+      "index": 3142,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3142.png",
+      "answer": "(0,2*sqrt(2))",
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+    {
+      "index": 3143,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3143.png",
+      "answer": "4*sqrt(3)/3",
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+      "index": 3144,
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+      "answer": "4",
+      "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/25 + y^2/16 = 1);PointOnCurve(P, G);Distance(P, RightDirectrix(G)) = 10",
+      "coords": {}
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+    {
+      "index": 3145,
+      "success": true,
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+      "error": null,
+      "params": {
+        "p": 0.125,
+        "direction": "up"
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3145.png",
+      "answer": "{(5/4, +oo), 0}",
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+    {
+      "index": 3146,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 3147,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 2.0,
+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3147.png",
+      "answer": "{5/2, pm*2}",
+      "fact_expr": "l: Line;G: Hyperbola;P: Point;Expression(G) = (x^2 - y^2/4 = 1);Coordinate(P) = (1, 1);PointOnCurve(P, l);Slope(l) = k;NumIntersection(l, G) = 1;k: Number",
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+      "index": 3148,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 3149,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3149.png",
+      "answer": "(\\sqrt{3} - 1)/2",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;c: Number;F1: Point;F2: Point;P: Point;e: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 2*c;AngleOf(F1, P, F2) = ApplyUnit(30, degree);Eccentricity(G) = e",
+      "coords": {}
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+    {
+      "index": 3150,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3150.png",
+      "answer": "sqrt(3)/2",
+      "fact_expr": "L: Line;Slope(L) = -1/2;C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;A: Point;B: Point;Intersection(L, C) = {A, B};P: Point;Coordinate(P) = (2, 1);MidPoint(LineSegmentOf(A, B)) = P",
+      "coords": {
+        "P": [
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+          1.0
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+    {
+      "index": 3151,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 3152,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "left"
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+      "answer": "sqrt(3)/3",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3176.png",
+      "answer": "pm*x-3*y=0",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3181.png",
+      "answer": "sqrt(13)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3183.png",
+      "answer": "2/3",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Eccentricity(G)=e;a <=sqrt(3)*b;e:Number",
+      "coords": {}
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+      "index": 3184,
+      "success": true,
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3184.png",
+      "answer": "3",
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+      "index": 3185,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 3186,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3186.png",
+      "answer": "7*sqrt(3)/2",
+      "fact_expr": "F1: Point;F2: Point;Focus(G) = {F1, F2};G: Ellipse;Expression(G) = (x^2/12 + y^2/3 = 1);P: Point;PointOnCurve(P, G);PointOnCurve(MidPoint(LineSegmentOf(P, F1)), yAxis)",
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3187.png",
+      "answer": "sqrt(13)/2",
+      "fact_expr": "E: Hyperbola;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(E) = F1;RightFocus(E) = F2;A: Point;PointOnCurve(A, LeftPart(E));AngleOf(F1, A, F2) = ApplyUnit(120, degree);Abs(LineSegmentOf(A, F2)) = 3*Abs(LineSegmentOf(A, F1))",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3188.png",
+      "answer": "x^2+(y-1)^2=2",
+      "fact_expr": "G: Parabola;H: Circle;L: Line;Center(H)=Focus(G);Expression(G) = (x^2 = 4*y);Expression(L) = (y = x + 3);IsTangent(L,H)",
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+    {
+      "index": 3189,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3189.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(RightFocus(G), H);Inclination(H)=ApplyUnit(60,degree);NumIntersection(H,G)=1",
+      "coords": {}
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+    {
+      "index": 3190,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 3191,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 3192,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "y_coef": 16.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3192.png",
+      "answer": "15",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);M: Point;Coordinate(M) = (6, 4)",
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+      "index": 3193,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 3194,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 3195,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "y_coef": 5.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3195.png",
+      "answer": "6",
+      "fact_expr": "G: Ellipse;P: Point;F: Point;Expression(G) = (x^2/9 + y^2/5 = 1);PointOnCurve(P, G);LeftFocus(G) = F;Distance(P, F) = 2",
+      "coords": {}
+    },
+    {
+      "index": 3196,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 2.0,
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+        "y_coef": 4.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3196.png",
+      "answer": "4*sqrt(5)/5-1",
+      "fact_expr": "G: Ellipse;H: Circle;P: Point;Q: Point;Expression(G) = (x^2/9 + y^2/4 = 1);Expression(H) = (y^2 + (x - 1)^2 = 1);PointOnCurve(P, H);PointOnCurve(Q, G)",
+      "coords": {}
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+    {
+      "index": 3197,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3197.png",
+      "answer": "y=pm*((sqrt(15)/5)*x)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;A: Point;PointOnCurve(A, G);O: Origin;Slope(LineOf(O, A)) = sqrt(3)/3;PointOnCurve(Vertex(G), PerpendicularBisector(LineSegmentOf(O, A)))",
+      "coords": {}
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+    {
+      "index": 3198,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3198.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "C: Hyperbola;F: Point;Expression(C) = (x^2/5 - y^2/3 = 1);RightFocus(C) = F",
+      "coords": {}
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+    {
+      "index": 3199,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3199.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 3200,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 3201,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
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+          2.0
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+        "radius": 1.0
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_3201.png",
+      "answer": "(x^2 + (y-1)^2 = 1)&(3/2 < y <= 2)",
+      "fact_expr": "l: Line;G: Circle;A: Point;B: Point;O: Origin;M: Point;Expression(G) = (-4*y + x^2 + y^2 + 3 = 0);PointOnCurve(O, l);Intersection(l, G) = {A, B};IsChordOf(LineSegmentOf(A, B), G);MidPoint(LineSegmentOf(A,B)) = M",
+      "coords": {}
+    },
+    {
+      "index": 3202,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3202.png",
+      "answer": "y=pm*sqrt(2)*x",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;FocalLength(C) = sqrt(3)*Length(RealAxis(C))",
+      "coords": {}
+    },
+    {
+      "index": 3203,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3203.png",
+      "answer": "5",
+      "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (x^2/4 - y^2/m = 1);FocalLength(G) = 6",
+      "coords": {}
+    },
+    {
+      "index": 3204,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "up"
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+      "optimization": {
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+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3204.png",
+      "answer": "4/9",
+      "fact_expr": "M: Parabola;Expression(M) = (x^2 = 2*p*y);p: Number;p>0;F: Point;Focus(M) = F;PointOnCurve(F, l) = True;Slope(l) = 5/12;l: Line;Intersection(l, M) = {A, B};A: Point;B: Point;Quadrant(A) = 2",
+      "coords": {}
+    },
+    {
+      "index": 3205,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3205.png",
+      "answer": "8",
+      "fact_expr": "C: Parabola;G: Line;A: Point;B: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F, G);Slope(G) = 1;Intersection(G, C) = {A, B}",
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+    },
+    {
+      "index": 3206,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3206.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
+      "fact_expr": "G: Parabola;A: Point;B: Point;M: Point;Expression(G) = (y = x^2);IsChordOf(LineSegmentOf(A, B), G);Length(LineSegmentOf(A, B)) = 2;MidPoint(LineSegmentOf(A, B)) = M",
+      "coords": {}
+    },
+    {
+      "index": 3207,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 4.47213595499958,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3207.png",
+      "answer": "14",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/20 = 1);M: Point;PointOnCurve(M, G);F1: Point;F2: Point;OneOf(Focus(G)) = F1;OneOf(Focus(G)) = F2;Negation(F1=F2);Distance(M, F1) = 6",
+      "coords": {}
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+    {
+      "index": 3208,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 3.0,
+        "y_coef": 2.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3208.png",
+      "answer": "pm*sqrt(6)/3",
+      "fact_expr": "H: Line;Expression(H) = (y = k*x + 2);k: Number;G: Ellipse;Expression(G) = (x^2/3 + y^2/2 = 1);NumIntersection(H, G) = 1;Slope(H) = k",
+      "coords": {}
+    },
+    {
+      "index": 3209,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3210,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3210.png",
+      "answer": "y=pm*(2*sqrt(2)/3)*x",
+      "fact_expr": "H: Ellipse;Expression(H) = (y^2/b^2 + x^2/a^2 = 1);Eccentricity(H) = 1/3;b: Number;a: Number;a > b;b > 0;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 3211,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3211.png",
+      "answer": "1/2",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;A: Point;B: Point;F: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(A)=(-a,0);Coordinate(B) = (0, b);LeftFocus(G)=F;In(A,Vertex(G));Distance(F, LineSegmentOf(A, B)) = b/sqrt(7);In(B,Vertex(G))",
+      "coords": {}
+    },
+    {
+      "index": 3212,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3213,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "y",
+        "x_coef": 1.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3213.png",
+      "answer": "2",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2 + y^2/16 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 3214,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3215,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3216,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3217,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3217.png",
+      "answer": "(1,1)",
+      "fact_expr": "G: Parabola;Expression(G) = (y = x^2);H: Line;Expression(H) = (2*x - y - 4 = 0);P: Point;PointOnCurve(P,G) = True;WhenMin(Distance(P,H)) = True",
+      "coords": {}
+    },
+    {
+      "index": 3218,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+      "answer": "81/4",
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+      "index": 3241,
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+      "answer": "1/9",
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+      "answer": "y=pm*sqrt(3)*x",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3249.png",
+      "answer": "sqrt(10)/2",
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+      "index": 3250,
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+      "answer": "(0,2)",
+      "fact_expr": "G: Parabola;Expression(G) = (y = x^2/8)",
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+      "answer": "sqrt(2)/2",
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+      "answer": "64",
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+      "answer": "32/3",
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+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3262.png",
+      "answer": "2*sqrt(3)",
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+    {
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+      "answer": "(-4, 9/5)",
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+      "index": 3264,
+      "success": false,
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+    {
+      "index": 3265,
+      "success": true,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3265.png",
+      "answer": "sqrt(21)/7",
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+      "index": 3266,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3266.png",
+      "answer": "4",
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+      "index": 3268,
+      "success": true,
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+      "error": null,
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+        "direction": "up"
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+        "note": "using explicit params"
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+      "answer": "8",
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+      "answer": "2*pi/3",
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3270.png",
+      "answer": "sqrt(3)",
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+      "index": 3271,
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+        "y_coef": 9.0
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3271.png",
+      "answer": "18",
+      "fact_expr": "C: Ellipse;A: Point;F1: Point;F2: Point;Expression(C) = (x^2/25 + y^2/9 = 1);LeftFocus(C)=F1;RightFocus(C)=F2;PointOnCurve(A,C)",
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+      "index": 3272,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3272.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "C: Ellipse;A: Point;F: Point;B: Point;l: Line;Expression(C) = (x^2/2 + y^2 = 1);RightFocus(C) = F;RightDirectrix(C) = l;In(A, l);Intersection(LineSegmentOf(A, F), C) = B;VectorOf(F, A) = 3*VectorOf(F, B)",
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+      "index": 3273,
+      "success": true,
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+      "error": null,
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3273.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;H: Line;PointOnCurve(F, H);Slope(H) = 1;A: Point;B: Point;Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, F))*Abs(LineSegmentOf(B, F)) = 8",
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+    {
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+      "error": null,
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+      "answer": "x^2/3 - y^2/6 = 1",
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+    {
+      "index": 3275,
+      "success": true,
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+      "error": null,
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+        "b": 3.0,
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+        "y_coef": 9.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3275.png",
+      "answer": "3*sqrt(3)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);M: Point;PointOnCurve(M, G);F1: Point;F2: Point;Focus(G) = {F1, F2};AngleOf(F1, M, F2) = ApplyUnit(60, degree)",
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+    {
+      "index": 3276,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 3277,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "a": 1.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3277.png",
+      "answer": "3",
+      "fact_expr": "G: Hyperbola;A: Point;B: Point;F1: Point;Expression(G) = (x^2 - y^2/3 = 1);LeftFocus(G) = F1;PointOnCurve(F1,LineOf(A,B));Intersection(LineOf(A,B),G)={A,B};Inclination(LineOf(A,B))=pi/6",
+      "coords": {}
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+    {
+      "index": 3278,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3278.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(P) = (2, 0);Distance(P, OneOf(Asymptote(G))) = sqrt(2)",
+      "coords": {
+        "P": [
+          2.0,
+          0.0
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+    {
+      "index": 3279,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3279.png",
+      "answer": "1/2",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = y)",
+      "coords": {}
+    },
+    {
+      "index": 3280,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3280.png",
+      "answer": "3*sqrt(21)/14",
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+      "answer": "(pm*6,0)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/100 + y^2/64 = 1)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3282.png",
+      "answer": "9",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3283.png",
+      "answer": "sqrt(5)/2",
+      "fact_expr": "G: Parabola;H: Circle;C:Line;P: Point;F: Point;Expression(G) = (y^2 = 4*x);Expression(C) = (x + y = 2);Focus(G)=F;PointOnCurve(P,G);Quadrant(P)=1;IsDiameter(LineSegmentOf(P,F),H);PointOnCurve(Center(H),C)",
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+    {
+      "index": 3284,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3284.png",
+      "answer": "2*sqrt(5)-1",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3285.png",
+      "answer": "16/3",
+      "fact_expr": "l: Line;G: Parabola;Expression(G) = (y^2 = 8*x);F: Point;Focus(G) = F;PointOnCurve(F, l) = True;Intersection(l,G) = {A,B};B: Point;A: Point;Intersection(l,Directrix(G)) = C;C: Point;MidPoint(LineSegmentOf(A, C)) = F",
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+    {
+      "index": 3286,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3286.png",
+      "answer": "{5/4, 3}",
+      "fact_expr": "C: Parabola;p: Number;P: Point;F: Point;p>0;m:Number;Expression(C) = (y^2 = 2*(p*x));Coordinate(P) = (2, m);Focus(C) = F;PointOnCurve(P, C);O:Origin;d:Number;Distance(P,O)=d;d/Abs(LineSegmentOf(P, F)) = 8/7",
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+    {
+      "index": 3287,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3287.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/6 + y^2/3 = 1)",
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+      "answer": "(5*sqrt(3))/2",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/9 + y^2/5 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C);AngleOf(P, F1, F2) = ApplyUnit(60, degree)",
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+    {
+      "index": 3289,
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+        "iterations": 89
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3289.png",
+      "answer": "(pm*4,0)\ny=pm*sqrt(3)*x",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 16*x);Eccentricity(G) = 2;OneOf(Focus(G)) = Focus(H)",
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+    {
+      "index": 3290,
+      "success": true,
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3290.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);K: Point;PointOnCurve(K, G);H: Point;Coordinate(H) = (0, 2)",
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+          2.0
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+    {
+      "index": 3291,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3291.png",
+      "answer": "4",
+      "fact_expr": "G: Ellipse;F1: Point;F2:Point;P: Point;O: Origin;M: Point;Expression(G) = (y^2/16 + y^2/16 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);MidPoint(LineSegmentOf(F1, P)) = M;Abs(LineSegmentOf(O, M)) = 3",
+      "coords": {}
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+    {
+      "index": 3292,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3292.png",
+      "answer": "(1/4, -1)",
+      "fact_expr": "G: Parabola;P: Point;Q: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;Coordinate(Q) = (2, -1);PointOnCurve(P, G);WhenMin(Abs(LineSegmentOf(P, Q)) + Abs(LineSegmentOf(P, F)))",
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+          -1.0
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+        "b": 1.4142135623730951,
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+      "optimization": {
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3293.png",
+      "answer": "4/3",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/2 = 1);P:Point;PointOnCurve(P,G);L1:Line;L2:Line;Asymptote(G)={L1,L2};Distance(P,L1)=d1;Distance(P,L2)=d2;d1:Number;d2:Number",
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+    {
+      "index": 3294,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "p": 1.0,
+        "direction": "up"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3294.png",
+      "answer": "x^2=2*y-2",
+      "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;Expression(G) = (x^2 = 4*y);Focus(G)=F;PointOnCurve(F,l);Intersection(l,G) = {A,B};IsChordOf(LineSegmentOf(A,B),G);MidPoint(LineSegmentOf(A,B))=M;M:Point;F:Point",
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+    {
+      "index": 3295,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 3296,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 1.5,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3296.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;a: Number;H: Circle;a>0;Expression(G) = (-y^2/3 + x^2/a^2 = 1);Expression(H) = (y^2 + (x - 2)^2 = 4);Length(InterceptChord(OneOf(Asymptote(G)),H))=2",
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+    {
+      "index": 3297,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 3298,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3298.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;c: Number;FocalLength(G) = 2*c;F: Point;RightFocus(G) = F;O: Origin;P: Point;PointOnCurve(P, G);Abs(LineSegmentOf(P, O)) = c;Area(TriangleOf(P, O, F)) = (1/2)*(a*b)",
+      "coords": {}
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+    {
+      "index": 3299,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
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+        "final_loss": 0.0,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3299.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "G: Ellipse;C2: Circle;P: Point;M: Point;A: Point;Expression(G) = (x^2/25 + y^2/16 = 1);Expression(C2) = ((x-3)^2+y^2=1);Coordinate(A) = (3, 0);PointOnCurve(P, G);PointOnCurve(M, C2);IsPerpendicular(LineSegmentOf(P, M), LineSegmentOf(A, M))",
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+          0.0
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+    {
+      "index": 3300,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 3301,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3301.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "H: Ellipse;Expression(H) = (y^2/4 + x^2/a^2 = 1);F1: Point;F2: Point;Focus(H) = {F1, F2};a: Number;a>0;P: Point;PointOnCurve(P, H);G: Parabola;Expression(G) = (y^2 = x);PointOnCurve(P, G);F3: Point;Focus(G) = F3;Abs(LineSegmentOf(P, F3)) = 9/4",
+      "coords": {}
+    },
+    {
+      "index": 3302,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3302.png",
+      "answer": "x=3",
+      "fact_expr": "l: Line;G: Parabola;Expression(G) = (y^2 = 4*x);IsPerpendicular(l,xAxis);Length(InterceptChord(l,G))=4*sqrt(3)",
+      "coords": {}
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+    {
+      "index": 3303,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3303.png",
+      "answer": "(x-2)^2+(y-4)^2=16",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;P: Point;Coordinate(P) = (2, m);m: Number;m>0;PointOnCurve(P, C);F: Point;Focus(C)=F;Distance(P,F) = 4;G: Circle;Center(G) = P;IsTangent(G, Directrix(C))",
+      "coords": {}
+    },
+    {
+      "index": 3304,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3304.png",
+      "answer": "2",
+      "fact_expr": "G: Ellipse;a: Number;b: Number;O: Origin;D: Point;A: Point;B: Point;C: Point;E: Point;F: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (1, 0);Eccentricity(G) = sqrt(2)/2;RightFocus(G) = F;PointOnCurve(Vertex(TriangleOf(A, B, C)), G);MidPoint(LineSegmentOf(A, B)) = D;MidPoint(LineSegmentOf(B, C)) = E;MidPoint(LineSegmentOf(A, C)) = F;k1: Number;k2: Number;k3: Number;Slope(OverlappingLine(LineSegmentOf(A, B))) = k1;Slope(OverlappingLine(LineSegmentOf(B, C))) = k2;Slope(OverlappingLine(LineSegmentOf(A, C))) = k3;Negation(k1*k2*k3 = 0);Slope(LineOf(O, D)) + Slope(LineOf(O, E)) + Slope(LineOf(O, F)) = -1",
+      "coords": {
+        "F": [
+          1.0,
+          0.0
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+      }
+    },
+    {
+      "index": 3305,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3305.png",
+      "answer": "3",
+      "fact_expr": "C: Parabola;p: Number;F: Point;l: Line;A: Point;B: Point;p > 0;Expression(C) = (y^2 = 2*p*x);Focus(C) = F;PointOnCurve(F,l);Intersection(l,C) = {A,B};Min(Abs(LineSegmentOf(A,B))) = 6",
+      "coords": {}
+    },
+    {
+      "index": 3306,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3306.png",
+      "answer": "5/3",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2 + y^2 = a^2);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, RightPart(C));IsTangent(LineOf(P, F1), G);Abs(LineSegmentOf(P, F2)) = Abs(LineSegmentOf(F1, F2))",
+      "coords": {}
+    },
+    {
+      "index": 3307,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3308,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "vertical"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3308.png",
+      "answer": "x^2/8 - y^2/6 = 1",
+      "fact_expr": "H: Point;Coordinate(H) = (3, 2);G: Hyperbola;Expression(G) = (-x^2/3 + y^2/4 = 1);Z: Hyperbola;Asymptote(Z) = Asymptote(G);PointOnCurve(H, Z)",
+      "coords": {
+        "H": [
+          3.0,
+          2.0
+        ]
+      }
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+      "answer": "-1/2",
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+      "answer": "8",
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+      "answer": "sqrt(3)/2",
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+      "answer": "sqrt(2)",
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+      "answer": "x^2/12 + y^2/4 = 1",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3318.png",
+      "answer": "1/3",
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+      "index": 3319,
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+        "direction": "up"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3319.png",
+      "answer": "pm*4",
+      "fact_expr": "G: Parabola;P: Point;F: Point;Expression(G) = (x^2 = 4*y);PointOnCurve(P, G);Focus(G) = F;Distance(P, F) = 5",
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+      "error": null,
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+        "b": 1.5,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3320.png",
+      "answer": "3*sqrt(2)/4",
+      "fact_expr": "G: Hyperbola;Expression(G) = (y^2/9 - x^2/m^2 = 1);m: Number;m>0;Distance(OneOf(Vertex(G)), OneOf(Asymptote(G))) = 1",
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+      "index": 3321,
+      "success": true,
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+      "error": null,
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+        "a": 1.0,
+        "b": 2.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3321.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/4 = 1)",
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+      "index": 3322,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 3323,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 3324,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 4.0,
+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3324.png",
+      "answer": "y=pm*(4/5)*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/25 - y^2/16 = 1)",
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+    {
+      "index": 3325,
+      "success": true,
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+      "error": null,
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+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3325.png",
+      "answer": "b^2",
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+      "index": 3326,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 3327,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3327.png",
+      "answer": "9/2",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*x);F: Point;Focus(C) = F;B: Point;A: Point;PointOnCurve(A, C);PointOnCurve(B, C);Negation(A = B);PointOnCurve(F, LineOf(A, B))",
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+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3328.png",
+      "answer": "4*a",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 3330,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3330.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, C) = {A, B};M: Point;Intersection(Directrix(C), xAxis) = M;Abs(LineSegmentOf(A, B)) = 8",
+      "coords": {}
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+    {
+      "index": 3331,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3331.png",
+      "answer": "(-4, 2)",
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+          3.0
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+      "index": 3332,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "p": 1.0,
+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3332.png",
+      "answer": "2",
+      "fact_expr": "l: Line;C: Parabola;p: Number;A: Point;F: Point;B: Point;M: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;PointOnCurve(F,l);Intersection(l, C) = {A,B};Intersection(l, yAxis) = M;Abs(LineSegmentOf(A, F)) = Abs(LineSegmentOf(B, M));Abs(LineSegmentOf(A, B)) = 2 + sqrt(5)",
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+      "index": 3333,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "down"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3333.png",
+      "answer": "(0,-1)",
+      "fact_expr": "G: Parabola;Expression(G) = (y = -x^2/4)",
+      "coords": {}
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+      "error": null,
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+        "b": 1.7320508075688772,
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+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3334.png",
+      "answer": "x^2/16+y^2/4=1",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;A: Point;B: Point;F1: Point;F2: Point;e:Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Focus(G) = {F1, F2};PointOnCurve(F2, LineSegmentOf(A, B));IsChordOf(LineSegmentOf(A, B), G);Perimeter(TriangleOf(A, F1, B)) = 16;Eccentricity(G) = e;e=sqrt(3)/2",
+      "coords": {}
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+    {
+      "index": 3335,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3335.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
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+      "coords": {}
+    },
+    {
+      "index": 3336,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3336.png",
+      "answer": "x + 2*y - 2 = 0",
+      "fact_expr": "G: Ellipse;H: LineSegment;M: Point;Expression(G) = (x^2/4 + y^2 = 1);Coordinate(M) = (1, 1/2);IsChordOf(H, G);MidPoint(H) = M",
+      "coords": {
+        "M": [
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+          0.5
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+    {
+      "index": 3337,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3337.png",
+      "answer": "(1,+\\infty)",
+      "fact_expr": "G: Parabola;B: Point;A: Point;P: Point;Expression(G) = (y^2 = 2*x);x0: Number;Coordinate(P) = (x0, 0);PointOnCurve(A, G);PointOnCurve(B, G);Intersection(PerpendicularBisector(LineSegmentOf(A,B)), xAxis) = P",
+      "coords": {}
+    },
+    {
+      "index": 3338,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3338.png",
+      "answer": "(\\sqrt{2}/2, 1)",
+      "fact_expr": "C1: Ellipse;Eccentricity(C1) = (x^2/(m + 2) - y^2/n = 1);m: Number;n: Number;C2: Hyperbola;Expression(C2) = (y^2/n + x^2/m = 1);Focus(C1) = Focus(C2);e1: Number;Eccentricity(C1) = e1",
+      "coords": {}
+    },
+    {
+      "index": 3339,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3340,
+      "success": true,
+      "conic_type": "parabola",
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+      "answer": "8/5",
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+      "answer": "x=-1",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3364.png",
+      "answer": "-1/3",
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+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "answer": "sqrt(5)",
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+      "index": 3366,
+      "success": false,
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+      "index": 3367,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 1.0,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3367.png",
+      "answer": "x-y-3=0",
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+      "index": 3368,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 3369,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 3370,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+      "answer": "x^2/4+y^2/3=1",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3390.png",
+      "answer": "2*sqrt(7)/7",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/7 + y^2/3 = 1)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3391.png",
+      "answer": "sqrt(2)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3392.png",
+      "answer": "(0, pm*sqrt(13))",
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+      "error": "Unsupported or unparseable expression"
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+      "index": 3395,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3395.png",
+      "answer": "arccos15/17",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3396.png",
+      "answer": "2*sqrt(5)",
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+      "index": 3397,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3397.png",
+      "answer": "{(11*sqrt(5)/15-2)*a, (11/6-sqrt(5))*b}",
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+    {
+      "index": 3398,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3398.png",
+      "answer": "{sqrt(5), sqrt(5)/2}",
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+    {
+      "index": 3399,
+      "success": true,
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3399.png",
+      "answer": "pm*2*sqrt(2)",
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+    {
+      "index": 3400,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3400.png",
+      "answer": "2",
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+      "index": 3401,
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3401.png",
+      "answer": "sqrt(2)-1",
+      "fact_expr": "C: Ellipse;a: Number;b: Number;G: Circle;P: Point;F1: Point;a > b;b > 0;k:Number;k>=1;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2 + y^2 = a^2 - b^2);PointOnCurve(P, C);LeftFocus(C) = F1;PointOnCurve(MidPoint(LineSegmentOf(P,F1)), G);Slope(LineOf(P,F1)) = k",
+      "coords": {}
+    },
+    {
+      "index": 3402,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3402.png",
+      "answer": "ApplyUnit(60, degree)",
+      "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (16*x^2 - 9*y^2 = 144);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1))*Abs(LineSegmentOf(P, F2)) = 64",
+      "coords": {}
+    },
+    {
+      "index": 3403,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3403.png",
+      "answer": "{-1, 2}",
+      "fact_expr": "C: Parabola;G: Line;P: Point;A: Point;B: Point;k: Number;Expression(C) = (y = x^2/4);Coordinate(P) = (2, -2);PointOnCurve(Focus(C), G);Slope(G) = k;Intersection(G, C) = {A, B};DotProduct(VectorOf(P, A), VectorOf(P, B)) = 25",
+      "coords": {
+        "P": [
+          2.0,
+          -2.0
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+      }
+    },
+    {
+      "index": 3404,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 2.449489742783178,
+        "major_axis": "x",
+        "x_coef": 11.0,
+        "y_coef": 6.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3404.png",
+      "answer": "2*sqrt(11)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/11 + y^2/6 = 1);H: Point;PointOnCurve(H, G);F1: Point;F2: Point;Focus(G) = {F1, F2}",
+      "coords": {}
+    },
+    {
+      "index": 3405,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "y",
+        "x_coef": 1.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3405.png",
+      "answer": "pm*3*sqrt(5)/5",
+      "fact_expr": "l: Line;G: Ellipse;H: Circle;B: Point;A: Point;Expression(G) = (x^2 + y^2/2 = 1);Expression(H) = (x^2 + y^2 = 1);Expression(l) = (m + x - y = 0);Intersection(l, G) = {A, B};PointOnCurve(MidPoint(LineSegmentOf(A,B)), H);m:Number",
+      "coords": {}
+    },
+    {
+      "index": 3406,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3406.png",
+      "answer": "1/2",
+      "fact_expr": "G: Parabola;H: Ellipse;m: Number;Expression(G) = (y^2 = 4*x);Expression(H) = (m*x^2 + y^2 = 1);OneOf(Focus(H)) = Focus(G)",
+      "coords": {}
+    },
+    {
+      "index": 3407,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.23606797749979,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 5.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3407.png",
+      "answer": "1/2",
+      "fact_expr": "G: Ellipse;F: Point;M: Point;A: Point;l: Line;Expression(G) = (x^2/5 + y^2/4 = 1);RightFocus(G) = F;RightDirectrix(G) = l;UpperVertex(G)=A;PointOnCurve(A,LineSegmentOf(A,M));IsPerpendicular(LineSegmentOf(A,M),l);FootPoint(LineSegmentOf(A,M),l)=M",
+      "coords": {}
+    },
+    {
+      "index": 3408,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3408.png",
+      "answer": "x^2=16*y",
+      "fact_expr": "C1: Hyperbola;Expression(C1) = ((x^2/a^2) - (y^2/b^2) = 1);a: Number;b: Number;a>0;Eccentricity(C1)=2;C2: Parabola;Expression(C2)=(x^2=2*p*y);p:Number;p>0;Distance(Focus(C2),Asymptote(C1))=2;b>0",
+      "coords": {}
+    },
+    {
+      "index": 3409,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3409.png",
+      "answer": "4",
+      "fact_expr": "M: Point;Coordinate(M) = (4, 0);P: Point;PointOnCurve(P, G);G: Curve;Expression(G) = (y^2 = 8*x);Q: Point;G1:Curve;Expression(G1) = (y^2 + (x - 2)^2 = 1);PointOnCurve(Q, G1)",
+      "coords": {
+        "M": [
+          4.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 3410,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3410.png",
+      "answer": "2+sqrt(5)",
+      "fact_expr": "G: Parabola;P: Point;x1:Number;y1:Number;Expression(G) = (y^2 = 8*x);Coordinate(P) = (x1, y1);PointOnCurve(P, G)",
+      "coords": {}
+    },
+    {
+      "index": 3411,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3412,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.4142135623730951,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3412.png",
+      "answer": "{3, 7}",
+      "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2 - y^2/2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 5",
+      "coords": {}
+    },
+    {
+      "index": 3413,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3413.png",
+      "answer": "4",
+      "fact_expr": "C: Parabola;A: Point;F: Point;P: Point;Expression(C) = (y = x^2/4);Coordinate(A) = (4, 1);Focus(C)=F;PointOnCurve(P, C)",
+      "coords": {
+        "A": [
+          4.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 3414,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.366584062576294,
+        "b": 2.364173412322998,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.00017838670464698225,
+        "converged": true,
+        "iterations": 89
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3414.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;G: Parabola;Expression(G) = (y^2 = 8*x);OneOf(Focus(C)) = Focus(G);Eccentricity(C) = 2",
+      "coords": {}
+    },
+    {
+      "index": 3415,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3415.png",
+      "answer": "9/2",
+      "fact_expr": "G: Parabola;A: Point;P: Point;M: Point;Expression(G) = (y^2 = 2*x);Coordinate(A) = (7/2, -4);PointOnCurve(P, G);Projection(P, yAxis) = M",
+      "coords": {
+        "A": [
+          3.5,
+          -4.0
+        ]
+      }
+    },
+    {
+      "index": 3416,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3417,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3417.png",
+      "answer": "1/2",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;I: Point;a>0;b>0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G) = F2;Incenter(TriangleOf(P,F1,F2))=I;Area(TriangleOf(I,P,F1))+Area(TriangleOf(I,P,F2))=2*Area(TriangleOf(I,F1,F2))",
+      "coords": {}
+    },
+    {
+      "index": 3418,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3418.png",
+      "answer": "-8*sqrt(3)/9",
+      "fact_expr": "G: Parabola;p: Number;A: Point;B: Point;O: Origin;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;IsChordOf(LineSegmentOf(A,B),G);PointOnCurve(F, LineSegmentOf(A, B));DotProduct(VectorOf(O, A), VectorOf(O, B)) =-3;Inclination(LineSegmentOf(A,B))=ApplyUnit(60,degree)",
+      "coords": {}
+    },
+    {
+      "index": 3419,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3419.png",
+      "answer": "{2*sqrt(3)/3, 2}",
+      "fact_expr": "C: Hyperbola;a: Number;b: Number;F1: Point;F2: Point;P: Point;A: Point;B: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Focus(C) = {F1, F2};PointOnCurve(P, C);l: Line;l1: Line;l2: Line;Asymptote(C) = {l1, l2};PointOnCurve(P, l);IsPerpendicular(l, l1);IsPerpendicular(l, l2);FootPoint(l, l1) = A;FootPoint(l, l2) = B;3*Abs(LineSegmentOf(F1, F2))^2 = (64*Abs(LineSegmentOf(P, A)))*Abs(LineSegmentOf(P, B))",
+      "coords": {}
+    },
+    {
+      "index": 3420,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          -3.0,
+          0.0
+        ],
+        "radius": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_3420.png",
+      "answer": "y=x^2/2",
+      "fact_expr": "G: Parabola;a: Number;C: Circle;l: Line;a>0;Expression(G) = (y = a*x^2);Expression(C) = (y^2 + (x - 3)^2 = 1);Directrix(G) = l;Length(InterceptChord(l,C))=sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 3421,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "left"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3421.png",
+      "answer": "2*sqrt(13)",
+      "fact_expr": "G: Parabola;A: Point;F: Point;P: Point;O: Origin;Expression(G) = (y^2 = -8*x);Focus(G) = F;PointOnCurve(P,Directrix(G));PointOnCurve(A, G);Abs(LineSegmentOf(A, F)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 3422,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3422.png",
+      "answer": "5/2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);IsChordOf(LineSegmentOf(A, B), G) = True;A: Point;B: Point;Length(LineSegmentOf(A, B)) = 9",
+      "coords": {}
+    },
+    {
+      "index": 3423,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3423.png",
+      "answer": "x^2/20+y^2/16=1",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;PointOnCurve(Focus(G), xAxis);I: Point;Coordinate(I) = (2, 1);L1: Line;L2: Line;TangentOfPoint(I, H) = {L1, L2};H: Circle;Expression(H) = (x^2 + y^2 = 4);TangentPoint(L1, H) = A;TangentPoint(L2, H) = B;A: Point;B: Point;PointOnCurve(RightFocus(G), LineOf(A, B));PointOnCurve(UpperVertex(G), LineOf(A, B))",
+      "coords": {
+        "I": [
+          2.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 3424,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3424.png",
+      "answer": "[3/2, +\\infty)",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;M: Point;F1: Point;F2: Point;O: Origin;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) =F1;RightFocus(C) = F2;PointOnCurve(M, C);Abs(LineSegmentOf(M, F1)) + Abs(LineSegmentOf(M, F2)) = 3*Abs(LineSegmentOf(M, O))",
+      "coords": {}
+    },
+    {
+      "index": 3425,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3426,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3427,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3427.png",
+      "answer": "5",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/4 = 1);P: Point;PointOnCurve(P, G);F1: Point;F2: Point;Focus(G) = {F1, F2}",
+      "coords": {}
+    },
+    {
+      "index": 3428,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3428.png",
+      "answer": "2*sqrt(5)/5",
+      "fact_expr": "C: Hyperbola;P: Point;A: Point;Expression(C) = (x^2/4 - y^2 = 1);Coordinate(A) = (3, 0);PointOnCurve(P, C)",
+      "coords": {
+        "A": [
+          3.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 3429,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3430,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3431,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3431.png",
+      "answer": "(-2,1/2)",
+      "fact_expr": "G: Parabola;M: Point;P: Point;F: Point;Expression(G) = (y = x^2/8);Coordinate(M) = (-2, 4);Focus(G)=F;PointOnCurve(P,G);WhenMin(Abs(LineSegmentOf(P,M))+Abs(LineSegmentOf(P,F)))",
+      "coords": {
+        "M": [
+          -2.0,
+          4.0
+        ]
+      }
+    },
+    {
+      "index": 3432,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "y",
+        "x_coef": 9.0,
+        "y_coef": 25.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3432.png",
+      "answer": "y^2 - x^2/15 = 1",
+      "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/9 + y^2/25 = 1);Focus(H) = Focus(G);Eccentricity(G) + Eccentricity(H) = 24/5",
+      "coords": {}
+    },
+    {
+      "index": 3433,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3433.png",
+      "answer": "y^2=4*sqrt(2)*x",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;F: Point;Focus(G) = F;M: Point;PointOnCurve(M, G);N: Point;Intersection(Directrix(G),axis)=N;Abs(LineSegmentOf(M,F))=(7/2)*p;Area(TriangleOf(M,N,F))=4*sqrt(6)",
+      "coords": {}
+    },
+    {
+      "index": 3434,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3434.png",
+      "answer": "8",
+      "fact_expr": "E: Point;PointOnCurve(E, yAxis);F: Point;Focus(G) = F;G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;M: Point;N: Point;Intersection(LineOf(E, F), G) = {M, N};MidPoint(LineSegmentOf(E, F)) = M;Abs(LineSegmentOf(N, F)) = 12",
+      "coords": {}
+    },
+    {
+      "index": 3435,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3435.png",
+      "answer": "pi/2",
+      "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 - y^2/16 = 1);Focus(G)={F1,F2};PointOnCurve(P, G);Abs(LineSegmentOf(P, F1))*Abs(LineSegmentOf(P, F2)) = 32",
+      "coords": {}
+    },
+    {
+      "index": 3436,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3436.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;PointOnCurve(LeftFocus(G), L) = True;IsPerpendicular(L, xAxis) = True;L: Line;Intersection(L, G) = {M, N};M: Point;N: Point;Q: Circle;IsDiameter(LineSegmentOf(M, N), Q) = True;PointOnCurve(RightVertex(G), Q) = True",
+      "coords": {}
+    },
+    {
+      "index": 3437,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3437.png",
+      "answer": "3-(3*sqrt(3)/2)",
+      "fact_expr": "C: Ellipse;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/4 + y^2 = 1);Focus(C) = {F1, F2};PointOnCurve(P,C);Quadrant(P)=1;XCoordinate(P)=1",
+      "coords": {}
+    },
+    {
+      "index": 3438,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3438.png",
+      "answer": "y^2 = 6*x",
+      "fact_expr": "C: Parabola;p: Number;M: Point;H: Point;F: Point;l: Line;l1: Line;p>0;Expression(C) = (y^2 = 2*p*x);Focus(C) = F;Directrix(C) = l;PointOnCurve(M, C);PointOnCurve(M, l1);IsPerpendicular(l,l1);Intersection(l,l1)=H;Abs(LineSegmentOf(M, H)) = 2;AngleOf(H, F, M) = ApplyUnit(30, degree)",
+      "coords": {}
+    },
+    {
+      "index": 3439,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 3440,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3440.png",
+      "answer": "7",
+      "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/12 + y^2/3 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);PointOnCurve(MidPoint(LineSegmentOf(P,F1)), yAxis);LineSegmentOf(P, F1) = t*LineSegmentOf(P, F2);t:Number",
+      "coords": {}
+    },
+    {
+      "index": 3441,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3442,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3442.png",
+      "answer": "sqrt(15)/2",
+      "fact_expr": "G: Parabola;A: Point;M: Point;Coordinate(A) = (4, 0);Expression(G) = (y^2 = x);PointOnCurve(M, G);Distance(A,M)=Abs(LineSegmentOf(M,A))",
+      "coords": {
+        "A": [
+          4.0,
+          0.0
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+      }
+    },
+    {
+      "index": 3443,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 3.0,
+        "direction": "left"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3443.png",
+      "answer": "24",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = -12*x);l: Line;PointOnCurve(Focus(G), l);A: Point;B: Point;Intersection(l, G) = {A, B};XCoordinate(MidPoint(LineSegmentOf(A, B))) = -9",
+      "coords": {}
+    },
+    {
+      "index": 3444,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_ecc_mismatch",
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+        "b": 1.7993911322594118,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 2.5576271215184017e-15,
+        "converged": true,
+        "iterations": 2
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3444.png",
+      "answer": "3",
+      "fact_expr": "C1: Ellipse;Expression(C1) = (x^2/4 + y^2/b^2 = 1);b: Number;0<b;b<2;Eccentricity(C1) = 1/2;F1: Point;F2: Point;LeftFocus(C1) = F1;RightFocus(C1) = F2;M: Point;PointOnCurve(M, C1);N: Point;PointOnCurve(N, OverlappingLine(LineSegmentOf(F1, M)));Abs(LineSegmentOf(M, N)) = Abs(LineSegmentOf(M, F2));P: Point;MidPoint(LineSegmentOf(F2, N)) = P",
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+      "validation_reasons": [
+        "ellipse_ecc_mismatch"
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+    {
+      "index": 3445,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3446,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3446.png",
+      "answer": "32",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);H: Line;P: Point;Coordinate(P) = (4, 0);PointOnCurve(P, H);A: Point;B: Point;x1: Number;x2: Number;y1: Number;y2: Number;Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);Intersection(H, G) = {A, B}",
+      "coords": {
+        "P": [
+          4.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 3447,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3448,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3448.png",
+      "answer": "12",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 8*y);F: Point;Focus(G) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, G) = {A, B};M: Point;MidPoint(LineSegmentOf(A, B)) = M;YCoordinate(M) = 4",
+      "coords": {}
+    },
+    {
+      "index": 3449,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3449.png",
+      "answer": "(y^{\\frac{9}{4}}) - \\frac{x^2}{4} = 1",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2/16 - y^2/9 = 1);C1: Hyperbola;Asymptote(C) = Asymptote(C1);H: Point;Coordinate(H) = (2*sqrt(3), -3);PointOnCurve(H, C1)",
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+        "H": [
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+          -3.0
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+    },
+    {
+      "index": 3450,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.4641016151377544,
+        "b": 2.449489742783178,
+        "major_axis": "x",
+        "x_coef": 12.0,
+        "y_coef": 6.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3450.png",
+      "answer": "{x^2/9+y^2/(9/2)=1,x^2/3+y^2/6=1}",
+      "fact_expr": "G: Ellipse;C:Ellipse;N: Point;Expression(G) = (x^2/12 + y^2/6 = 1);Coordinate(N) = (1, 2);PointOnCurve(N,C);Eccentricity(G)=Eccentricity(C)",
+      "coords": {
+        "N": [
+          1.0,
+          2.0
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+      }
+    },
+    {
+      "index": 3451,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.4142135623730951,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3451.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
+      "fact_expr": "G: Hyperbola;H: Point;C: ConicSection;Expression(G) = (x^2/4 - y^2/2 = 1);Coordinate(H) = (2, 1);PointOnCurve(H,C);Focus(G)=Focus(C)",
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+        "H": [
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+          1.0
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+    },
+    {
+      "index": 3452,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3452.png",
+      "answer": "(1/2,1)",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;P: Point;F: Point;A: Point;e: Number;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F;RightVertex(C)=A;PointOnCurve(P,C);IsPerpendicular(LineSegmentOf(P,F),LineSegmentOf(P,A));Eccentricity(C) = e",
+      "coords": {}
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+    {
+      "index": 3453,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3454,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3454.png",
+      "answer": "8",
+      "fact_expr": "C: Hyperbola;G: Circle;Expression(C) = (x^2/4 - y^2/9 = 1);OneOf(Focus(C))=Center(G);Radius(G)=5",
+      "coords": {}
+    },
+    {
+      "index": 3455,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3456,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3456.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "G: Hyperbola;a: Number;b: Number;P: Point;F2: Point;F1: Point;PointOnCurve(P,G);Focus(G)={F1,F2};Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);IsPerpendicular(LineSegmentOf(P, F2), LineSegmentOf(F1, F2));Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 3457,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3457.png",
+      "answer": "[b^2,a^2]",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F2: Point;F1: Point;a > b;b > 0;x1:Number;y1:Number;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(P) = (x1, y1);PointOnCurve(P, G);Focus(G)={F1,F2}",
+      "coords": {}
+    },
+    {
+      "index": 3458,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3458.png",
+      "answer": "3",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;A: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightVertex(C)=A;Distance(A, OneOf(Asymptote(C))) = a*(2*sqrt(2))/3",
+      "coords": {}
+    },
+    {
+      "index": 3459,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3459.png",
+      "answer": "sqrt(6)/3",
+      "fact_expr": "C: Ellipse;a: Number;G: Line;P: Point;A: Point;B: Point;a>0;Expression(C) = (y^2 + x^2/a^2 = 1);Expression(G) = (a*x + y - 4 = 0);PointOnCurve(P, G);IsTangent(LineOf(P, A), C);IsTangent(LineOf(P, B), C);IsPerpendicular(LineSegmentOf(P, A),LineSegmentOf(P, B))",
+      "coords": {}
+    },
+    {
+      "index": 3460,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3460.png",
+      "answer": "16",
+      "fact_expr": "E: Ellipse;Expression(E) = (x^2/25 + y^2/16 = 1);F1: Point;F2: Point;LeftFocus(E) = F1;RightFocus(E) = F2;P: Point;PointOnCurve(P, E)",
+      "coords": {}
+    },
+    {
+      "index": 3461,
+      "success": true,
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+      "answer": "x^2/18+y^2/9=1",
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+      "answer": "-3*sqrt(3)",
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+      "answer": "(pm*sqrt(5), 0)",
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+      "answer": "-83",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3502.png",
+      "answer": "sqrt(3)",
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+      "error": null,
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3504.png",
+      "answer": "(16,4)",
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+          2.0
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+      "success": true,
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+      "error": null,
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3505.png",
+      "answer": "2*sqrt(2)",
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+          0.0
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+      "success": true,
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+      "error": null,
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+        "direction": "up"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3506.png",
+      "answer": "1",
+      "fact_expr": "G: Parabola;a: Number;Expression(G) = (x^2 = a*y);Expression(Directrix(G)) = (y = -1/4)",
+      "coords": {}
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+      "index": 3507,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.0,
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+        "y_coef": 1.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3508.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
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+      "index": 3509,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3509.png",
+      "answer": "x = -1/4",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = x)",
+      "coords": {}
+    },
+    {
+      "index": 3510,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3511,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3512,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.0,
+        "major_axis": "y",
+        "x_coef": 1.0,
+        "y_coef": 4.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3512.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "E: Ellipse;m: Number;A: Point;P: Point;B: Point;m>1;Expression(E) = (x^2 + y^2/m = 1);Vertex(MinorAxis(E)) = {A, B};PointOnCurve(P, E);Negation(P = A);Negation(P = B);Slope(LineOf(A, P))*Slope(LineOf(B, P)) = -4/m",
+      "coords": {}
+    },
+    {
+      "index": 3513,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
+      },
+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3513.png",
+      "answer": "pm*2*sqrt(2)/3",
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+      "coords": {}
+    },
+    {
+      "index": 3514,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3514.png",
+      "answer": "sqrt(2)",
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+      "coords": {}
+    },
+    {
+      "index": 3515,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 3516,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3516.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;M: Point;F: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(M,G);Focus(G)=F;Distance(M,F)=Abs(LineSegmentOf(M,F));Abs(LineSegmentOf(M,F))=5",
+      "coords": {}
+    },
+    {
+      "index": 3517,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3517.png",
+      "answer": "(0, 1)",
+      "fact_expr": "G: Parabola;p: Number;M: Point;F: Point;p>0;Expression(G) = (x^2 = 2*(p*y));y0: Number;Coordinate(M) = (4, y0);PointOnCurve(M, G);Focus(G) = F;Distance(M, F) = Abs(LineSegmentOf(M, F)) ;Abs(LineSegmentOf(M, F)) = (5/4)*y0",
+      "coords": {}
+    },
+    {
+      "index": 3518,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3519,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3520,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3520.png",
+      "answer": "ApplyUnit(45,degree)",
+      "fact_expr": "l: Line;C: Parabola;B: Point;A: Point;Expression(C) = (y^2 = 6*x);Intersection(l,C) = {A, B};YCoordinate(MidPoint(LineSegmentOf(A,B))) = 3",
+      "coords": {}
+    },
+    {
+      "index": 3521,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3521.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y^2 + (x - 2)^2 = 4);Length(InterceptChord(OneOf(Asymptote(C)),G))=2",
+      "coords": {}
+    },
+    {
+      "index": 3522,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3522.png",
+      "answer": "3",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;P: Point;PointOnCurve(P, C);LineSegmentOf(P, F) = 3;G: Line;Expression(G) = (x = -1)",
+      "coords": {}
+    },
+    {
+      "index": 3523,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.449489742783178,
+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 6.0,
+        "y_coef": 2.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3523.png",
+      "answer": "1/3",
+      "fact_expr": "C1: Curve;C2: Curve;F1: Point;P: Point;F2: Point;Expression(C1)= (x^2/6 + y^2/2 = 1);Expression(C2)=( x^2/3 - y^2 = 1);Focus(C1)={F1,F2};OneOf(Intersection(C1,C2))=P",
+      "coords": {}
+    },
+    {
+      "index": 3524,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3524.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "H: Ellipse;Expression(H) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(H) = F1;RightFocus(H) = F2;G: Parabola;Expression(G) = (y^2 = 2*(b*x));F: Point;Focus(G) = F;VectorOf(F1, F) = (5/3)*VectorOf(F, F2)",
+      "coords": {}
+    },
+    {
+      "index": 3525,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3526,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3526.png",
+      "answer": "8",
+      "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (y = x - 1);Intersection(H, G) = {A, B};IsChordOf(LineSegmentOf(A,B),G)",
+      "coords": {}
+    },
+    {
+      "index": 3527,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3527.png",
+      "answer": "{2, 6}",
+      "fact_expr": "C: Parabola;p: Number;A: Point;B: Point;F: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Coordinate(A) = (0, sqrt(3));Focus(C) = F;PointOnCurve(B, C);IsPerpendicular(LineSegmentOf(A, B), LineSegmentOf(A, F));Abs(LineSegmentOf(B, F)) = 4",
+      "coords": {
+        "A": [
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+          1.7320508075688772
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+    },
+    {
+      "index": 3528,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3528.png",
+      "answer": "6",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 6*x);F: Point;Focus(G) = F;PointOnCurve(F, l) = True;l: Line;Intersection(l, G) = {A, B};B: Point;A: Point;O: Origin;VectorOf(O, M) = VectorOf(O, B)/4;VectorOf(O, A) = VectorOf(O, N)/4;L1: Line;L2: Line;PointOnCurve(M, L1) = True;PointOnCurve(N, L2) = True;M: Point;N: Point;IsPerpendicular(L1, Directrix(G)) = True;IsPerpendicular(L2, Directrix(G)) = True;FootPoint(L1, Directrix(G)) = C;FootPoint(L2, Directrix(G)) = D;C: Point;D: Point",
+      "coords": {}
+    },
+    {
+      "index": 3529,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3530,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.125,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3530.png",
+      "answer": "pm*4*sqrt(3)",
+      "fact_expr": "C: Parabola;G: Line;k: Number;A: Point;B: Point;N: Point;M: Point;Expression(C) = (y = 2*x^2);Expression(G) = (y = k*x + 2);Intersection(C, G) = {A, B};MidPoint(LineSegmentOf(A, B)) = M;l: Line;PointOnCurve(M, l);IsPerpendicular(l, xAxis);FootPoint(l, xAxis) = N;DotProduct(VectorOf(N, A), VectorOf(N, B)) = 0",
+      "coords": {}
+    },
+    {
+      "index": 3531,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3531.png",
+      "answer": "42",
+      "fact_expr": "G: Parabola;p: Number;A: Point;B: Point;F: Point;p>0;y1:Number;y2:Number;Expression(G) = (y^2 = 2*p*x);Coordinate(A) = (2, y1);Coordinate(B) = (1/2, y2);Focus(G) = F;Quadrant(A)=1;Quadrant(B)=4;Abs(LineSegmentOf(A, F)) = 10;PointOnCurve(A,G);PointOnCurve(B,G)",
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+    },
+    {
+      "index": 3532,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3532.png",
+      "answer": "x^2/2-y^2=1",
+      "fact_expr": "G: Hyperbola;H: Ellipse;Q: Point;Expression(H) = (x^2/4 + y^2 = 1);Coordinate(Q) = (2, 1);Focus(G)=Focus(H);PointOnCurve(Q,G)",
+      "coords": {
+        "Q": [
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+          1.0
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+    },
+    {
+      "index": 3533,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3534,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3535,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3535.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "P: Point;PointOnCurve(P, G);F1: Point;F2: Point;Focus(G) = {F1, F2};G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2));Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 3536,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3537,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 3.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3537.png",
+      "answer": "sqrt(3)/2",
+      "fact_expr": "G: Parabola;H: Ellipse;k: Number;Expression(G) = (y^2 = 12*x);k>0;Expression(H) = (k*y^2 + x^2 = 3*k);OneOf(Focus(H))=Focus(G)",
+      "coords": {}
+    },
+    {
+      "index": 3538,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3538.png",
+      "answer": "4",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 + y^2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G)",
+      "coords": {}
+    },
+    {
+      "index": 3539,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3539.png",
+      "answer": "16/3",
+      "fact_expr": "l: Line;C: Parabola;A: Point;B: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F,l);Slope(l)=sqrt(3);Intersection(l, C) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 3540,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3540.png",
+      "answer": "2",
+      "fact_expr": "G:Hyperbola;p: Number;H: Parabola;Expression(G) = (-x^2 + 2*y^2 = 2*p^2);p>0;Expression(H) = (y^2 = 2*p*x);Distance(Focus(H),OneOf(Focus(G))) = sqrt(13)",
+      "coords": {}
+    },
+    {
+      "index": 3541,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3541.png",
+      "answer": "sqrt(3)/2",
+      "fact_expr": "G: Hyperbola;a: Number;H: Ellipse;e: Number;Expression(G) = (-y^2/2 + x^2/a = 1);Expression(H) = (x^2/4 + y^2/a^2 = 1);Focus(H) = Focus(G);Eccentricity(H) = e",
+      "coords": {}
+    },
+    {
+      "index": 3542,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3542.png",
+      "answer": "5/3",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;l: Line;PointOnCurve(F2, l) ;Intersection(l, RightPart(C)) = {A, B};A: Point;B: Point;Abs(LineSegmentOf(A, F1)) = Abs(LineSegmentOf(F1, F2));Abs(LineSegmentOf(B, F2)) = 2*Abs(LineSegmentOf(A, F2))",
+      "coords": {}
+    },
+    {
+      "index": 3543,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3543.png",
+      "answer": "y = pm*(x- 1)",
+      "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;Expression(G) = (y^2 = 4*x);Abs(LineSegmentOf(A, B)) = 8;NumIntersection(H, G) = 2;IsChordOf(LineSegmentOf(A, B), G);PointOnCurve(Focus(G), H)",
+      "coords": {}
+    },
+    {
+      "index": 3544,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3544.png",
+      "answer": "(2*sqrt(3))/3",
+      "fact_expr": "G: Hyperbola;a: Number;b: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Distance(OneOf(Focus(G)),OneOf(Asymptote(G)))=1/4*FocalLength(G)",
+      "coords": {}
+    },
+    {
+      "index": 3545,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3545.png",
+      "answer": "2",
+      "fact_expr": "C: Parabola;H: Line;A: Point;F: Point;B: Point;K: Point;Expression(C)=(y^2=4*x);Focus(C)=F;Intersection(Directrix(C), xAxis) = K;PointOnCurve(F,H);Intersection(H, C) = {A, B};Abs(LineSegmentOf(A, F)) - Abs(LineSegmentOf(B, F)) = 3/2",
+      "coords": {}
+    },
+    {
+      "index": 3546,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3546.png",
+      "answer": "2*sqrt(3)/3",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Expression(OneOf(Asymptote(C))) = (x - sqrt(3)*y = 0)",
+      "coords": {}
+    },
+    {
+      "index": 3547,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3547.png",
+      "answer": "5/4",
+      "fact_expr": "G: Parabola;A: Point;B: Point;F: Point;M: Point;Expression(G) = (y^2 = x);Focus(G) = F;PointOnCurve(F, LineOf(A, B));Intersection(LineOf(A, B), G) = {A, B};Abs(LineSegmentOf(A, B)) = 3;MidPoint(LineSegmentOf(A, B)) = M",
+      "coords": {}
+    },
+    {
+      "index": 3548,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3548.png",
+      "answer": "3",
+      "fact_expr": "G: Parabola;P: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(P, G);O:Origin;Distance(P,O)=2*sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 3549,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3549.png",
+      "answer": "(2,+oo)",
+      "fact_expr": "M: Point;Coordinate(M) = (x0, y0);x0: Number;y0: Number;C: Parabola;Expression(C) = (x^2 = 8*y);PointOnCurve(M, C) = True;Focus(C) = F;F:Point;G: Circle;Center(G) = F;Radius(G) = Abs(LineSegmentOf(F, M));IsIntersect(G , Directrix(C)) = True",
+      "coords": {}
+    },
+    {
+      "index": 3550,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          0.0,
+          0.0
+        ],
+        "radius": 1.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_3550.png",
+      "answer": "(\\sqrt{2}-1)/2",
+      "fact_expr": "P: Point;m: Number;H: Circle;G: Parabola;Expression(G) = (x^2 = 4*(y-m));Expression(H) = (x^2 + y^2 = 1);Quadrant(P) = 1;Intersection(H, G)=P;IsPerpendicular(TangentOnPoint(P, G), TangentOnPoint(P, H))",
+      "coords": {}
+    },
+    {
+      "index": 3551,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 2.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3551.png",
+      "answer": "1",
+      "fact_expr": "G: Hyperbola;M: Point;P: Point;Q: Point;L:Line;Expression(G) = (x^2 - y^2/4 = 1);PointOnCurve(M,G);PointOnCurve(M,L);IsParallel(L,RealAxis(G));Intersection(L,Asymptote(G))={P,Q}",
+      "coords": {}
+    },
+    {
+      "index": 3552,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3552.png",
+      "answer": "y^2 = 8*x",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p<0;F: Point;Focus(G) = F;PointOnCurve(F, LineSegmentOf(A, B)) = True;PointOnCurve(F, LineSegmentOf(C, D)) = True;IsChordOf(LineSegmentOf(A, B), G)  = True;IsChordOf(LineSegmentOf(C, D), G) = True;IsPerpendicular(LineSegmentOf(A, B), LineSegmentOf(C, D)) = True;A: Point;B: Point;C: Point;D: Point;Min(Area(TriangleOf(A, C, F)) + Area(TriangleOf(B, D, F))) = 32",
+      "coords": {}
+    },
+    {
+      "index": 3553,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 3.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3553.png",
+      "answer": "8",
+      "fact_expr": "C: Parabola;G: Line;F: Point;P: Point;Q: Point;Expression(C) = (x^2 = 12*y);Expression(G) = (y=-3);Focus(C) = F;PointOnCurve(P, C);Intersection(LineOf(F,P),G) = Q;VectorOf(P, Q) = 2*VectorOf(F, Q)",
+      "coords": {}
+    },
+    {
+      "index": 3554,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3555,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3555.png",
+      "answer": "sqrt(3) - 1",
+      "fact_expr": "C1: Ellipse;Expression(C1) = (x^2/a^2+y^2/b^2 = 1);a: Number;b: Number;a>b;b>0;C2: Circle;Expression(C2) = (y^2+x^2 = a^2-b^2);P: Point;OneOf(Intersection(C1,C2)) = P;2*AngleOf(P,F1,F2) = AngleOf(P,F2,F1);F1: Point;F2: Point;LeftFocus(C1) = F1;RightFocus(C1) = F2",
+      "coords": {}
+    },
+    {
+      "index": 3556,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3557,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3558,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3558.png",
+      "answer": "(0,4]",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);D: Point;Coordinate(D) = (4, 0);F: Point;Focus(C) = F;PointOnCurve(F, l);Slope(l) = k;k: Number;l: Line;Intersection(l, C) = {A, B};A: Point;B: Point;DotProduct(VectorOf(A, D), VectorOf(B, D)) <= 1",
+      "coords": {
+        "D": [
+          4.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 3559,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3559.png",
+      "answer": "{1, 17}",
+      "fact_expr": "G: Hyperbola;F1: Point;F2: Point;P: Point;Expression(G) = (x^2/16 - y^2/9 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);Distance(P, F1) = 9",
+      "coords": {}
+    },
+    {
+      "index": 3560,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3561,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3562,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3562.png",
+      "answer": "3/2",
+      "fact_expr": "G: Parabola;M: Point;O: Origin;Expression(G) = (y^2 = 2*x);PointOnCurve(M, G);Distance(M, O) = sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 3563,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3563.png",
+      "answer": "{y^2 = 8*x, y^2 = -16*x}",
+      "fact_expr": "G: Parabola;m: Number;H: Line;Expression(G) = (y^2 = m*x);Expression(H) = (x = 1);Distance(Directrix(G), H) = 3",
+      "coords": {}
+    },
+    {
+      "index": 3564,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3564.png",
+      "answer": "sqrt(17)/2",
+      "fact_expr": "G: Parabola;Q: Point;P: Point;Expression(G) = (y^2 = 2*x);Coordinate(Q) = (0, 2);PointOnCurve(P,G)",
+      "coords": {
+        "Q": [
+          0.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 3565,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3565.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1);P: Point;PointOnCurve(P, LeftPart(G));F: Point;RightFocus(G) = F;M: Point;MidPoint(LineSegmentOf(F, P)) = M;O: Origin;Distance(M, O) = 7",
+      "coords": {}
+    },
+    {
+      "index": 3566,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3567,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3567.png",
+      "answer": "1/2",
+      "fact_expr": "C: Ellipse;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/4 + y^2/3 = 1);Focus(C) = {F1, F2};Quadrant(P) = 1;PointOnCurve(P, C);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 9/4",
+      "coords": {}
+    },
+    {
+      "index": 3568,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3568.png",
+      "answer": "x=-1",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;H: Line;Slope(H) = 1;PointOnCurve(Focus(G), H);A: Point;B: Point;Intersection(H, G) = {A, B};YCoordinate(MidPoint(LineSegmentOf(A, B))) = 2",
+      "coords": {}
+    },
+    {
+      "index": 3569,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 0.5,
+        "major_axis": "y",
+        "x_coef": 0.25,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3569.png",
+      "answer": "1 - sqrt(3)/2",
+      "fact_expr": "G: Ellipse;P: Point;F: Point;Expression(G) = (4*x^2 + y^2 = 1);PointOnCurve(P, G);OneOf(Focus(G))=F",
+      "coords": {}
+    },
+    {
+      "index": 3570,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3570.png",
+      "answer": "sqrt(6)-sqrt(3)",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;P: Point;F1: Point;O: Origin;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C)=F1;RightFocus(C)=F2;PointOnCurve(P,C);DotProduct(VectorOf(P,F1),(VectorOf(O,F1)+VectorOf(O,P)))=0;Abs(LineSegmentOf(P,F1))=sqrt(2)*Abs(LineSegmentOf(P,F2))",
+      "coords": {}
+    },
+    {
+      "index": 3571,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3571.png",
+      "answer": "[0, 1]",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2/m^2 = 1);m: Number;0 < m;m < 2;F1: Point;F2: Point;Focus(C) = {F1, F2};P: Point;x0: Number;y0: Number;Coordinate(P) = (x0, y0);PointOnCurve(P, C);Area(TriangleOf(P, F1, F2)) = sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 3572,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3572.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;p: Number;H: Line;M: Point;N: Point;A: Point;B: Point;p>0;Expression(G) = (x^2 = 2*(p*y));Coordinate(M) = (0, 1);PointOnCurve(M, H);Intersection(H, G) = {A, B};Negation(A=B);Intersection(TangentOnPoint(A,G),TangentOnPoint(B,G))=N;DotProduct(VectorOf(N, A), VectorOf(N, B)) = 0",
+      "coords": {
+        "M": [
+          0.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 3573,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3574,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3574.png",
+      "answer": "5/3",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;H: Parabola;Expression(H) = (y^2 = 20*x);OneOf(Focus(G)) = Focus(H);Distance(OneOf(Focus(G)), Asymptote(G)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 3575,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3575.png",
+      "answer": "pi/2",
+      "fact_expr": "E: Point;Coordinate(E) = (2, 2);C: Parabola;Expression(C) = (y^2 = 2*p*x);p: Number;H: Point;Coordinate(H) = (2, 0);l: Line;PointOnCurve(H, l) = True;Intersection(l, C) = {A, B};A: Point;B: Point;Negation(A = E);Negation(B = E);G: Line;Expression(G) = (x = -2);Intersection(LineOf(E, A), G) = M;M: Point;Intersection(LineOf(E, B), G) = N;N: Point;O: Origin",
+      "coords": {
+        "E": [
+          2.0,
+          2.0
+        ],
+        "H": [
+          2.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 3576,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.463192105293274,
+        "b": 2.070901870727539,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 1.4661259228887502e-06,
+        "converged": true,
+        "iterations": 106
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3576.png",
+      "answer": "x^2/(1/3)-y^2/(2/3)=1",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 4*x);Eccentricity(G) = sqrt(3);OneOf(Focus(G)) = Focus(H)",
+      "coords": {}
+    },
+    {
+      "index": 3577,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3577.png",
+      "answer": "y^2 = 8*x",
+      "fact_expr": "l: Line;G: Parabola;a: Number;O: Origin;A: Point;F: Point;a>0;Expression(G) = (y^2 = a*x);Slope(l) = 2;Focus(G) = F;PointOnCurve(F, l);Intersection(l,yAxis) = A;Area(TriangleOf(O, A, F)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 3578,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3578.png",
+      "answer": "y^2=20*x",
+      "fact_expr": "H: Parabola;Expression(H) = (y^2 = 4*(m*x));m: Number;m>0;G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1);Distance(Focus(H), OneOf(Asymptote(G))) = 3",
+      "coords": {}
+    },
+    {
+      "index": 3579,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "y",
+        "x_coef": 1.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3579.png",
+      "answer": "-3",
+      "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (x^2 + y^2/m = 1);Eccentricity(G) = 2",
+      "coords": {}
+    },
+    {
+      "index": 3580,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3580.png",
+      "answer": "sqrt(13)",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;F1: Point;F2: Point;A: Point;B: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;IsDiameter(LineSegmentOf(F1,F2),G);Intersection(G,LeftPart(C))=A;Intersection(LineSegmentOf(A,F2),RightPart(C))=B;Cos(AngleOf(F1, B, F2)) = -3/5",
+      "coords": {}
+    },
+    {
+      "index": 3581,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3581.png",
+      "answer": "{(sqrt(3),0),(-sqrt(3),0)}\nsqrt(6)/2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - 2*y^2 = 2)",
+      "coords": {}
+    },
+    {
+      "index": 3582,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "y",
+        "x_coef": 1.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3582.png",
+      "answer": "4*sqrt(5)/5",
+      "fact_expr": "G: Ellipse;H1: Line;H2: Line;P: Point;d1: Number;d2: Number;Expression(G) = (x^2 + y^2/4 = 1);Expression(H1) = (2*x + y = 0);Expression(H2) = (2*x - y = 0);PointOnCurve(P, G);Distance(P, H1) = d1;Distance(P, H2) = d2",
+      "coords": {}
+    },
+    {
+      "index": 3583,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3583.png",
+      "answer": "sqrt(5)/2",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(C) = F;L: Line;PointOnCurve(F,L) = True;Intersection(L,C) = {A,B};A: Point;B: Point;Quadrant(A) = 1;VectorOf(A, F) = 5*VectorOf(F, B)",
+      "coords": {}
+    },
+    {
+      "index": 3584,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3584.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Hyperbola;b: Number;H: Parabola;Expression(G) = (x^2/8 - y^2/b^2 = 1);Expression(H) = (y^2 = 8*x);OneOf(Directrix(G))=Directrix(H)",
+      "coords": {}
+    },
+    {
+      "index": 3585,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3586,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "vertical"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3586.png",
+      "answer": "42",
+      "fact_expr": "G: Hyperbola;P: Point;Expression(G) = (-x^2/36 + y^2/64 = 1);PointOnCurve(P, G);Distance(P, OneOf(Focus(G))) = 3;F1:Point;F2:Point;Focus(G)={F1,F2}",
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+      "answer": "25/8",
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+      "answer": "17/2",
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+      "answer": "x^2/5 - y^2/20 = 1",
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+        "note": "using explicit params"
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+      "answer": "y=pm*(sqrt(3)/3)*x",
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+        "note": "using explicit params"
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+      "answer": "sqrt(5)*2/5",
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+        "note": "using explicit params"
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+      "answer": "(2, 2*sqrt(2))",
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+      "answer": "x^2/9 - y^2/16 = 1",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3602.png",
+      "answer": "y=pm*sqrt(2)*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/2 - y^2/4 = -1)",
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+        "note": "using explicit params"
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+      "answer": "2*sqrt(3)",
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+      "index": 3613,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 3614,
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+      "answer": "y=pm*(1/2)*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3615.png",
+      "answer": "y = pm*sqrt(3)*x",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = 2",
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+      "answer": "[\\sqrt{3}, +\\infty)",
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+    {
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+        "b": 2.0,
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3617.png",
+      "answer": "y=pm*2*x",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2 - y^2/4 = 1)",
+      "coords": {}
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+    {
+      "index": 3618,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3618.png",
+      "answer": "2",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;c: Number;F: Point;a>0;b>0;L:Line;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y^2 + (-a + x)^2 = c^2/16);RightFocus(C) = F;TangentOfPoint(F,G)=L;IsPerpendicular(L,OneOf(Asymptote(C)))",
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+      "index": 3619,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3619.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "C: Parabola;p: Number;G: Line;Q: Point;P: Point;F: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;PointOnCurve(F,G);Intersection(G, C) = {P, Q};Quadrant(Q)=1;2*VectorOf(P, F) = VectorOf(F, Q)",
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+    {
+      "index": 3620,
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+      "conic_type": "parabola",
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+        "iterations": 88
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3620.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;p: Real;H: Point;Expression(G) = (y^2 = 2*(p*x));Coordinate(H) = (2, 4);PointOnCurve(H, G)",
+      "coords": {
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+          4.0
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+    {
+      "index": 3621,
+      "success": true,
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+        "direction": "up"
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3621.png",
+      "answer": "7",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 12*y);A: Point;P: Point;B: Point;Coordinate(A) = (0, 5);PointOnCurve(P,G);l1:Line;Expression(l1)=(y=-3);l2:Line;IsPerpendicular(l2,l1);PointOnCurve(P,l2);FootPoint(l1,l2)=B;Abs(LineSegmentOf(P, B)) = Abs(LineSegmentOf(P, A))",
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+    {
+      "index": 3622,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3622.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;HalfFocalLength(G)=c;c:Number;O:Origin;l:Line;Expression(l) = (a*y + b*x = a*b);Distance(O,l)=(c/4+1)",
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+    {
+      "index": 3623,
+      "success": true,
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+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3623.png",
+      "answer": "2*sqrt(7)",
+      "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (x^2/4 - y^2/m = 1);Expression(Asymptote(G)) = (y = pm*(x*(sqrt(3)/2)))",
+      "coords": {}
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+    {
+      "index": 3624,
+      "success": true,
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+        "b": 4.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3624.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 3625,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "up"
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3625.png",
+      "answer": "-3",
+      "fact_expr": "l: Line;C: Parabola;P: Point;Q: Point;A: Point;B: Point;lambda:Real;mu:Real;Expression(C) = (x^2 = y);Expression(l) = (y = x + 1);Intersection(l, C) = {A, B};Coordinate(P) = (0, 1);Coordinate(Q) = (-1, 0);VectorOf(P, Q) = lambda*VectorOf(Q, A);lambda*VectorOf(Q, A) = mu*VectorOf(Q, B)",
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+          1.0
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+        "Q": [
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+          0.0
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+    {
+      "index": 3626,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 3.0,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3626.png",
+      "answer": "(5,0)",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/25 - y^2/9 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 9*Abs(LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 3627,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 3628,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "left"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3628.png",
+      "answer": "6*sqrt(2)",
+      "fact_expr": "G: Parabola;H: Line;P: Point;d1: Number;d2: Number;Expression(G) = (y^2 = -8*x);Expression(H) = (x + y - 10 = 0);PointOnCurve(P, G);Distance(P, Directrix(G)) = d1;Distance(P, H) = d2",
+      "coords": {}
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+    {
+      "index": 3629,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3629.png",
+      "answer": "5/7",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;G: Line;A: Point;F: Point;B: Point;O: Origin;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F;PointOnCurve(O, G);Intersection(C, G) = {A, B};LineSegmentOf(A, B) = 15;LineSegmentOf(B, F) = 12;Sin(AngleOf(A, B, F)) = 3/5",
+      "coords": {}
+    },
+    {
+      "index": 3630,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.7320508075688772,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3630.png",
+      "answer": "-4",
+      "fact_expr": "G: Hyperbola;H: Parabola;p: Real;Expression(G) = (x^2/3 - y^2 = 1);Expression(H) = (y^2 = 2*(p*x));Focus(H) = LeftFocus(G)",
+      "coords": {}
+    },
+    {
+      "index": 3631,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3631.png",
+      "answer": "x^2/3-y^2=1",
+      "fact_expr": "C1: Hyperbola;Expression(C1) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;FocalLength(C1) = 4;C2: Circle;Expression(C2) = ((x - 2)^2 + y^2 = 1);IsTangent(Asymptote(C1), C2)",
+      "coords": {}
+    },
+    {
+      "index": 3632,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3633,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "vertical"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3633.png",
+      "answer": "3*sqrt(5)/5",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;A: Point;P: Point;B: Point;a>0;b>0;Expression(G) = (-x^2/b^2 + y^2/a^2 = 1);Intersection(LeftDirectrix(G), xAxis) = B;Coordinate(A) = (0, b);VectorOf(A, P) = 3*VectorOf(A, B);PointOnCurve(P, G)",
+      "coords": {}
+    },
+    {
+      "index": 3634,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3634.png",
+      "answer": "(\\sqrt{2}/2, 1)",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);DotProduct(VectorOf(P, F1), VectorOf(P, F2))<0",
+      "coords": {}
+    },
+    {
+      "index": 3635,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3635.png",
+      "answer": "sqrt(5)/2",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;O: Origin;F: Point;M: Point;N: Point;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;IsDiameter(LineSegmentOf(O,F),G);Intersection(OneOf(Asymptote(C)),G)=M;Negation(M=O);L:Line;PointOnCurve(F,L);IsPerpendicular(L,xAxis);Intersection(L,OneOf(Asymptote(C)))=N;Area(TriangleOf(O,M,F))=4*Area(TriangleOf(M,N,F))",
+      "coords": {}
+    },
+    {
+      "index": 3636,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3636.png",
+      "answer": "y=pm*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/2 - y^2/m = 1);m: Number;H: Circle;Expression(H) = (-4*y - 3*x + x^2 + y^2 + 2 = 0);F: Point;RightFocus(G) = F;PointOnCurve(F, H)",
+      "coords": {}
+    },
+    {
+      "index": 3637,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3637.png",
+      "answer": "3*x-2*y-3=0",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);l: Line;PointOnCurve(Focus(G), l);H: Line;Expression(H) = (3*x - 2*y = 0);IsParallel(H, l)",
+      "coords": {}
+    },
+    {
+      "index": 3638,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3638.png",
+      "answer": "x^2/(4*a^2)+y^2/(4*b^2)=1",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;O: Origin;Q: Point;P: Point;F1: Point;F2:Point;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P,G);Focus(G)={F1,F2};VectorOf(O, Q) = VectorOf(P, F1) + VectorOf(P, F2)",
+      "coords": {}
+    },
+    {
+      "index": 3639,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.3484079837799072,
+        "b": 1.8638464212417603,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 4.620347226591548e-06,
+        "converged": true,
+        "iterations": 88
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3639.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "L: Hyperbola;b: Number;a: Number;F1: Point;F2: Point;P: Point;Expression(L) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-3, 0);Coordinate(F2) = (3, 0);LeftFocus(L) = F1;RightFocus(L) = F2;PointOnCurve(P, L);4*Abs(LineSegmentOf(P, F1)) = 5*Abs(LineSegmentOf(P, F2));DotProduct(VectorOf(P, F2), VectorOf(F1, F2)) = 0",
+      "coords": {
+        "F1": [
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+          0.0
+        ],
+        "F2": [
+          3.0,
+          0.0
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+      }
+    },
+    {
+      "index": 3640,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3640.png",
+      "answer": "8",
+      "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;M: Point;N: Point;F:Point;Expression(G) = (y^2 = 8*x);PointOnCurve(F,H);Focus(G)=F;Intersection(H,G)={A,B};MidPoint(LineSegmentOf(A,B))=M;L:Line;PointOnCurve(M,L);IsPerpendicular(L,yAxis);Intersection(L,yAxis)=N;Abs(LineSegmentOf(M, N)) = 2",
+      "coords": {}
+    },
+    {
+      "index": 3641,
+      "success": false,
+      "error": "Circle from diameter: missing point coordinates"
+    },
+    {
+      "index": 3642,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 2.0511727333068848,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 4.657696081267204e-06,
+        "converged": true,
+        "iterations": 116
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3642.png",
+      "answer": "3",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2/4 - y^2/m = 1);m: Number;m>0;Expression(Asymptote(C)) = (y = pm*(sqrt(2)*x));F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, RightPart(C)) = True;Abs(LineSegmentOf(P, F1)) = m - 1",
+      "coords": {}
+    },
+    {
+      "index": 3643,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3643.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G:Line;k: Number;A: Point;F2: Point;B: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y = k*x);RightFocus(C) = F2;Intersection(G, C) = {A, B};AngleOf(A, F2, B) = (2*pi)/3;Area(TriangleOf(A,F2,B))=2*sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 3644,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3644.png",
+      "answer": "(x+1)^2+(y-1/2)^2=1/2",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 2*y);H: Circle;PointOnCurve(Center(H),G) = True;I: Line;Expression(I) = (2*x + 2*y + 3 = 0);IsTangent(I,H) = True;WhenMin(Area(H))",
+      "coords": {}
+    },
+    {
+      "index": 3645,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3646,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3646.png",
+      "answer": "2",
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+      "answer": "sqrt(17)/2",
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+      "answer": "{0, 1}",
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+      "answer": "(a-p)/2",
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+      "answer": "x^2/5+y^2=1\n2*sqrt(5)/5",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3662.png",
+      "answer": "sqrt(2)/2",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3663.png",
+      "answer": "x^2/4 - y^2/2 = 1",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;O: Origin;F: Point;a > b;b > 0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;PointOnCurve(P,L);Expression(L)=(x+sqrt(2)*y=0);Abs(LineSegmentOf(O,F))=Abs(LineSegmentOf(P,F));Area(TriangleOf(P, O, F)) = 2*sqrt(2);L:Line;OneOf(Asymptote(C))=L",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_3664.png",
+      "answer": "[2*sqrt(3)-1,2*sqrt(3)+1]",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3665.png",
+      "answer": "4*sqrt(2)",
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+    {
+      "index": 3666,
+      "success": true,
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+        "b": 1.5,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3666.png",
+      "answer": "(1, \\sqrt{2}+1)",
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+      "index": 3667,
+      "success": true,
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+      "error": null,
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+        "direction": "up"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3667.png",
+      "answer": "y = 1",
+      "fact_expr": "G: Parabola;Expression(G) = (y = x^2/4);l:Line;PointOnCurve(Focus(G),l);IsPerpendicular(l,SymmetryAxis(G))",
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+      "index": 3668,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 3669,
+      "success": true,
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3669.png",
+      "answer": "(p, +\\infty)",
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+    {
+      "index": 3670,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 3671,
+      "success": true,
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3671.png",
+      "answer": "sqrt(29)",
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+      "index": 3672,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3672.png",
+      "answer": "4*sqrt(3)",
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+      "index": 3673,
+      "success": true,
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+        "b": 1.0,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3673.png",
+      "answer": "11",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2/9 - y^2 = 1);F1: Point;F2: Point;F1 = LeftFocus(C);F2 = RightFocus(C);A: Point;PointOnCurve(A, LeftPart(C));Abs(LineSegmentOf(A, F1)) = 5",
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+      "index": 3674,
+      "success": true,
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+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3674.png",
+      "answer": "2",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;F1: Point;P: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Focus(C) = {F1, F2};PointOnCurve(P, C);AngleOf(F1, P, F2) = ApplyUnit(120, degree);Area(TriangleOf(F1, P, F2)) = 4*sqrt(3)",
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+    {
+      "index": 3675,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "x_coef": 3.0,
+        "y_coef": 2.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3675.png",
+      "answer": "3*sqrt(2)/2",
+      "fact_expr": "G: Ellipse;A: Point;B: Point;C: Point;Expression(G) = (x^2/3 + y^2/2 = 1);LeftFocus(G) = A;UpperVertex(G)=B;PointOnCurve(C, G)",
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+    {
+      "index": 3676,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3676.png",
+      "answer": "(pm*3,0)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1)",
+      "coords": {}
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+    {
+      "index": 3677,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+      "answer": "4+sqrt(5)",
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+      "answer": "3*sqrt(3)",
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+      "answer": "2",
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+        "note": "using explicit params"
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+      "answer": "3*sqrt(3)",
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+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
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+      "answer": "16/5",
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+      "answer": "y^2=4*x",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3717.png",
+      "answer": "1/3",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+        "direction": "left"
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+        "note": "using explicit params"
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+      "answer": "(-6, \\pm 6\\sqrt{2})",
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+      "success": true,
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+      "error": null,
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+        "b": 1.0,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3721.png",
+      "answer": "2*x-y-15=0",
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+      "index": 3722,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "p": 0.0625,
+        "direction": "up"
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3722.png",
+      "answer": "65/16",
+      "fact_expr": "l: Line;G: Parabola;Expression(G) = (y = 4*x^2);Directrix(G) = l;A: Point;Coordinate(A) = (1, 4)",
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+          4.0
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+        "b": 3.4641016151377544,
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3723.png",
+      "answer": "(pm*4, 0)\nsqrt(3)*x+pm*y=0",
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+      "index": 3724,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3724.png",
+      "answer": "1",
+      "fact_expr": "G: Hyperbola;a: Number;H: Ellipse;Expression(G) = (-y^2/2 + x^2/a = 1);Expression(H) = (x^2/4 + y^2/a^2 = 1);Focus(H) = Focus(G)",
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+      "index": 3725,
+      "success": false,
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+      "error": "validation: ellipse_ecc_mismatch",
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+        "iterations": 2
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3725.png",
+      "answer": "3",
+      "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2/4 + y^2/m = 1);PointOnCurve(Focus(G), xAxis);Eccentricity(G)=1/2",
+      "coords": {},
+      "validation_reasons": [
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+      "success": true,
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+      "error": null,
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3726.png",
+      "answer": "sqrt(21)/7",
+      "fact_expr": "C: Ellipse;G: Line;A: Point;F1: Point;B: Point;F2: Point;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, G);Intersection(G, C) = {A, B};Abs(LineSegmentOf(A, F1)) = 4*Abs(LineSegmentOf(B, F1));AngleOf(A, F2, F1) = ApplyUnit(90, degree);b:Number;a:Number",
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+    {
+      "index": 3727,
+      "success": true,
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+        "x_coef": 2.0,
+        "y_coef": 4.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3727.png",
+      "answer": "x^2/2+y^2=1",
+      "fact_expr": "G: Ellipse;Expression(G) = (2*x^2 + y^2 = 4);D: Point;P: Point;M: Point;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P,D),xAxis);FootPoint(LineSegmentOf(P,D),xAxis)=D;MidPoint(LineSegmentOf(P,D))=M;PointOnCurve(P,LineSegmentOf(P,D))",
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+    {
+      "index": 3728,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3729,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3729.png",
+      "answer": "4/9",
+      "fact_expr": "l: Line;G: Hyperbola;M: Point;O: Origin;A: Point;B: Point;k1: Number;k2: Number;Expression(G) = (x^2/9 - y^2/4 = 1);Intersection(l, G) = {A, B};MidPoint(LineSegmentOf(A, B)) = M;Slope(l) = k1;Negation(k1 = 0);Slope(LineOf(O, M)) = k2",
+      "coords": {}
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+    {
+      "index": 3730,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3730.png",
+      "answer": "{-14,2}",
+      "fact_expr": "G: Parabola;a: Real;H: Circle;Expression(G) = (y^2 = 2*(a*x));Expression(H) = (-6*x + x^2 + y^2 - 7 = 0);IsTangent(Directrix(G),H)",
+      "coords": {}
+    },
+    {
+      "index": 3731,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3731.png",
+      "answer": "(\\sqrt{5}+1)/2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;O: Origin;M: Point;N: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(LeftFocus(G), H);IsPerpendicular(H,xAxis);Intersection(H, G) = {M, N};Center(G) = O;DotProduct(VectorOf(O, M), VectorOf(O, N)) = 0",
+      "coords": {}
+    },
+    {
+      "index": 3732,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3732.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "Gamma: Parabola;Expression(Gamma) = (y^2 = 4*x);A: Point;B: Point;PointOnCurve(A, Gamma);PointOnCurve(B, Gamma);M: Point;PointOnCurve(M, Directrix(Gamma));PointOnCurve(MidPoint(LineSegmentOf(M, A)), Gamma);PointOnCurve(MidPoint(LineSegmentOf(M, B)), Gamma);N: Point;n: Number;Coordinate(N) = (0, n);Intersection(LineOf(A, B), yAxis) = N",
+      "coords": {}
+    },
+    {
+      "index": 3733,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3734,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3734.png",
+      "answer": "y = pm*2*x",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;OneOf(Focus(C)) = F1;A1: Point;A2: Point;Vertex(C) = {A1,A2};M: Point;PointOnCurve(M, C);G: Circle;IsDiameter(LineSegmentOf(A1,A2),G);TangentPoint(LineSegmentOf(M, F1), G) = N;MidPoint(LineSegmentOf(M, F1)) = N;N:Point",
+      "coords": {}
+    },
+    {
+      "index": 3735,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3736,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
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+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3736.png",
+      "answer": "x^2/8+y^2/2=1",
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+      "answer": "{y^2=4*x,y^2=36*x}",
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+      "index": 3738,
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+        "b": 1.5,
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3738.png",
+      "answer": "5/3",
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+        "y_coef": 4.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3739.png",
+      "answer": "3/5",
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+    {
+      "index": 3740,
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3740.png",
+      "answer": "[\\sqrt{14}/2,2)",
+      "fact_expr": "C: Circle;Expression(C) = (y^2 + (x - 3)^2 = 1);M: Point;T: Parabola;Expression(T) = (y^2 = 4*x);PointOnCurve(M, T);l1: Line;l2: Line;TangentPoint(l1, C) = A;TangentPoint(l2, C) = B;A: Point;B: Point;TangentOfPoint(M,C)={l1,l2}",
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+    {
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+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3741.png",
+      "answer": "sqrt(2)",
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3742.png",
+      "answer": "6",
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+          -3.0
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+      "index": 3743,
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+        "direction": "up"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3743.png",
+      "answer": "8",
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+      "coords": {}
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+    {
+      "index": 3744,
+      "success": true,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3744.png",
+      "answer": "x^2/6 + y^2/2 = 1",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3745.png",
+      "answer": "sqrt(2)",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3746.png",
+      "answer": "sqrt(3)",
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+        "note": "using explicit params"
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+      "answer": "x=3",
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+    {
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+        "note": "using explicit params"
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+      "answer": "4",
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+      "index": 3749,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3750.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;M: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(M, G);Distance(M, Focus(G)) = 5",
+      "coords": {}
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3751.png",
+      "answer": "8/3",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3752.png",
+      "answer": "4",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3753.png",
+      "answer": "sqrt(2)",
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+        "x_coef": 0.1111111111111111,
+        "y_coef": 1.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3755.png",
+      "answer": "-9",
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+        "note": "using explicit params"
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+      "answer": "8",
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+        "note": "using explicit params"
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+      "answer": "y=pm*sqrt(3)*x",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3759.png",
+      "answer": "16",
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+    {
+      "index": 3760,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 3761,
+      "success": true,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3761.png",
+      "answer": "9",
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+    {
+      "index": 3762,
+      "success": true,
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+      "error": null,
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+        "direction": "left"
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+      "optimization": {
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3762.png",
+      "answer": "(x+3/2)^2+y^2=9",
+      "fact_expr": "G: Parabola;H: Circle;Expression(G) = (y^2 = -6*x);Focus(G) = Center(H);IsTangent(Directrix(G),H)",
+      "coords": {}
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+    {
+      "index": 3763,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_point_off_curve",
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+        "iterations": 83
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3763.png",
+      "answer": "8",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Eccentricity(G) = sqrt(2);H: Point;Coordinate(H) = (3, 1);PointOnCurve(H, G)",
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+      "validation_reasons": [
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+    {
+      "index": 3764,
+      "success": true,
+      "conic_type": "ellipse",
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+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3764.png",
+      "answer": "2/3",
+      "fact_expr": "C: Ellipse;l: Line;O: Origin;A: Point;B: Point;Expression(C) = (x^2/2 + y^2 = 1);Expression(l) = (y = x - 1);PointOnCurve(RightFocus(C),l);Intersection(l, C) = {A, B}",
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+      "index": 3765,
+      "success": true,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3766.png",
+      "answer": "sqrt(6)/2",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/4 + x^2/a^2 = 1);a: Number;FocalLength(C) = 4*sqrt(3)",
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+        "note": "using explicit params"
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+      "answer": "pm*sqrt(2)/2",
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+        "note": "using explicit params"
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+      "answer": "9",
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+      "error": "Unsupported or unparseable expression"
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+      "answer": "{(x^2/8+y^2/2=1),(x^2/3-y^2/3=1)}",
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+      "answer": "9",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3775.png",
+      "answer": "9*sqrt(3)",
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+        "direction": "left"
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+        "note": "using explicit params"
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+      "answer": "2*sqrt(2) - 2",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3779.png",
+      "answer": "x^2/4 - y^2/12 = 1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Expression(OneOf(Asymptote(G))) = (y = sqrt(3)*x);H: Parabola;Expression(H) = (y^2 = 16*x);OneOf(Focus(G)) = Focus(H)",
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+        "note": "using explicit params"
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+      "answer": "6*sqrt(2)/7",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_3781.png",
+      "answer": "4",
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+        "direction": "up"
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+        "note": "using explicit params"
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+      "answer": "4*sqrt(2)",
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+      "answer": "5",
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+        "note": "using explicit params"
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+      "answer": "[-2,1]",
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+      "conic_type": "parabola",
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3786.png",
+      "answer": "2",
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+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3787.png",
+      "answer": "3/2",
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+      "success": false,
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+      "error": "validation: ellipse_point_off_curve",
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+        "converged": true,
+        "note": "using explicit params"
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+      "answer": "{(x^2/10+y^2/8=1),(41*y^2/4+41*x^2/5=1)}",
+      "fact_expr": "C: Curve;Expression(C) = (x^2/5 + y^2/4 = 1);G: Ellipse;Eccentricity(C) = Eccentricity(G);H: Point;Coordinate(H) = (sqrt(5), 2);PointOnCurve(H, G) = True",
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+        "direction": "up"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3789.png",
+      "answer": "(-\\infty,-1)+(1,+\\infty)",
+      "fact_expr": "Gamma:Parabola;H: Line;A: Point;B: Point;t: Real;Expression(Gamma) = (x^2=2*y);Coordinate(A) = (0,-2);Coordinate(B) = (t, 0);PointOnCurve(A,H);PointOnCurve(B,H);NumIntersection(H,Gamma)=0",
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+      "index": 3790,
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+        "iterations": 95
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3790.png",
+      "answer": "{sqrt(5),sqrt(5)/2}",
+      "fact_expr": "C: Hyperbola;O: Origin;Expression(Asymptote(C)) = (y = pm*2*x);PointOnCurve(Focus(C),axis);Center(C)=O",
+      "coords": {}
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+    {
+      "index": 3791,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3791.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;A: Point;B: Point;D: Point;C: Point;L1:Line;L2:Line;H:LineSegment;Expression(G) = (y^2 = 4*x);Focus(G)=F;PointOnCurve(F,H);Intersection(H,G)={A,B};PointOnCurve(A,L1);PointOnCurve(B,L2);IsPerpendicular(L1,yAxis);IsPerpendicular(L2,yAxis);FootPoint(L1,yAxis)=C;FootPoint(L2,yAxis)=D;IsChordOf(H,G);F:Point",
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+    {
+      "index": 3792,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 3793,
+      "success": true,
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+      "answer": "5/2",
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+      "success": false,
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+      "index": 3809,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 3810,
+      "success": true,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3810.png",
+      "answer": "{pi/3, 2*pi/3}",
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+        "direction": "up"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3811.png",
+      "answer": "(0,1)",
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+        "direction": "up"
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3812.png",
+      "answer": "15/16",
+      "fact_expr": "G: Parabola;Expression(G) = (y = 4*x^2);M: Point;PointOnCurve(M, G);Distance(M, Focus(G)) = 1",
+      "coords": {}
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+    {
+      "index": 3813,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 2.449489742783178,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3813.png",
+      "answer": "24",
+      "fact_expr": "G: Hyperbola;M: Point;F2: Point;F1: Point;N: Point;Expression(G) = (x^2/4 - y^2/6 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(M,RightPart(G));DotProduct(VectorOf(M,F1),VectorOf(M,F2))=0;OneOf(Intersection(LineOf(M,F2),G))=N",
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+    {
+      "index": 3814,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 3.0,
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+        "y_coef": 9.0
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3814.png",
+      "answer": "4",
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+        "b": 1.5,
+        "orientation": "vertical"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3815.png",
+      "answer": "5/4",
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+      "success": true,
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+          0.0
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_3816.png",
+      "answer": "x^2/9+y^2/8=1",
+      "fact_expr": "C1: Circle;Expression(C1) = ((x+1)^2 + y^2 = 1);C2: Circle;Expression(C2) = ((x-1)^2 + y^2 = 25);M: Circle;IsOutTangent(M, C1) ;IsInTangent(M, C2);M1: Point;Center(M) = M1",
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+    {
+      "index": 3817,
+      "success": true,
+      "conic_type": "parabola",
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3817.png",
+      "answer": "(1/2, 0)  \nx = -1/2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x)",
+      "coords": {}
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+    {
+      "index": 3818,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 3.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3818.png",
+      "answer": "2*sqrt(10)",
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+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "orientation": "horizontal"
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+        "converged": true,
+        "iterations": 2
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3819.png",
+      "answer": "y=pm*3/4*x",
+      "fact_expr": "E: Hyperbola;a: Number;b: Number;a>0;b>0;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(E) = 5/4",
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+    {
+      "index": 3820,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3820.png",
+      "answer": "-3/4",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);M: Point;N: Point;Endpoint(MajorAxis(G)) = {M, N};P: Point;Negation(P=M);Negation(P=N);PointOnCurve(P, G)",
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+    },
+    {
+      "index": 3821,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3822,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "a": 2.0,
+        "b": 3.4641016151377544,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3822.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/12 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 3823,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3823.png",
+      "answer": "x^2/2+y^2=1",
+      "fact_expr": "C: Ellipse;O: Origin;Center(C) = O;PointOnCurve(Focus(C), xAxis);Eccentricity(C) = sqrt(2)/2;G: Parabola;Expression(G) = (x^2 = 4*y);OneOf(Vertex(C)) = Focus(G)",
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+    {
+      "index": 3824,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3824.png",
+      "answer": "y=-1/4",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = y)",
+      "coords": {}
+    },
+    {
+      "index": 3825,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3826,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_focus_mismatch",
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3826.png",
+      "answer": "x^2/5+y^2/4=1",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;PointOnCurve(Focus(G), xAxis);H: Circle;Expression(H) = (x^2 + y^2 = 1);F: Point;Coordinate(F) = (1, 1/2);L1: Line;L2: Line;TangentOfPoint(F, H) = {L1, L2};A: Point;B: Point;TangentPoint(L1, H) = A;TangentPoint(L2, H) = B;PointOnCurve(RightFocus(G), LineOf(A, B));PointOnCurve(UpperVertex(G), LineOf(A, B))",
+      "coords": {
+        "F": [
+          1.0,
+          0.5
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+      "validation_reasons": [
+        "ellipse_focus_mismatch"
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+    },
+    {
+      "index": 3827,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3827.png",
+      "answer": "6",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;A: Point;PointOnCurve(A, C);Distance(A, Focus(C)) = 12;Distance(A, yAxis) = 9",
+      "coords": {}
+    },
+    {
+      "index": 3828,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3828.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;B: Point;A: Point;F: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F;RightVertex(G)=A;OneOf(Endpoint(ImageinaryAxis(G)))=B;Distance(F,LineOf(A,B)) = b/2",
+      "coords": {}
+    },
+    {
+      "index": 3829,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3830,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3830.png",
+      "answer": "2",
+      "fact_expr": "E: Hyperbola;b: Number;a: Number;T: Ellipse;a>0;b>0;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);Expression(T) = (x^2/4 + y^2/3 = 1);LeftVertex(E) = LeftFocus(T);RightVertex(E) = RightFocus(T);LeftVertex(T) = LeftFocus(E);RightVertex(T) = RightFocus(E)",
+      "coords": {}
+    },
+    {
+      "index": 3831,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3831.png",
+      "answer": "8",
+      "fact_expr": "G: Ellipse;H: Line;F2: Point;A: Point;B: Point;F1: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Focus(G) = {F1, F2};PointOnCurve(F1, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(F2, A)) + Abs(LineSegmentOf(F2, B)) = 12",
+      "coords": {}
+    },
+    {
+      "index": 3832,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 6.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 36.0,
+        "y_coef": 9.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3832.png",
+      "answer": "x+2*y-8=0",
+      "fact_expr": "M: Point;Coordinate(M) = (4, 2);l: Line;G: Ellipse;Expression(G) = (x^2 + 4*y^2 = 36);A: Point;B: Point;MidPoint(LineSegmentOf(A, B)) = M;InterceptChord(l, G) = LineSegmentOf(A, B)",
+      "coords": {
+        "M": [
+          4.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 3833,
+      "success": false,
+      "error": "Circle from diameter: missing point coordinates"
+    },
+    {
+      "index": 3834,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.4142135623730951,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3834.png",
+      "answer": "3/4",
+      "fact_expr": "C: Hyperbola;P: Point;F1: Point;F2: Point;Expression(C) = (x^2 - y^2 = 2);LeftFocus(C) =F1;RightFocus(C)=F2;PointOnCurve(P, C);Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 3835,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 2.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3835.png",
+      "answer": "{2, 18}",
+      "fact_expr": "G: Hyperbola;P: Point;Expression(G) = (x^2/16 - y^2/4 = 1);PointOnCurve(P, G);Distance(P, F1) = 10;F1: Point;F2: Point;OneOf(Focus(G)) = F1;OneOf(Focus(G)) = F2;Negation(F1 = F2)",
+      "coords": {}
+    },
+    {
+      "index": 3836,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3836.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "C1:Parabola;C2:Hyperbola;F: Point;A:Point;a:Number;b:Number;a>0;b>0;Expression(C1)=(y^2=4*x);Expression(C2)=(x^2/a^2-y^2/b^2=1);Focus(C1)=F;OneOf(Intersection(C1,OneOf(Asymptote(C2))))=A;IsPerpendicular(LineSegmentOf(A,F),xAxis)",
+      "coords": {}
+    },
+    {
+      "index": 3837,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 2.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3837.png",
+      "answer": "y=pm*sqrt(2)*(x-sqrt(2))",
+      "fact_expr": "l: Line;G: Ellipse;H: Circle;A: Point;B: Point;F1: Point;F2: Point;O: Origin;Expression(G) = (x^2/4 + y^2/2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2,l);Intersection(l, G) = {A,B};IsDiameter(LineSegmentOf(A, B),H);PointOnCurve(O, H)",
+      "coords": {}
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+    {
+      "index": 3838,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3838.png",
+      "answer": "sqrt(5)/2",
+      "fact_expr": "E: Hyperbola;b: Number;a: Number;P: Point;F: Point;Q: Point;a>0;b>0;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(E) = F;L1:Line;L2:Line;L:Line;OneOf(Asymptote(E))=L1;OneOf(Asymptote(E))=L2;Negation(L1=L2);PointOnCurve(F,L);IsPerpendicular(L1,L);FootPoint(L1,L)=P;Intersection(L2,L)=Q;5*VectorOf(P, F) = 3*VectorOf(F, Q)",
+      "coords": {}
+    },
+    {
+      "index": 3839,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3839.png",
+      "answer": "1",
+      "fact_expr": "C: Parabola;p: Number;G: Circle;F: Point;A: Point;B: Point;D: Point;l: Line;p>0;Expression(C) = (y^2 = 2*p*x);Focus(C) = F;Directrix(C) = l;PointOnCurve(A, C);Center(G)=F;Radius(G)=LineSegmentOf(F,A);Intersection(G,l)={B,D};AngleOf(B, F, D) = ApplyUnit(120, degree);Area(TriangleOf(A,B,D))=2*sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 3840,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_point_off_curve",
+      "params": {
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+        "b": 1.4970015287399292,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "iterations": 2
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3840.png",
+      "answer": "2*sqrt(13)",
+      "fact_expr": "G: Hyperbola;a: Number;H: Point;Expression(G) = (-y^2/4 + x^2/a^2 = 1);Coordinate(H) = (-3*sqrt(2), 2);PointOnCurve(H, G)",
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+          2.0
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+        "hyperbola_point_off_curve"
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+    {
+      "index": 3841,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3842,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3843,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3844,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3844.png",
+      "answer": "(\\sqrt{5}-1)/2",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;A: Point;B: Point;F: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftVertex(G)=A;Endpoint(MinorAxis(G))=B;RightFocus(G)=F;IsPerpendicular(LineSegmentOf(A, B), LineSegmentOf(B, F))",
+      "coords": {}
+    },
+    {
+      "index": 3845,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3846,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3846.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
+      "fact_expr": "G: Hyperbola;C:Hyperbola;A: Point;Expression(G) = (x^2/16 - y^2/9 = 1);Coordinate(A) = (2*sqrt(3), -3);Asymptote(G) = Asymptote(C);PointOnCurve(A, C)",
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+    {
+      "index": 3847,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3847.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F2: Point;F1: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, LeftPart(G));LeftFocus(G) = F1;RightFocus(G) = F2;PerpendicularBisector(LineSegmentOf(P, F2)) = OneOf(Asymptote(G))",
+      "coords": {}
+    },
+    {
+      "index": 3848,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3848.png",
+      "answer": "2*sqrt(2)-sqrt(6)",
+      "fact_expr": "G: Ellipse;O: Origin;A: Point;B: Point;E: Point;P: Point;Expression(G) = (x^2/4 + y^2 = 1);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);x1:Number;x2:Number;y1:Number;y2:Number;Coordinate(E) = (sqrt(6), 0);PointOnCurve(A, G);PointOnCurve(B, G);H:Line;C:Line;PointOnCurve(A,H);PointOnCurve(B,C);Slope(H)=x1/(4*y1);Slope(C)=x2/(4*y2);Intersection(H,C) = P;Slope(LineOf(O,A))*Slope(LineOf(O,B))=-1/4",
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+          0.0
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+    },
+    {
+      "index": 3849,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_point_off_curve",
+      "params": {
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+        "b": 1.6118527266061986,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 1.4665974334810584e-06,
+        "converged": true,
+        "iterations": 90
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3849.png",
+      "answer": "2",
+      "fact_expr": "C1: Parabola;Expression(C1) = (y = a*x^2);a: Number;a>0;F: Point;Focus(C1) = F;C2: Ellipse;Expression(C2) = (y^2/4 + x^2/b^2 = 1);b: Number;b>0;OneOf(Focus(C2)) = F;M: Point;PointOnCurve(M, C1);P: Point;Coordinate(P) = (3/2, 1);PointOnCurve(P, C2)",
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+        "P": [
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+          1.0
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+      "validation_reasons": [
+        "ellipse_point_off_curve"
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+    {
+      "index": 3850,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3851,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3851.png",
+      "answer": "sqrt(3)/3",
+      "fact_expr": "A: Point;Coordinate(A) = (0, b);b: Number;B: Point;G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);a: Number;a > b;b > 0;Intersection(LeftDirectrix(G), xAxis) = B;MidPoint(LineSegmentOf(A, B)) = C;C: Point;PointOnCurve(C, G) = True",
+      "coords": {}
+    },
+    {
+      "index": 3852,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3852.png",
+      "answer": "sqrt(10)/3",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;LeftFocus(G) = F;l: Line;PointOnCurve(F, l);IsPerpendicular(OneOf(Asymptote(G)), l);A: Point;B: Point;Intersection(l, LeftPart(G)) = {A, B};VectorOf(A, F) = 2*VectorOf(F, B);Negation(A=B)",
+      "coords": {}
+    },
+    {
+      "index": 3853,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3854,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3855,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3856,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.366584062576294,
+        "b": 2.364173412322998,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.00017838670464698225,
+        "converged": true,
+        "iterations": 89
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3856.png",
+      "answer": "1",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;p: Number;A: Point;O: Origin;B: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);p>0;Expression(H) = (y^2 = 2*(p*x));Eccentricity(G) = 2;L1: Line;L2: Line;Asymptote(G) = {L1, L2};Intersection(L1, Directrix(H)) = A;Intersection(L2, Directrix(H)) = B;Area(TriangleOf(A, O, B)) = sqrt(3)/4",
+      "coords": {}
+    },
+    {
+      "index": 3857,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3857.png",
+      "answer": "6",
+      "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;Expression(G) = (x^2 = 4*y);Coordinate(A) = (-1, 5);Focus(G) = F;PointOnCurve(P, G)",
+      "coords": {
+        "A": [
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+          5.0
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+    },
+    {
+      "index": 3858,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3858.png",
+      "answer": "2/3",
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+    {
+      "index": 3859,
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+      "answer": "sqrt(10)",
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+      "answer": "y^2=16*x",
+      "fact_expr": "G: Hyperbola;H: Parabola;Expression(G) = (x^2/6 - y^2/10 = 1);Vertex(H) = Center(G);Focus(H) = RightFocus(G)",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3861.png",
+      "answer": "sqrt(2)-1",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3864.png",
+      "answer": "sqrt(5)/3",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F1: Point;O: Origin;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(P,G);DotProduct(VectorOf(P,F1),(VectorOf(O,F1)+VectorOf(O,P)))=0;Abs(LineSegmentOf(P,F1))=2*Abs(LineSegmentOf(P,F2))",
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+      "index": 3866,
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+        "y_coef": 3.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3866.png",
+      "answer": "{4, 1}",
+      "fact_expr": "G: Ellipse;m: Number;Expression(G) = (y^2/2 + x^2/m = 1);Eccentricity(G) = sqrt(2)/2",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3867.png",
+      "answer": "3/2",
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+    {
+      "index": 3868,
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+        "y_coef": 3.0
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3868.png",
+      "answer": "(\\sqrt{5}-1)/2",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;A: Point;B: Point;F: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);RightVertex(G) = A;UpperVertex(G) = B;LeftFocus(G) = F;AngleOf(A, B, F) = ApplyUnit(90, degree)",
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+    {
+      "index": 3869,
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3869.png",
+      "answer": "1/2",
+      "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;M: Point;k: Number;x0: Number;Expression(G) = (y^2 = 4*x);Coordinate(M) = (x0, 4);Intersection(l, G) = {A, B};MidPoint(LineSegmentOf(A, B)) = M;Slope(l) = k;Negation(A = B)",
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+    {
+      "index": 3870,
+      "success": true,
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+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3870.png",
+      "answer": "3*sqrt(2)/4",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;H:Line;F1: Point;e: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y =(b/(3*a))*x);Intersection(H,LeftPart(G))=P;LeftFocus(G) = F1;IsPerpendicular(LineSegmentOf(P,F1),xAxis);Eccentricity(G) = e",
+      "coords": {}
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+    {
+      "index": 3871,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3871.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;L:Line;A: Point;B: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = 1 - x);Expression(L) = (x - 2*y = 0);Intersection(H, G) = {A, B};PointOnCurve(MidPoint(LineSegmentOf(A,B)), L)",
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+    {
+      "index": 3872,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3872.png",
+      "answer": "(1,0)\n7",
+      "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (6, 3);Focus(G)=F;PointOnCurve(P,G)",
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+          3.0
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+      "index": 3873,
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+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3873.png",
+      "answer": "12",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);F: Point;Focus(G) = F;A: Point;B: Point;C: Point;PointOnCurve(A, G);PointOnCurve(B, G);PointOnCurve(C, G);VectorOf(F, A) + VectorOf(F, B) + VectorOf(F, C) = 0",
+      "coords": {}
+    },
+    {
+      "index": 3874,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3874.png",
+      "answer": "y=pm*(sqrt(3)/3)*x",
+      "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;N: Point;F1: Point;M: Point;F2: Point;a > b;b > 0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1,l);L1:Line;L2:Line;Asymptote(C)={L1,L2};IsPerpendicular(l,L1);Intersection(l,L1)=M;Intersection(l,L2)=N;Abs(LineSegmentOf(N, F1)) = 2*Abs(LineSegmentOf(M, F1))",
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+    {
+      "index": 3875,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3875.png",
+      "answer": "(1,1)",
+      "fact_expr": "C: Parabola;M: Point;F: Point;Expression(C) = (y^2 = x);Focus(C) = F;Quadrant(M)=1;PointOnCurve(M,C);Abs(LineSegmentOf(M, F)) = 5/4",
+      "coords": {}
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+    {
+      "index": 3876,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3876.png",
+      "answer": "y^2 = 8*x",
+      "fact_expr": "C: Parabola;p: Number;G: Point;y0: Number;Expression(C) = (y^2 = 2*(p*x));Coordinate(G) = (1, y0);PointOnCurve(G, C);Distance(G, Focus(C)) = 3",
+      "coords": {}
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+    {
+      "index": 3877,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3877.png",
+      "answer": "9",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);P: Point;PointOnCurve(P, G);F1: Point;F2: Point;Focus(G) = {F1, F2};IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2))",
+      "coords": {}
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+    {
+      "index": 3878,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 3879,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3880,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 3881,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "up"
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3881.png",
+      "answer": "12*sqrt(2)",
+      "fact_expr": "G: Parabola;Expression(G) = (y = x^2/8);P: Point;M: Point;FootPoint(l, Directrix(G)) = M;F: Point;Abs(LineSegmentOf(P, M)) = 6;Focus(G) = F;l:Line;PointOnCurve(P,l);IsPerpendicular(l,Directrix(G));PointOnCurve(P,G)",
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+    },
+    {
+      "index": 3882,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_3882.png",
+      "answer": "(x^2/4-y^2/12=1)&(x>=2)",
+      "fact_expr": "M: Circle;C1: Circle;C2: Circle;M1: Point;Expression(C1)=((x+4)^2+y^2=4);Expression(C2)=((x-4)^2+y^2=4);IsOutTangent(M,C1);IsInTangent(M,C2);Center(M)=M1;Locus(M1)=C;C:Curve",
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+    {
+      "index": 3883,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 3884,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "vertical"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3884.png",
+      "answer": "(0,15)",
+      "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (y^2/5 - x^2/m = 1);Eccentricity(G)=e;e:Number;In(e,(1,2))",
+      "coords": {}
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+    {
+      "index": 3885,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 4.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3885.png",
+      "answer": "2",
+      "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/16 + y^2/4 = 1);PointOnCurve(P, G);F1:Point;F2:Point;OneOf(Focus(G))=F1;OneOf(Focus(G))=F2;Negation(F1=F2);Distance(P, F1) = 6",
+      "coords": {}
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+    {
+      "index": 3886,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3886.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;m: Number;H: Ellipse;Expression(G) = (-y^2/5 + x^2/m = 1);Expression(H) = (x^2/25 + y^2/16 = 1);Focus(H) = Focus(G)",
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+    {
+      "index": 3887,
+      "success": true,
+      "conic_type": "parabola",
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3887.png",
+      "answer": "-4",
+      "fact_expr": "G: Parabola;p: Number;H: Line;A: Point;B: Point;p>0;x1:Number;y1:Number;x2:Number;y2:Number;Expression(G) = (y^2 = 2*(p*x));Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);PointOnCurve(Focus(G), H);Intersection(H, G) = {A,B}",
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+    {
+      "index": 3888,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3888.png",
+      "answer": "6\nx=-3",
+      "fact_expr": "C: Parabola;p: Real;G: Line;Expression(C) = (y^2 = 2*(p*x));Expression(G) = (x + y - 3 = 0);PointOnCurve(Focus(C), G)",
+      "coords": {}
+    },
+    {
+      "index": 3889,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 2.23606797749979,
+        "major_axis": "y",
+        "x_coef": 5.0,
+        "y_coef": 16.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3889.png",
+      "answer": "8",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/5 + y^2/16 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G)",
+      "coords": {}
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+    {
+      "index": 3890,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "up"
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+      "optimization": {
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+      "answer": "x^2/4+y^2/3=1",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3904.png",
+      "answer": "8*sqrt(2)/3",
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+      "answer": "1",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G) = True;LineSegmentOf(P, F1)*LineSegmentOf(P, F2) = 0",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3906.png",
+      "answer": "15*sqrt(7)/7",
+      "fact_expr": "P: Point;Q: Point;H: Circle;Expression(H) = (x^2 + (y + 1)^2 = 7);G: Ellipse;Expression(G) = (x^2/8 + y^2 = 1);PointOnCurve(P, H) = True;PointOnCurve(Q, G) = True",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3907.png",
+      "answer": "16",
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+      "index": 3908,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3908.png",
+      "answer": "[-3*sqrt(10)/2,3*sqrt(10)/2]&Negation(b=0)",
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+      "index": 3909,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3909.png",
+      "answer": "9/4",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2 + x^2/a^2 = 1);a: Number;a>1;A: Point;UpperVertex(G) = A;B: Point;LeftVertex(G) = B;P: Point;PointOnCurve(P, G);Max(Area(TriangleOf(P, A, B))) = 1 + sqrt(2);M: Point;Coordinate(M) = (-sqrt(3), 0);N: Point;Coordinate(N) = (sqrt(3), 0);Q: Point;PointOnCurve(Q, G)",
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+      "index": 3910,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 3911,
+      "success": true,
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+      "answer": "2*sqrt(3)/3",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;A: Point;F: Point;B: Point;L1: Line;L2: Line;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;L:Line;OneOf(Asymptote(C))=L1;OneOf(Asymptote(C))=L2;Negation(L1=L2);PointOnCurve(F,L);IsPerpendicular(L,L1);FootPoint(L,L1)=A;Intersection(L,L2)=B;2*VectorOf(A, F) = VectorOf(F, B)",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3912.png",
+      "answer": "11/4",
+      "fact_expr": "G: Parabola;Expression(G) = (y = x^2);B: Point;A: Point;Length(LineSegmentOf(A,B)) = 6;PointOnCurve(Endpoint(LineSegmentOf(A,B)), G);MidPoint(LineSegmentOf(A,B)) = M;M: Point",
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+    {
+      "index": 3913,
+      "success": true,
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+      "error": null,
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3913.png",
+      "answer": "16",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l1: Line;l2: Line;PointOnCurve(F, l1);PointOnCurve(F, l2);IsPerpendicular(l1, l2);A: Point;B: Point;Intersection(l1, C) = {A, B};D: Point;E: Point;Intersection(l2, C) = {D, E}",
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+    {
+      "index": 3914,
+      "success": true,
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+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3914.png",
+      "answer": "sqrt(2)",
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+    {
+      "index": 3915,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 1.5,
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3915.png",
+      "answer": "{sqrt(2),sqrt(17)}",
+      "fact_expr": "l: Line;G: Hyperbola;b: Number;a: Number;F1: Point;P: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(F1,l);Intersection(l,G)=P;Sin(AngleOf(F1, P, F2)) = 4/5",
+      "coords": {}
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+    {
+      "index": 3916,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_point_off_curve",
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+        "b": 2.298015331604694,
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+        "iterations": 91
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+      "answer": "16*sqrt(3)",
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+      "answer": "y = x",
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+      "answer": "3/5",
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+      "answer": "4",
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+        "note": "using explicit params"
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+      "answer": "(2,2)",
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+      "index": 3931,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3932.png",
+      "answer": "4 + 2*sqrt(2)",
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+    {
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3933.png",
+      "answer": "-2*sqrt(2)",
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+        "y_coef": 3.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3934.png",
+      "answer": "1/2",
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+      "index": 3935,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3936.png",
+      "answer": "pi/2",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3937.png",
+      "answer": "{42, 22}",
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+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3938.png",
+      "answer": "sqrt(3)",
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+      "success": false,
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+        "note": "using explicit params"
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+      "answer": "y=pm*x",
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+        "converged": true
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+      "answer": "6",
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+    {
+      "index": 3942,
+      "success": true,
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3942.png",
+      "answer": "16",
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+    {
+      "index": 3943,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3943.png",
+      "answer": "-8; 2*sqrt(2)",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;P: Point;Coordinate(P) = (2, 0);H: Line;PointOnCurve(P,H);A: Point;B: Point;Intersection(H, G) = {A, B};x1: Number;x2: Number;y1: Number;y2: Number;Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2)",
+      "coords": {
+        "P": [
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+          0.0
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+    },
+    {
+      "index": 3944,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "p": 0.125,
+        "direction": "up"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3944.png",
+      "answer": "-1/4",
+      "fact_expr": "G: Parabola;H:Line;m: Number;Expression(G) = (y = 2*x^2);Expression(H) = (m + x + 2*y = 0);PointOnCurve(Focus(G),H)",
+      "coords": {}
+    },
+    {
+      "index": 3945,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_focus_mismatch",
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3945.png",
+      "answer": "(1, \\sqrt{5}]",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;Coordinate(F) = (3*sqrt(5), 0);RightFocus(C) = F;N: Point;Coordinate(N) = (0, 2);M: Point;PointOnCurve(M, LeftPart(C));Negation(Perimeter(TriangleOf(M, N, F)) < 20)",
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+        "N": [
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+          2.0
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+      "validation_reasons": [
+        "hyperbola_focus_mismatch"
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+    {
+      "index": 3946,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3946.png",
+      "answer": "3/2",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2 - y^2/3 = 1);F: Point;OneOf(Focus(C))=F;P: Point;PointOnCurve(P, C);O: Origin;Abs(LineSegmentOf(O, P)) = Abs(LineSegmentOf(O, F))",
+      "coords": {}
+    },
+    {
+      "index": 3947,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_focus_mismatch",
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3947.png",
+      "answer": "[2,+oo)",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;F: Point;O: Origin;P: Point;Q: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = ((x + 2)^2 + (y - 4)^2 = 4);Coordinate(F) = (2, 0);PointOnCurve(P, G);RightFocus(C) = F;PointOnCurve(Q,Asymptote(C));VectorOf(O,P)+VectorOf(O,F)=2*VectorOf(O,Q)",
+      "coords": {
+        "F": [
+          2.0,
+          0.0
+        ]
+      },
+      "validation_reasons": [
+        "hyperbola_focus_mismatch"
+      ]
+    },
+    {
+      "index": 3948,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3948.png",
+      "answer": "17/4",
+      "fact_expr": "G: Parabola;A: Point;M: Point;F:Point;Coordinate(A) = (4, 1);Expression(G) = (y^2 = x);PointOnCurve(M, G);Focus(G)=F",
+      "coords": {
+        "A": [
+          4.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 3949,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3950,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "vertical"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3950.png",
+      "answer": "14",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-x^2/64 + y^2/16 = 1);F1: Point;F2: Point;Focus(C) = {F1, F2};P: Point;PointOnCurve(P, C);Abs(LineSegmentOf(P, F1)) = 6",
+      "coords": {}
+    },
+    {
+      "index": 3951,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3952,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3953,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3954,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3955,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3955.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "P: Point;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;PointOnCurve(P, RightPart(G)) = True;Negation(Vertex(G) = P);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;M: Point;Incenter(TriangleOf(P, F1, F2)) = M;Area(TriangleOf(M, P, F1)) - Area(TriangleOf(M, P, F2)) = sqrt(2)*Area(TriangleOf(M, F1, F2))/2",
+      "coords": {}
+    },
+    {
+      "index": 3956,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3957,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3957.png",
+      "answer": "sqrt(3)/3",
+      "fact_expr": "l: Line;C: Ellipse;b: Number;a: Number;A: Point;F1: Point;B: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;FocalLength(C) = 2;PointOnCurve(F2, l);Intersection(l, C) = {A, B};Perimeter(TriangleOf(A, F1, B)) = 4*sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 3958,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3958.png",
+      "answer": "5/3",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, RightPart(C));Abs(LineSegmentOf(P, F2)) = Abs(LineSegmentOf(F1, F2));Distance(F2,LineOf(P,F1)) = 2*a",
+      "coords": {}
+    },
+    {
+      "index": 3959,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3959.png",
+      "answer": "2*sqrt(6)/3",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;B: Point;A: Point;F: Point;l1: Line;l2: Line;e: Number;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Asymptote(C) = {l1, l2};RightFocus(C) = F;L: Line;PointOnCurve(F, L);IsPerpendicular(L, l1);FootPoint(L, l1) = A;Intersection(L, l2) = B;VectorOf(B, A) = 3*VectorOf(A, F);Eccentricity(C) = e",
+      "coords": {}
+    },
+    {
+      "index": 3960,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3961,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3962,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3963,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3963.png",
+      "answer": "[2,+oo)+{0}",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;F1:Point;Intersection(Directrix(C),xAxis)=F1;A: Point;B: Point;PointOnCurve(A,C);PointOnCurve(B,C);PointOnCurve(F1,LineOf(A,B));l1:Line;l2:Line;PointOnCurve(F,l1);PointOnCurve(F,l2);IsParallel(l1,LineOf(O,A));IsParallel(l2,LineOf(O,B));O: Origin;P:Point;Intersection(l1,l2)=P;E: Curve;Locus(P)=E;H:Line;k:Number;k>=0;Expression(H)=(y=k*x);NumIntersection(H,E)=0",
+      "coords": {}
+    },
+    {
+      "index": 3964,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 3.4641016151377544,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3964.png",
+      "answer": "12*sqrt(3)",
+      "fact_expr": "G: Hyperbola;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 - y^2/12 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(60, degree)",
+      "coords": {}
+    },
+    {
+      "index": 3965,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3965.png",
+      "answer": "5/8",
+      "fact_expr": "l: Line;G: Parabola;A: Point;O: Origin;B: Point;F: Point;Expression(G) = (y^2 = 2*x);Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B};Abs(LineSegmentOf(A, F)) = 4*Abs(LineSegmentOf(B, F))",
+      "coords": {}
+    },
+    {
+      "index": 3966,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_point_off_curve",
+      "params": {
+        "a": 1.9960020780563354,
+        "b": 1.4970015287399292,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "iterations": 2
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3966.png",
+      "answer": "y=pm*(sqrt(3)/2)*x",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Parabola;H: Circle;P: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y^2 = -16*x);Expression(H) = (y^2 - 2*a*x + x^2 = 0);Coordinate(P) = (4, 3);IsTangent(Directrix(G),H);PointOnCurve(P, C)",
+      "coords": {
+        "P": [
+          4.0,
+          3.0
+        ]
+      },
+      "validation_reasons": [
+        "hyperbola_point_off_curve"
+      ]
+    },
+    {
+      "index": 3967,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3967.png",
+      "answer": "sqrt(6)/3",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;A: Point;B: Point;F: Point;C: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (0, b/2);RightFocus(G)=F;PointOnCurve(A,H);IsPerpendicular(H,yAxis);Intersection(H,G)={B,C};AngleOf(B, F, C) = ApplyUnit(90, degree)",
+      "coords": {}
+    },
+    {
+      "index": 3968,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3969,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3969.png",
+      "answer": "2*x+pm*sqrt(2)*y-1=0",
+      "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;P: Point;F: Point;Expression(G) = (y^2 = 2*x);Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B};MidPoint(LineSegmentOf(A,B))=M;L:Line;PointOnCurve(M,L);IsPerpendicular(L,yAxis);Intersection(L,G)=P;Abs(LineSegmentOf(P, F)) = 3/2;M:Point",
+      "coords": {}
+    },
+    {
+      "index": 3970,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3971,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3971.png",
+      "answer": "8",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = a*x);a: Number;a>0;F:Point;Focus(G)=F;l: Line;PointOnCurve(F, l);Slope(l)=2;A:Point;Intersection(l,yAxis)=A;O: Origin;Area(TriangleOf(O, A, F))=4",
+      "coords": {}
+    },
+    {
+      "index": 3972,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3973,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3973.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
+      "fact_expr": "G: Ellipse;I: Line;A: Point;B: Point;F1: Point;F2: Point;Expression(G) = (x^2/16 + y^2/9 = 1);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, I);Intersection(I, G) = {A, B};Area(InscribedCircle(TriangleOf(A,B,F2)))=pi;x1:Number;x2:Number;y1:Number;y2:Number",
+      "coords": {}
+    },
+    {
+      "index": 3974,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 3975,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3975.png",
+      "answer": "2*sqrt(6)",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x);H: Line;Expression(H) = (-x + y + 1 = 0);A: Point;B: Point;Intersection(G, H) = {A, B};IsChordOf(LineSegmentOf(A, B), G)",
+      "coords": {}
+    },
+    {
+      "index": 3976,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3976.png",
+      "answer": "(-12, 0)",
+      "fact_expr": "G: Hyperbola;k: Number;Expression(G) = (x^2/4 + y^2/k = 1);e: Number;Eccentricity(G) = e ;In(e, (1, 2))",
+      "coords": {}
+    },
+    {
+      "index": 3977,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3977.png",
+      "answer": "2",
+      "fact_expr": "A: Point;F: Point;G: Parabola;O: Origin;M: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(A, G);AngleOf(O, F, A) = ApplyUnit(120, degree);MidPoint(LineSegmentOf(A, F)) = M",
+      "coords": {}
+    },
+    {
+      "index": 3978,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 4.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_3978.png",
+      "answer": "x^2/4 - y^2/4 = 1",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2 - y^2 = a^2);a: Number;G: Parabola;Expression(G) = (y^2 = 16*x);A: Point;B: Point;O: Origin;Center(C) = O;PointOnCurve(Focus(C), xAxis);Intersection(C, Directrix(G)) = {A,B};Abs(LineSegmentOf(A, B)) = 4*sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 3979,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_3979.png",
+      "answer": "sqrt(3)/2",
+      "fact_expr": "E: Ellipse;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;H: Point;Coordinate(H) = (4, 0);L: Line;PointOnCurve(H, L) = True;Intersection(L, E) = {A, B};A: Point;B: Point;Coordinate(MidPoint(LineSegmentOf(A,B))) = (2,-1)",
+      "coords": {
+        "H": [
+          4.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 3980,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3980.png",
+      "answer": "9",
+      "fact_expr": "G: Hyperbola;H: Circle;Z: Circle;P: Point;M: Point;N: Point;Expression(G) = (x^2/9 - y^2/16 = 1);Expression(H) = (y^2 + (x + 5)^2 = 4);Expression(Z) = (y^2 + (x - 5)^2 = 1);PointOnCurve(P, RightPart(G));PointOnCurve(M, H);PointOnCurve(N, Z)",
+      "coords": {}
+    },
+    {
+      "index": 3981,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_3981.png",
+      "answer": "ApplyUnit(90, degree)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1))*Abs(LineSegmentOf(P, F2)) = 32",
+      "coords": {}
+    },
+    {
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+      "answer": "y=pm*(sqrt(2)/2)*x",
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+        "M": [
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+      "index": 4006,
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+      "answer": "sqrt(2)*3/4",
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+      "index": 4007,
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+      "error": "Unsupported or unparseable expression"
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+      "index": 4008,
+      "success": true,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4008.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
+      "fact_expr": "A: Point;a: Real;Coordinate(A) = (a, 0);G: Curve;Expression(G) = (y^2 = 2*x);P: Point;PointOnCurve(P, G)",
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "answer": "2+4*sqrt(5)/5",
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+      "answer": "(0, 1/2]",
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+      "answer": "x^2/24 + y^2/8 = 1",
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+      "answer": "x^2/9 - y^2/16 = 1",
+      "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/40 + y^2/15 = 1);Focus(G)=Focus(H);Eccentricity(G)=5/3",
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+      "answer": "1/16",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4022.png",
+      "answer": "[1,2)",
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+      "index": 4023,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 4024,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4024.png",
+      "answer": "(2*sqrt(3),sqrt(3))",
+      "fact_expr": "G: Ellipse;A: Point;M: Point;F: Point;Expression(G) = (x^2/16 + y^2/12 = 1);Coordinate(A) = (-2, sqrt(3));RightFocus(G) = F;PointOnCurve(M, G);WhenMin(Abs(LineSegmentOf(M, A)) + 2*Abs(LineSegmentOf(M, F)))",
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+      "index": 4025,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4025.png",
+      "answer": "(\\sqrt{2}/2, 1)",
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+    {
+      "index": 4026,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4026.png",
+      "answer": "a^2/b^2",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;P: Point;PointOnCurve(P, G);M: Point;IsPerpendicular(LineSegmentOf(P, M), xAxis);FootPoint(LineSegmentOf(P, M), xAxis) = M;Negation(M=P);A1: Point;A2: Point;MajorAxis(G) = LineSegmentOf(A1, A2);PointOnCurve(P, LineSegmentOf(P, M))",
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+      "index": 4027,
+      "success": true,
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+          0.0
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_4027.png",
+      "answer": "x^2/16+y^2/12=1",
+      "fact_expr": "G: Circle;Expression(G) = (y^2 + (x + 2)^2 = 64);M: Point;Center(G) = M;A: Point;PointOnCurve(A, G);N: Point;Coordinate(N) = (2, 0);P: Point;Intersection(PerpendicularBisector(LineSegmentOf(A, N)), LineSegmentOf(M, A)) = P",
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+      "index": 4028,
+      "success": true,
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+      "error": null,
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4028.png",
+      "answer": "2",
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+    {
+      "index": 4029,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 4030,
+      "success": true,
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+        "y_coef": 4.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4030.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 2*(p*y));p: Number;p>0;H: Ellipse;Expression(H) = (x^2/3 + y^2/4 = 1);Focus(G) = OneOf(Vertex(H))",
+      "coords": {}
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+    {
+      "index": 4031,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4031.png",
+      "answer": "3*sqrt(5)",
+      "fact_expr": "C1: Circle;C: Point;Center(C1) = C;G: Parabola;Expression(G) = (y^2 = 2*x);PointOnCurve(C, G) = True;TangentPoint(C1, yAxis) = A;A: Point;Intersection(C1, xAxis) = {P, Q};P: Point;Q: Point;O: Origin;DotProduct(VectorOf(O, C), VectorOf(O, A)) = 9",
+      "coords": {}
+    },
+    {
+      "index": 4032,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 4.898979485566356,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4032.png",
+      "answer": "24",
+      "fact_expr": "G: Hyperbola;P: Point;H: Ellipse;F1: Point;F2: Point;Expression(G) = (x^2 - y^2/24 = 1);Expression(H) = (x^2/49 + y^2/24 = 1);Focus(G) = {F1, F2};OneOf(Intersection(G, H)) = P",
+      "coords": {}
+    },
+    {
+      "index": 4033,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 2.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4033.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2 - y^2/4 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, RightPart(G));Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 4034,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4034.png",
+      "answer": "3",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1);F: Point;LeftFocus(G) = F;A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G);Inclination(LineSegmentOf(A, B)) = pi/6;PointOnCurve(F, LineSegmentOf(A, B))",
+      "coords": {}
+    },
+    {
+      "index": 4035,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4035.png",
+      "answer": "{15/4, 17/4}",
+      "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2/4 + y^2/m = 1);FocalLength(G) = 1",
+      "coords": {}
+    },
+    {
+      "index": 4036,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4036.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
+      "fact_expr": "C: Parabola;p: Number;G: Line;P: Point;F: Point;Q: Point;A: Point;l: Line;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;Directrix(C) = l;PointOnCurve(F, G);Intersection(G, C) = {P, Q};Intersection(G, l) = A;VectorOf(P, F) = 3*VectorOf(F, Q)",
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+    },
+    {
+      "index": 4037,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4037.png",
+      "answer": "5/3",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4038.png",
+      "answer": "x^2/9 - y^2/16 = 1",
+      "fact_expr": "G: Hyperbola;F1: Point;F2: Point;Coordinate(F1) = (5, 0);Coordinate(F2) = (-5, 0);Focus(G)={F1,F2};PointOnCurve(P,G);Abs(Distance(P,F1)-Distance(P,F2))=6;P:Point",
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+      "answer": "9*sqrt(3)",
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+      "answer": "3/16",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4041.png",
+      "answer": "4/5",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/9 + x^2/a^2 = 1);a: Number;G: Line;Expression(G) = (y = x - 4);PointOnCurve(OneOf(Focus(C)),G) = True",
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+    {
+      "index": 4042,
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+        "b": 2.8284271247461903,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4042.png",
+      "answer": "12*sqrt(6)",
+      "fact_expr": "G: Hyperbola;A: Point;P: Point;F: Point;Expression(G) = (x^2 - y^2/8 = 1);Coordinate(A) = (0, 6*sqrt(6));RightFocus(G) = F;PointOnCurve(P, LeftPart(G));WhenMin(Perimeter(TriangleOf(A, P, F)))",
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4043.png",
+      "answer": "2",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, C) = {A, B};Abs(LineSegmentOf(A, F)) - Abs(LineSegmentOf(B, F)) = 3/2",
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+      "index": 4044,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 4045,
+      "success": true,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4045.png",
+      "answer": "1+3*sqrt(6)/2",
+      "fact_expr": "P: Point;H: Circle;Expression(H) = (x^2 + (y - 2)^2 = 1);PointOnCurve(P, H);G: Ellipse;Expression(G) = (x^2/9 + y^2 = 1);Q: Point;PointOnCurve(Q, G)",
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+      "index": 4046,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4047,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4047.png",
+      "answer": "(1, 2)",
+      "fact_expr": "C: Parabola;P: Point;F: Point;K: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;Intersection(Directrix(C), xAxis) = K;PointOnCurve(P, C);Quadrant(P) = 1;WhenMin(Abs(LineSegmentOf(P, F))/Abs(LineSegmentOf(P, K)))",
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+    {
+      "index": 4048,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4049,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4050,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4051,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 4.0,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4051.png",
+      "answer": "32/15",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);A: Point;RightVertex(G)=A;F: Point;RightFocus(G)=F;H: Line;PointOnCurve(F,H);IsParallel(H,OneOf(Asymptote(G)));B: Point;Intersection(H,G)=B",
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+    {
+      "index": 4052,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4053,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4053.png",
+      "answer": "sqrt(3)/3",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;P: Point;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;L:Line;PointOnCurve(F1, L);IsPerpendicular(L,xAxis);Intersection(L, G) = P;AngleOf(F1, P, F2) = ApplyUnit(60, degree)",
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+    {
+      "index": 4054,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4055,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4055.png",
+      "answer": "12",
+      "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;x1:Number;x2:Number;y2:Number;Expression(G) = (y^2 = 8*x);Coordinate(A) = (x1,x2);Coordinate(B) = (x2,y2);PointOnCurve(Focus(G),H);Intersection(H, G) = {A,B};Abs(LineSegmentOf(A,B))=16",
+      "coords": {}
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+    {
+      "index": 4056,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4057,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4057.png",
+      "answer": "7/5",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;P: Point;F1: Point;F2: Point;Q: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2, H);Intersection(H, RightPart(G)) = {P, Q};Abs(LineSegmentOf(P, F1)) = Abs(LineSegmentOf(F1, F2));3*Abs(LineSegmentOf(P, F2)) = 2*Abs(LineSegmentOf(Q, F2))",
+      "coords": {}
+    },
+    {
+      "index": 4058,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4059,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4059.png",
+      "answer": "4+2*sqrt(2)",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2 - y^2 = 1);F1: Point;F2: Point;Focus(C)={F1,F2};P: Point;PointOnCurve(P, C);Abs(LineSegmentOf(P, F1)) = 3*Abs(LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 4060,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4060.png",
+      "answer": "1",
+      "fact_expr": "l: Line;C: Parabola;p: Number;A: Point;F: Point;B: Point;p>0;Expression(C) = (y^2 = 2*p*x);Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};Abs(LineSegmentOf(A, F)) = 2/3;Abs(LineSegmentOf(B, F)) = 2",
+      "coords": {}
+    },
+    {
+      "index": 4061,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 4.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4061.png",
+      "answer": "4*x+9*y-13=0",
+      "fact_expr": "G: Ellipse;A: Point;B: Point;M: Point;H:Line;Expression(G) = (x^2/9 + y^2/4 = 1);Coordinate(M) = (1, 1);PointOnCurve(M,H);Intersection(H,G)={A,B};IsChordOf(LineSegmentOf(A,B),G);MidPoint(LineSegmentOf(A, B)) = M",
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+        "M": [
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+          1.0
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+    {
+      "index": 4062,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4062.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
+      "fact_expr": "G: Ellipse;H: Line;P: Point;Q: Point;F2: Point;F1: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Focus(G)={F1,F2};PointOnCurve(F1,H);Intersection(H,G)={P,Q}",
+      "coords": {}
+    },
+    {
+      "index": 4063,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "vertical"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4063.png",
+      "answer": "{2,18}",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/9 + y^2/16 = 1);P: Point;PointOnCurve(P, G) = True;P1: Point;P2: Point;OneOf(Focus(G))=P1;OneOf(Focus(G))=P2;Negation(P1=P2);Distance(P, P1) = 10",
+      "coords": {}
+    },
+    {
+      "index": 4064,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4064.png",
+      "answer": "sqrt(13)/2",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);A: Point;RightVertex(G) = A;F: Point;RightFocus(G) = F;H: Circle;Center(H) = A;R: Number;Radius(H) = R;B: Point;C: Point;Intersection(H, G) = {B, C};PointOnCurve(F,LineOf(B,C)) = True",
+      "coords": {}
+    },
+    {
+      "index": 4065,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4065.png",
+      "answer": "1",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);F2: Point;F1: Point;LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G) = True;P: Point;A: Point;Coordinate(A) = ((-1)*4, 4)",
+      "coords": {
+        "A": [
+          -4.0,
+          4.0
+        ]
+      }
+    },
+    {
+      "index": 4066,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4066.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;p: Number;t:Number;H: Point;I: Point;Expression(G) = (y^2 = 2*(p*x));Coordinate(H) = (2, t);Coordinate(I) = (p/2, 0);PointOnCurve(H, G);Distance(H, I) = 3",
+      "coords": {}
+    },
+    {
+      "index": 4067,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4067.png",
+      "answer": "3",
+      "fact_expr": "G: Parabola;C: Circle;M: Point;A: Point;F: Point;Expression(G) = (y^2 = 4*x);Expression(C) = ((x - 3)^2 + (y - 1)^2 = 1);PointOnCurve(M, G);Focus(G) = F;PointOnCurve(A, C)",
+      "coords": {}
+    },
+    {
+      "index": 4068,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4068.png",
+      "answer": "(pm*(4*sqrt(3))/3,-1/3)",
+      "fact_expr": "G: Ellipse;A: Point;P: Point;Expression(G) = (x^2 + 4*y^2 = 4);Coordinate(A) = (0, 1);PointOnCurve(A, G);PointOnCurve(P, G);IsChordOf(LineSegmentOf(A,P),G);WhenMax(Length(LineSegmentOf(A,P)))",
+      "coords": {
+        "A": [
+          0.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 4069,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
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+        "note": "using explicit params"
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+      "answer": "3+sqrt(2)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4081.png",
+      "answer": "sqrt(2)",
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+      "answer": "6/5",
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+      "answer": "2*x**2-2*y**2-2*x-2*y-1=0",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "success": true,
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+      "answer": "2",
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+      "answer": "6",
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+        "note": "using explicit params"
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+      "answer": "pm*sqrt(3)/3",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4089.png",
+      "answer": "[4, 5)",
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+      "answer": "8",
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+        "note": "using explicit params"
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+      "answer": "y^2=2*x-2",
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+      "index": 4092,
+      "success": true,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4092.png",
+      "answer": "sqrt(2)-1",
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+      "index": 4093,
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+        "y_coef": 1.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4093.png",
+      "answer": "4*sqrt(3)/3",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);P: Point;PointOnCurve(P, G);B: Point;UpperVertex(G) = B",
+      "coords": {}
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+    {
+      "index": 4094,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4095,
+      "success": true,
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+      "error": null,
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+        "b": 3.0,
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+        "y_coef": 9.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4095.png",
+      "answer": "6",
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+      "index": 4096,
+      "success": true,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4096.png",
+      "answer": "(1, \\sqrt{2}]",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;c: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);HalfFocalLength(G) = c",
+      "coords": {}
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+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 4.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4097.png",
+      "answer": "-8/9",
+      "fact_expr": "G: Ellipse;H: LineSegment;I: Point;Expression(G) = (x^2/9 + y^2/4 = 1);Coordinate(I) = (2, 1);IsChordOf(H,G);MidPoint(H)=I;Slope(OverlappingLine(H)) = k;k:Number",
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+    {
+      "index": 4098,
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+        "x_coef": 49.0,
+        "y_coef": 24.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4098.png",
+      "answer": "4*pi",
+      "fact_expr": "C: Ellipse;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/49 + y^2/24 = 1);LeftFocus(C)=F1;RightFocus(C)=F2;PointOnCurve(P,C);DotProduct(VectorOf(P,F1),VectorOf(P,F2))=0",
+      "coords": {}
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+        "note": "using explicit params"
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+      "answer": "5",
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4100.png",
+      "answer": "x^2/4 - y^2/2 = 1",
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+      "index": 4101,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "error": "Unsupported or unparseable expression"
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+      "answer": "x^2/4 + y^2/2 = 1",
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+      "answer": "(4, 0)",
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+      "answer": "8",
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+      "answer": "9*pi/16",
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+      "answer": "5/4",
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4111.png",
+      "answer": "sqrt(10)/2",
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+      "answer": "x^2 - y^2 / 3 = 1",
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+      "index": 4113,
+      "success": true,
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+        "direction": "up"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4113.png",
+      "answer": "2+pm*sqrt(2)",
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+      "index": 4114,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 4115,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 4116,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 4117,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4117.png",
+      "answer": "sqrt(3)",
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+    {
+      "index": 4118,
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4118.png",
+      "answer": "1/3",
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+      "index": 4119,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 4120,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4120.png",
+      "answer": "6",
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+    {
+      "index": 4121,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4121.png",
+      "answer": "9/2",
+      "fact_expr": "C: Parabola;p: Number;A: Point;B: Point;F: Point;O: Origin;L:Line;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;PointOnCurve(F, L)=True;Intersection(C, L) = {A, B};VectorOf(A, B) = 3*VectorOf(F, B);Area(TriangleOf(O, A, B)) = (sqrt(2)/3)*Abs(LineSegmentOf(A, B))",
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+    {
+      "index": 4122,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4122.png",
+      "answer": "1",
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+    {
+      "index": 4123,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4123.png",
+      "answer": "sqrt(3)/3",
+      "fact_expr": "E: Parabola;p: Number;P: Point;A: Point;F: Point;p>0;Expression(E) = (y^2 = 2*(p*x));Focus(E) = F;Intersection(Directrix(E),axis)=A;PointOnCurve(P, E);AngleOf(P, A, F) = ApplyUnit(30, degree)",
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+    {
+      "index": 4124,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
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+          -0.0
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+        "radius": 1.0
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_4124.png",
+      "answer": "x^2/3-y^2=1",
+      "fact_expr": "G: Hyperbola;H: Circle;Expression(H) = (-4*x + x^2 + y^2 + 3 = 0);Center(H)=F;RightFocus(G)=F;IsTangent(Asymptote(G),H);F: Point",
+      "coords": {}
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+    {
+      "index": 4125,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4125.png",
+      "answer": "sqrt(5)/5",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;H: Line;Expression(H) = (y = 2*x - 1);PointOnCurve(OneOf(Vertex(G)),H) = True;PointOnCurve(OneOf(Focus(G)),H)",
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+    {
+      "index": 4126,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4127,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4128,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_point_off_curve",
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+        "converged": true,
+        "iterations": 2
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4128.png",
+      "answer": "2",
+      "fact_expr": "C: Hyperbola;a: Number;b: Number;G: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(G) = (2, 3);PointOnCurve(G, C);FocalLength(C) = 4",
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+    {
+      "index": 4129,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 10.0,
+        "b": 8.0,
+        "major_axis": "x",
+        "x_coef": 100.0,
+        "y_coef": 64.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4129.png",
+      "answer": "100",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/100 + y^2/64 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G)",
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+    {
+      "index": 4130,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4130.png",
+      "answer": "{0,1,-1}",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);l: Line;H: Point;Coordinate(H) = (-1, 0);PointOnCurve(H, l) = True;NumIntersection(l, G) = 1",
+      "coords": {
+        "H": [
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+          0.0
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+    {
+      "index": 4131,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 0.7071067811865476,
+        "major_axis": "y",
+        "x_coef": 0.5,
+        "y_coef": 1.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4131.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
+      "fact_expr": "G: Ellipse;P: Point;Coordinate(P) = (x1, y1);x1:Number;y1:Number;Expression(G)=(2*x^2+y^2=1)",
+      "coords": {}
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+    {
+      "index": 4132,
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4132.png",
+      "answer": "y^2 = -6*x",
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+      "index": 4133,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 4134,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4134.png",
+      "answer": "pm*((2*sqrt(2))/3)",
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+      "coords": {}
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+    {
+      "index": 4135,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4135.png",
+      "answer": "7/4",
+      "fact_expr": "G: Parabola;M: Point;Expression(G) = (y^2 = x);PointOnCurve(M, G);Distance(M, Focus(G)) = 2",
+      "coords": {}
+    },
+    {
+      "index": 4136,
+      "success": true,
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+        "b": 4.0,
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4136.png",
+      "answer": "16/3",
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+    {
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4137.png",
+      "answer": "5/3",
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+    },
+    {
+      "index": 4138,
+      "success": true,
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+        "b": 1.7320508075688772,
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+        "y_coef": 3.0
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4138.png",
+      "answer": "sqrt(3)/2",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;A: Point;B: Point;PointOnCurve(A, G);PointOnCurve(B, G);Z: Circle;Expression(Z) = (x^2 + y^2 = R^2);IsDiameter(LineSegmentOf(A, B),Z);R: Number;R>0;M: Point;PointOnCurve(M, G);Negation(M=A);Negation(M=B);Slope(LineOf(M, A))*Slope(LineOf(M, B)) = -1/4",
+      "coords": {}
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+    {
+      "index": 4139,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4140,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
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+        "y_coef": 3.0
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4140.png",
+      "answer": "x^2/4+y^2/3=1",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;A: Point;UpperVertex(G) = A;Coordinate(A) = (0, sqrt(3));Area(InscribedCircle(TriangleOf(A, F1, F2))) = (1/3)*pi",
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+    {
+      "index": 4141,
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+        "b": 1.5,
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4141.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;O: Origin;P: Point;F: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x = -a);RightFocus(G) = F;PointOnCurve(P, H);AngleOf(O, P, F) = ApplyUnit(30, degree)",
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+    {
+      "index": 4142,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4142.png",
+      "answer": "4",
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+      "coords": {}
+    },
+    {
+      "index": 4143,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4144,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4145,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4145.png",
+      "answer": "(1/2,0)",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x)",
+      "coords": {}
+    },
+    {
+      "index": 4146,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 9.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4146.png",
+      "answer": "8",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/16 + y^2/9 = 1);Focus(G)={F1,F2};PointOnCurve(P,G)",
+      "coords": {}
+    },
+    {
+      "index": 4147,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 4.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4147.png",
+      "answer": "sqrt(13)",
+      "fact_expr": "G: Ellipse;A: Point;B: Point;Expression(G) = (x^2/9 + y^2/4 = 1);RightVertex(G)=A;UpperVertex(G)=B",
+      "coords": {}
+    },
+    {
+      "index": 4148,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 0.3333333333333333,
+        "b": 0.25,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4148.png",
+      "answer": "5/6",
+      "fact_expr": "G: Hyperbola;Expression(G) = (9*x^2 - 16*y^2 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 4149,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4149.png",
+      "answer": "18*sqrt(2)/7",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;F: Point;RightFocus(C) = F;A: Point;B: Point;LeftVertex(C) = A;RightVertex(C) = B;Abs(LineSegmentOf(F,A)) = 3;Abs(LineSegmentOf(F,B)) = 1;G: Line;Expression(G) = (y = x + 1/2)",
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+    },
+    {
+      "index": 4150,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4151,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4151.png",
+      "answer": "5/2",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l: Line;Directrix(C) = l;P: Point;PointOnCurve(P, l);Q: Point;OneOf(Intersection(LineOf(P, F), C)) = Q;VectorOf(P, F) = 4*VectorOf(F, Q)",
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+    },
+    {
+      "index": 4152,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4153,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4154,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4154.png",
+      "answer": "x^2/5 + y^2/4 = 1",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (2*x - y + 2 = 0);PointOnCurve(OneOf(Vertex(G)),H);PointOnCurve(OneOf(Focus(G)),H)",
+      "coords": {}
+    },
+    {
+      "index": 4155,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4155.png",
+      "answer": "sqrt(2)/3",
+      "fact_expr": "G:Ellipse;Expression(G)=(x^2/a^2+y^2/b^2=1);A:Point;B:Point;F:Point;a:Number;b:Number;a>b;b>0;H:Line;RightFocus(G)=F;PointOnCurve(F,H);Slope(H)=1;Intersection(H,G)={A,B};LineSegmentOf(A,F)=2*LineSegmentOf(F,B)",
+      "coords": {}
+    },
+    {
+      "index": 4156,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4156.png",
+      "answer": "5/3",
+      "fact_expr": "G: Hyperbola;a: Number;b: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, RightPart(G));Abs(LineSegmentOf(P, F1)) = 4*Abs(LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 4157,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4157.png",
+      "answer": "7",
+      "fact_expr": "G: Parabola;P: Point;A: Point;F: Point;Expression(G) = (y^2 = 8*x);Coordinate(A) = (5, 2);PointOnCurve(P, G);Focus(G) = F",
+      "coords": {
+        "A": [
+          5.0,
+          2.0
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+    {
+      "index": 4158,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4158.png",
+      "answer": "x^2 - y^2 / 3 = 1",
+      "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(l)=(x+sqrt(3)*y=0);IsPerpendicular(OneOf(Asymptote(C)),l);Distance(OneOf(Focus(C)), l) = 1",
+      "coords": {}
+    },
+    {
+      "index": 4159,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "down"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4159.png",
+      "answer": "(0,-1)",
+      "fact_expr": "G: Parabola;Expression(G) = (y = -x^2/4)",
+      "coords": {}
+    },
+    {
+      "index": 4160,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4160.png",
+      "answer": "x^2/5+y^2/4=1",
+      "fact_expr": "A: Point;B: Point;G: Ellipse;b: Number;a: Number;H: Circle;P: Point;l1: Line;l2:Line;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2 + y^2 = 1);Coordinate(P) = (1, 1/2);PointOnCurve(Focus(G), xAxis);TangentOfPoint(P,H)={l1,l2};TangentPoint(l1,H)=A;TangentPoint(l2,H)=B;PointOnCurve(RightFocus(G),LineOf(A,B));PointOnCurve(UpperVertex(G),LineOf(A,B))",
+      "coords": {
+        "P": [
+          1.0,
+          0.5
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+    },
+    {
+      "index": 4161,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4161.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "G: Parabola;P: Point;Q: Point;F: Point;l: Line;Expression(G) = (y^2 = 4*x);Focus(G) = F;Directrix(G) = l;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P,Q),xAxis);FootPoint(LineSegmentOf(P,Q),xAxis)=Q;LineSegmentOf(P,F)=3",
+      "coords": {}
+    },
+    {
+      "index": 4162,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4163,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "y",
+        "x_coef": 1.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4163.png",
+      "answer": "4",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2 + y^2/m = 1);m: Number;PointOnCurve(Focus(G), yAxis);Length(MajorAxis(G)) = 2*Length(MinorAxis(G))",
+      "coords": {}
+    },
+    {
+      "index": 4164,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4165,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+      "answer": "x = -2",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4166.png",
+      "answer": "(1, \\sqrt{6}/2)",
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+      "coords": {}
+    },
+    {
+      "index": 4167,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4167.png",
+      "answer": "10",
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+      "index": 4168,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 4169,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "answer": "16",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/64 - y^2/36 = 1);P: Point;PointOnCurve(P, G);Distance(P, RightFocus(G)) = 4",
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+      "index": 4171,
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+        "y_coef": 16.0
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+        "note": "using explicit params"
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+      "answer": "16",
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+      "coords": {}
+    },
+    {
+      "index": 4172,
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+      "conic_type": "hyperbola",
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+        "a": 4.453753471374512,
+        "b": 4.020467281341553,
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+      "optimization": {
+        "final_loss": 4.0108716348186135e-07,
+        "converged": true,
+        "iterations": 129
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4172.png",
+      "answer": "y=pm*2*sqrt(2)*x",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;F1: Point;F2: Point;N: Point;M: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-6, 0);Coordinate(F2) = (6, 0);Coordinate(N) = (0, 4);LeftFocus(C)=F1;RightFocus(C)=F2;PointOnCurve(M, RightPart(C));Min(Perimeter(TriangleOf(M, N, F1))) = 4 + 4*sqrt(13)",
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+          0.0
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+        "F2": [
+          6.0,
+          0.0
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+        "N": [
+          0.0,
+          4.0
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+    {
+      "index": 4173,
+      "success": true,
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+      "error": null,
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4173.png",
+      "answer": "4",
+      "fact_expr": "l: Line;C: Parabola;A: Point;B: Point;Expression(C) = (y^2 = 2*x);Expression(l) = (y = x - 1/2);Intersection(C, l) = {A, B}",
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+    {
+      "index": 4174,
+      "success": true,
+      "conic_type": "parabola",
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+        "direction": "right"
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4174.png",
+      "answer": "9",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;A: Point;B: Point;PointOnCurve(A, C);PointOnCurve(B, C);Negation(A=B);PointOnCurve(F, LineOf(A, B))",
+      "coords": {}
+    },
+    {
+      "index": 4175,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "a": 2.0,
+        "b": 1.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4175.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;H: Parabola;p: Number;Expression(G) = (x^2/4 - y^2 = 1);Expression(H) = (y^2 = 2*(p*x));Focus(H) = RightVertex(G)",
+      "coords": {}
+    },
+    {
+      "index": 4176,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4176.png",
+      "answer": "y^2=4*sqrt(2)*x",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(C) = F;L: Line;Expression(L) = (y = 4);Intersection(L,C) = P;P: Point;Intersection(L,yAxis) = Q;Q: Point;Abs(LineSegmentOf(P, F)) = (3/2)*Abs(LineSegmentOf(P, Q))",
+      "coords": {}
+    },
+    {
+      "index": 4177,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4178,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4178.png",
+      "answer": "3/4",
+      "fact_expr": "l: Line;C: Parabola;A: Point;M: Point;B: Point;Expression(C) = (x^2 = 4*y);Coordinate(M) = (3/2, -1);PointOnCurve(Focus(C), l);Intersection(l, C) = {A, B};IsTangent(LineSegmentOf(A, M), C)",
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+        "M": [
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+          -1.0
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+    {
+      "index": 4179,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4179.png",
+      "answer": "1/3",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;Focus(C) = {F1, F2};L1: Line;L2: Line;Asymptote(C) = {L1, L2};IsPerpendicular(L1, L2);P: Point;PointOnCurve(P, C);Abs(LineSegmentOf(P, F1)) = 3*Abs(LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 4180,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4181,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "a": 1.0,
+        "b": 1.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4181.png",
+      "answer": "(0, +\\infty)",
+      "fact_expr": "G: Hyperbola;O: Point;A: Point;P: Point;Expression(G) = (x^2 - y^2 = 1);Coordinate(O) = (0, 0);Coordinate(A) = (1, 1);PointOnCurve(P,RightPart(G))",
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+          0.0
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+        "A": [
+          1.0,
+          1.0
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+    {
+      "index": 4182,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4182.png",
+      "answer": "8/3",
+      "fact_expr": "C: Parabola;F: Point;P: Point;Q: Point;l: Line;Expression(C) = (y^2 = 8*x);Focus(C) = F;Directrix(C) = l;PointOnCurve(P, l);OneOf(Intersection(LineOf(P,F), C))=Q;VectorOf(F, P) = 3*VectorOf(F, Q)",
+      "coords": {}
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+    {
+      "index": 4183,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4183.png",
+      "answer": "{0}+[1,+oo)",
+      "fact_expr": "G: Parabola;H: Line;k: Number;Expression(G) = (y^2 = 4*x);Expression(H) = (y = k*x + 1);NumIntersection(H, G) <= 1",
+      "coords": {}
+    },
+    {
+      "index": 4184,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4185,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4185.png",
+      "answer": "x^2=16*y",
+      "fact_expr": "C1: Hyperbola;Expression(C1) = (x^2 - y^2/3 = 1);C2: Parabola;Expression(C2) = (x^2 = 2*p*y) ;p: Number;p>0;Distance(Focus(C2),Asymptote(C1)) = 2",
+      "coords": {}
+    },
+    {
+      "index": 4186,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4186.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;FocalLength(G) = 4*sqrt(2);l1: Line;l2: Line;Asymptote(G) = {l1, l2};IsPerpendicular(l1, l2) = True",
+      "coords": {}
+    },
+    {
+      "index": 4187,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4187.png",
+      "answer": "pi/3",
+      "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 - y^2/16 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);Abs(LineSegmentOf(P, F1))*Abs(LineSegmentOf(P, F2)) = 64",
+      "coords": {}
+    },
+    {
+      "index": 4188,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4188.png",
+      "answer": "2",
+      "fact_expr": "C: Hyperbola;b: Number;a: Real;G: Circle;O: Origin;F: Point;A: Point;e: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(C) = e;e = sqrt(5)/2;RightFocus(C) = F;IsDiameter(LineSegmentOf(O, F), G);Intersection(G, OneOf(Asymptote(C))) = {O, A};Area(TriangleOf(A, O, F)) = 1",
+      "coords": {}
+    },
+    {
+      "index": 4189,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4189.png",
+      "answer": "3*x+4*y-7=0",
+      "fact_expr": "G: Ellipse;H: Line;B: Point;A: Point;M: Point;Expression(G) = (x^2/4 + y^2/3 = 1);Coordinate(M) = (1, 1);PointOnCurve(M, H);Intersection(H, G) = {A, B};IsChordOf(LineSegmentOf(A,B),G);MidPoint(LineSegmentOf(A,B))=M",
+      "coords": {
+        "M": [
+          1.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 4190,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_focus_mismatch",
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4190.png",
+      "answer": "sqrt(5)/2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Focus(G) = F;F: Point;Coordinate(F) = (1, 0);Distance(F, Asymptote(G)) = sqrt(5)/5",
+      "coords": {
+        "F": [
+          1.0,
+          0.0
+        ]
+      },
+      "validation_reasons": [
+        "hyperbola_focus_mismatch"
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+    },
+    {
+      "index": 4191,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4191.png",
+      "answer": "-(b^2/a^2)",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;A: Point;B: Point;Endpoint(MinorAxis(G)) = {A, B};P: Point;PointOnCurve(P, G);Negation(P = A);Negation(P = B)",
+      "coords": {}
+    },
+    {
+      "index": 4192,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4192.png",
+      "answer": "5",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;PointOnCurve(F, H);H: Line;Intersection(H, G) = {A, B};A: Point;B: Point;l1: Line;l2: Line;PointOnCurve(A, l1) ;PointOnCurve(B, l2) ;Directrix(G) = l;l: Line;IsPerpendicular(l1, l) ;IsPerpendicular(l2, l) ;FootPoint(l1, l) = C;FootPoint(l2, l) = D;C: Point;D: Point;Abs(LineSegmentOf(A, F)) = 4*Abs(LineSegmentOf(B, F))",
+      "coords": {}
+    },
+    {
+      "index": 4193,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4194,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4195,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.23606797749979,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 5.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4195.png",
+      "answer": "4*sqrt(5)/5",
+      "fact_expr": "G: Ellipse;H: Circle;M: Point;N: Point;F: Point;Expression(G) = (x^2/5 + y^2 = 1);Expression(H) = (x^2 + y^2 = 1);RightFocus(G) = F;Quadrant(M) = 1;PointOnCurve(M, G);IsPerpendicular(LineSegmentOf(M,F),xAxis);TangentPoint(LineOf(M,N),H)=N",
+      "coords": {}
+    },
+    {
+      "index": 4196,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4197,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "answer": "2*sqrt(6)",
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+      "answer": "{x-2*y+1=0,y=1}",
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+      "answer": "2*sqrt(2)",
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+      "answer": "2",
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+      "answer": "[\\sqrt{2}-1,1)",
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+      "answer": "(0, 3)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4211.png",
+      "answer": "(1,5/3]",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4212.png",
+      "answer": "[4/5, 8/5]",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4213.png",
+      "answer": "5/2",
+      "fact_expr": "G: Parabola;p: Number;A: Point;Expression(G) = (y^2 = 2*p*x);Coordinate(A) = (2, 2);PointOnCurve(A, G)",
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+        "b": 2.0,
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4214.png",
+      "answer": "8",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 4);F1: Point;LeftFocus(G) = F1;P1: Point;P2: Point;PointOnCurve(P1, RightPart(G));PointOnCurve(P2, RightPart(G))",
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+    {
+      "index": 4215,
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+        "direction": "up"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4215.png",
+      "answer": "-4",
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+        "note": "using explicit params"
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+      "answer": "7",
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+        "b": 4.0,
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+        "y_coef": 16.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4217.png",
+      "answer": "-1/3",
+      "fact_expr": "G: Ellipse;H: Line;a: Real;Expression(G) = (x^2/25 + y^2/16 = 1);Expression(H) = (a*x - y + 1 = 0);PointOnCurve(RightFocus(G),H)",
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+    {
+      "index": 4218,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4218.png",
+      "answer": "1/2",
+      "fact_expr": "C: Ellipse;O: Origin;P: Point;F: Point;Expression(C) = (x^2/4 + y^2 = 1);RightFocus(C) = F;PointOnCurve(P, C);Abs(LineSegmentOf(O, P)) = Abs(LineSegmentOf(O, F))",
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+      "index": 4219,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "up"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4219.png",
+      "answer": "2",
+      "fact_expr": "C: Parabola;G: Curve;k: Number;P: Point;F: Point;Expression(C) = (y = x^2/4);k>0;Expression(G) = (y = k/x);Focus(C) = F;Intersection(G, C) = P;IsPerpendicular(LineSegmentOf(P, F), yAxis)",
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+      "index": 4220,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4220.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);M: Point;PointOnCurve(M, G) = True;Distance(M, Focus(G)) = 2",
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+      "index": 4221,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "up"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4221.png",
+      "answer": "4/3",
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+    {
+      "index": 4222,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4222.png",
+      "answer": "2*x+1=0",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = y)",
+      "coords": {}
+    },
+    {
+      "index": 4223,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4223.png",
+      "answer": "-1/2",
+      "fact_expr": "G: Hyperbola;P: Point;Expression(G) = (x^2 - y^2 = 1);Coordinate(P) = (a, b);PointOnCurve(P, LeftPart(G));L:Line;OneOf(Asymptote(G))=L;Expression(L)=(y = x);Distance(P, L) = sqrt(2);a:Number;b:Number",
+      "coords": {}
+    },
+    {
+      "index": 4224,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4224.png",
+      "answer": "8",
+      "fact_expr": "A: Point;B: Point;l: Line;G: Parabola;Expression(G) = (y^2 = 4*x);PointOnCurve(Focus(G), l);Intersection(l, G) = {A,B};XCoordinate(MidPoint(LineSegmentOf(A, B))) = 3",
+      "coords": {}
+    },
+    {
+      "index": 4225,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 4.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4225.png",
+      "answer": "6",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/4 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};H: Line;PointOnCurve(F1, H);Intersection(H, G) = {A, B};A: Point;B: Point;Abs(LineSegmentOf(A, B)) + Abs(LineSegmentOf(F2, A)) = 10",
+      "coords": {}
+    },
+    {
+      "index": 4226,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.463192105293274,
+        "b": 2.070901870727539,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 1.4661259228887502e-06,
+        "converged": true,
+        "iterations": 106
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4226.png",
+      "answer": "y=pm*sqrt(2)*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/2 + x^2/a^2 = 1);a: Number;Eccentricity(G) = sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 4227,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4227.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/m^2 = 1);m: Number;m > 0;H: Line;Expression(H) = (y = x*sqrt(2)/2);OneOf(Intersection(H, G)) = M;M: Point;Projection(M, xAxis) = F;F: Point;RightFocus(G) = F",
+      "coords": {}
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+    {
+      "index": 4228,
+      "success": true,
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+      "error": null,
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4228.png",
+      "answer": "x^2/6 + y^2/3 = 1",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;F1: Point;E: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;FocalLength(C) = 2*sqrt(3);PointOnCurve(E, C);IsPerpendicular(LineSegmentOf(E, F1), LineSegmentOf(E, F2));Slope(LineOf(E,F1))=b/c;c:Number;HalfFocalLength(C)=c",
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+      "error": null,
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+          -0.0
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_4229.png",
+      "answer": "14",
+      "fact_expr": "G: Parabola;p: Number;H: Circle;p>0;Expression(G) = (y^2 = -2*p*x);Expression(H) = (-6*x + x^2 + y^2 - 7 = 0);IsTangent(Directrix(G),H)",
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+    {
+      "index": 4230,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "up"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4230.png",
+      "answer": "{(x^2=2*y),(x^2=4*y)}",
+      "fact_expr": "M: Point;Coordinate(M) = (2, -2*p);p: Number;G: Parabola;Expression(G) = (x^2 = 2*p*y);p>0;l1: Line;l2: Line;TangentOfPoint(M, G) = {l1, l2};TangentPoint(l1, G) = A;TangentPoint(l2, G) = B;B: Point;A: Point;YCoordinate(MidPoint(LineSegmentOf(A, B))) = 6",
+      "coords": {}
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+    {
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4231.png",
+      "answer": "(x-5)^2+y^2=9",
+      "fact_expr": "G: Hyperbola;H: Parabola;C:Circle;Expression(G) = (x^2/16 - y^2/9 = 1);Expression(H) = (y^2 = 20*x);Center(C)=Focus(H);IsTangent(Asymptote(G),C)",
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+    {
+      "index": 4232,
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+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4232.png",
+      "answer": "pm*(5/2)",
+      "fact_expr": "G: Hyperbola;H: Circle;m: Number;Expression(G) = (x^2/16 - y^2/9 = 1);Expression(H) = (x^2 + (-m + y)^2 = 4);IsTangent(Asymptote(G),H)",
+      "coords": {}
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+    {
+      "index": 4233,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4234,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4235,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4235.png",
+      "answer": "4",
+      "fact_expr": "C: Parabola;p: Number;M: Point;F: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Coordinate(MidPoint(LineSegmentOf(M,F))) = (2, 2);PointOnCurve(M, C);Focus(C) = F",
+      "coords": {}
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+    {
+      "index": 4236,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.2352789464429823,
+        "b": 2.2352789464429823,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 1.4312591569844774e-06,
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+        "iterations": 89
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4236.png",
+      "answer": "x^2/9-y^2=1",
+      "fact_expr": "G: Hyperbola;F1: Point;F2: Point;M: Point;Coordinate(F1) = (-sqrt(10), 0);Coordinate(F2) = (sqrt(10), 0);Focus(G) = {F1,F2};PointOnCurve(M, G);DotProduct(VectorOf(M, F1),VectorOf(M,F2))=0;Abs(VectorOf(M, F1))*Abs(VectorOf(M, F2)) = 2",
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+        "F2": [
+          3.1622776601683795,
+          0.0
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+    {
+      "index": 4237,
+      "success": true,
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+      "error": null,
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4237.png",
+      "answer": "36",
+      "fact_expr": "l: Line;G: Parabola;Expression(G) = (y^2 = 4*x);Intersection(l, G) = {A, B};A: Point;B: Point;O: Origin;Slope(LineOf(O,A)) * Slope(LineOf(O,B)) = -1;C: Circle;Center(C) = MidPoint(LineSegmentOf(A,B));Radius(C) = sqrt(2);Intersection(l,C) = {P,Q};P: Point;Q: Point",
+      "coords": {}
+    },
+    {
+      "index": 4238,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "a": 1.0,
+        "b": 1.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4238.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "G: Hyperbola;H: Parabola;p: Number;Expression(G) = (x^2 - y^2 = 1);p>0;Expression(H) = (y^2 = 2*p*x);PointOnCurve(OneOf(Focus(G)), Directrix(H))",
+      "coords": {}
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+    {
+      "index": 4239,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4239.png",
+      "answer": "7/2",
+      "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;Expression(G) = (y^2 = 2*x);Coordinate(A) = (3, 2);PointOnCurve(P, G);Focus(G) = F",
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+          2.0
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+      "index": 4240,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4241,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 9.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4241.png",
+      "answer": "11",
+      "fact_expr": "C: Ellipse;H: Line;A: Point;B: Point;F: Point;G: Point;Expression(C) = (x^2/16 + y^2/9 = 1);Focus(C) = {F, G};PointOnCurve(G, H);Intersection(H, C) = {A, B};Abs(LineSegmentOf(A, B)) = 5",
+      "coords": {}
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+    {
+      "index": 4242,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4243,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4243.png",
+      "answer": "x^2/16-y^2/9=1;3",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);FocalLength(G)=10;Slope(OneOf(Asymptote(G)))=3/4",
+      "coords": {}
+    },
+    {
+      "index": 4244,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 6.0,
+        "b": 8.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4244.png",
+      "answer": "27",
+      "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/36 - y^2/64 = 1);PointOnCurve(P, G);Focus(G) = {F1, F2};Distance(P, F1) = 15",
+      "coords": {}
+    },
+    {
+      "index": 4245,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4245.png",
+      "answer": "{22,2}",
+      "fact_expr": "G: Hyperbola;F1: Point;F2: Point;P:Point;Expression(G) = (x^2/25 - y^2/9 = 1);Focus(G)={F1,F2};PointOnCurve(P,G);Distance(P,F1)=12",
+      "coords": {}
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+    {
+      "index": 4246,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4247,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4247.png",
+      "answer": "x^2/5 - y^2/20 = 1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;l: Line;Expression(l) = (y = 2*x + 10);IsParallel(OneOf(Asymptote(G)),l) = True;PointOnCurve(OneOf(Focus(G)),l) = True",
+      "coords": {}
+    },
+    {
+      "index": 4248,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4248.png",
+      "answer": "sqrt(2)/4",
+      "fact_expr": "l: Line;G: Ellipse;B: Point;A: Point;O: Origin;M: Point;N: Point;Expression(G) = (x^2/2 + y^2 = 1);Negation(PointOnCurve(O, l));Intersection(l, G) = {A, B};MidPoint(LineSegmentOf(A,B)) = M;Intersection(PerpendicularBisector(LineSegmentOf(A,B)), xAxis) = N",
+      "coords": {}
+    },
+    {
+      "index": 4249,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4249.png",
+      "answer": "sqrt(7)/4",
+      "fact_expr": "l: Line;E: Ellipse;b: Number;a: Number;P: Point;F: Point;Q: Point;O:Origin;a > b;b > 0;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(E) = F;PointOnCurve(O,l);Intersection(l, E) = {P, Q};Abs(LineSegmentOf(P, F)) = 3*Abs(LineSegmentOf(Q, F));AngleOf(P, F, Q) = ApplyUnit(120, degree)",
+      "coords": {}
+    },
+    {
+      "index": 4250,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4250.png",
+      "answer": "(2*sqrt(3))/3",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;l: Line;Directrix(G) = l;H: Line;PointOnCurve(F,H);A: Point;B: Point;Intersection(H, G) = {A, B};L1: Line;L2: Line;C: Point;D: Point;PointOnCurve(A, L1);PointOnCurve(B, L2);IsPerpendicular(L1, l);IsPerpendicular(L2, l);FootPoint(L1, l) = C;FootPoint(L2, l) = D;Abs(LineSegmentOf(A, F)) = 2*Abs(LineSegmentOf(B, F));Area(TriangleOf(C, D, F)) = sqrt(2)",
+      "coords": {}
+    },
+    {
+      "index": 4251,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4251.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(Asymptote(G)) = (y = pm*x)",
+      "coords": {}
+    },
+    {
+      "index": 4252,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4253,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4253.png",
+      "answer": "pi/2",
+      "fact_expr": "C: Parabola;p: Number;P: Point;M: Point;Q:Point;L1: Line;L2: Line;p>0;Expression(C) = (y^2 = 2*(p*x));Intersection(Directrix(C), xAxis) = M;TangentOfPoint(M, C) = {L1, L2};TangentPoint(L1,C)=P;TangentPoint(L2,C)=Q",
+      "coords": {}
+    },
+    {
+      "index": 4254,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4255,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4255.png",
+      "answer": "-12",
+      "fact_expr": "G: Parabola;H: Line;O: Origin;A: Point;B: Point;Expression(G) = (y^2 = 8*x);PointOnCurve(Focus(G), H);Intersection(G, H) = {A, B};Slope(H) = 1",
+      "coords": {}
+    },
+    {
+      "index": 4256,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4256.png",
+      "answer": "sqrt(6)",
+      "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;M: Point;N: Point;a>0;b>0;e:Number;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(C) = e;Expression(l)=(y=x);Intersection(l, C) = {M, N};Abs(LineSegmentOf(M, N)) = sqrt(2)*b",
+      "coords": {}
+    },
+    {
+      "index": 4257,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4257.png",
+      "answer": "x^2 - y^2/4 = 1",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Ellipse;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2/9 + y^2/4 = 1);Focus(C) = Focus(G);Expression(Asymptote(C)) = (y = pm*(2*x))",
+      "coords": {}
+    },
+    {
+      "index": 4258,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.4142135623730951,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4258.png",
+      "answer": "x^2 - y^2/(1/2) = 1",
+      "fact_expr": "P: Point;Coordinate(P) = (3, 2);PointOnCurve(P, Z);G: Hyperbola;Expression(G) = (x^2/4 - y^2/2 = 1);Z: Hyperbola;Expression(Asymptote(Z))=Expression(Asymptote(G))",
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+      "answer": "2*sqrt(2)-6",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/9 + y^2/5 = 1);B: Point;Coordinate(B) = (-2, 0);A: Point;Coordinate(A) = (4, 2);M: Point;PointOnCurve(M, C)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4260.png",
+      "answer": "9",
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+      "coords": {}
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+    {
+      "index": 4261,
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+      "answer": "y^2/4 - x^2/12 = 1",
+      "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/9 + y^2/25 = 1);FocalLength(G) = FocalLength(H);Eccentricity(G) + Eccentricity(H) = 14/5",
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+      "index": 4262,
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+      "answer": "pm*8",
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+      "answer": "x^2 - y^2 = 1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>b;b>0;Expression(Asymptote(G)) = (y = pm*x);H: Point;Coordinate(H) = (sqrt(2), 1);PointOnCurve(H,G) = True",
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+    {
+      "index": 4264,
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+        "direction": "up"
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+      "answer": "-4",
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+      "index": 4265,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 4266,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4266.png",
+      "answer": "3",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Focus(C)={F1,F2};PointOnCurve(P, C);IsPerpendicular(VectorOf(P, F1),VectorOf(P,F2));Area(TriangleOf(P, F1, F2)) = 9",
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+    {
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+        "note": "using explicit params"
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+      "answer": "5*sqrt(3)/3",
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+    },
+    {
+      "index": 4268,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4268.png",
+      "answer": "8/3",
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+    {
+      "index": 4269,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.0,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4269.png",
+      "answer": "1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 4270,
+      "success": true,
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+        "b": 2.0,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4270.png",
+      "answer": "x^2/2 - y^2/8 = 1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/4 = 1);H: Hyperbola;Asymptote(G) = Asymptote(H)",
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+    {
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4271.png",
+      "answer": "[3/4,3/2]",
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+    {
+      "index": 4272,
+      "success": true,
+      "conic_type": "ellipse",
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+        "y_coef": 5.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4272.png",
+      "answer": "5*sqrt(3)/6",
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+    {
+      "index": 4273,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4274,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4274.png",
+      "answer": "sqrt(3)",
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+      "index": 4275,
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+        "y_coef": 1.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4275.png",
+      "answer": "1/4",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4276.png",
+      "answer": "sqrt(5)/2",
+      "fact_expr": "G: Hyperbola;a: Number;Expression(G) = (-y^2/4 + x^2/a = 1);Expression(OneOf(Asymptote(G))) = (x - 2*y = 0)",
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+    {
+      "index": 4277,
+      "success": true,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4277.png",
+      "answer": "8",
+      "fact_expr": "H: Ellipse;e: Number;Eccentricity(H) = e;e = 1/2;G: Hyperbola;Expression(G) = (x^2/3 - y^2 = 1);Focus(H) = Focus(G);P: Point;PointOnCurve(P, H);F1: Point;F2: Point;Focus(H) = {F1, F2}",
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+    {
+      "index": 4278,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4278.png",
+      "answer": "2*sqrt(3)/3",
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+    {
+      "index": 4279,
+      "success": true,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4279.png",
+      "answer": "4*x - y -10 =0",
+      "fact_expr": "l: Line;G: Parabola;Expression(G) = (y^2 = 16*x);A: Point;B: Point;Intersection(l, G) = {A, B};IsChordOf(LineSegmentOf(A, B), G);P: Point;Coordinate(P) = (3, 2);MidPoint(LineSegmentOf(A, B)) = P",
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+      "index": 4280,
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+      "error": "validation: ellipse_point_off_curve",
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+        "iterations": 101
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4280.png",
+      "answer": "sqrt(5) + 2",
+      "fact_expr": "C: Ellipse;a: Number;b: Number;A: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (1, 2);PointOnCurve(A, C)",
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+    {
+      "index": 4281,
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+        "direction": "up"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4281.png",
+      "answer": "x^2=6*y",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4282.png",
+      "answer": "sqrt(13)",
+      "fact_expr": "G: Ellipse;A: Point;P: Point;Expression(G) = (x^2/12 + y^2/9 = 1);Coordinate(A) = (0, 1/2);PointOnCurve(P, G)",
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+      "index": 4283,
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+        "iterations": 95
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4283.png",
+      "answer": "6",
+      "fact_expr": "F1: Point;F2: Point;Coordinate(F1) = (-2, 0);Coordinate(F2) = (2, 0);H: Ellipse;Expression(H) = (x^2 + 2*y^2 = 8);G: Hyperbola;Expression(G) = (x^2 - y^2 = 2);P: Point;OneOf(Intersection(H, G)) = P",
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+        "F2": [
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+      "index": 4284,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4284.png",
+      "answer": "sqrt(3)-1",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;O: Origin;l: Line;PointOnCurve(O, l);Inclination(l) = ApplyUnit(60, degree);M: Point;OneOf(Intersection(l, C)) = M;Abs(VectorOf(M, F1) + VectorOf(M, F2)) = Abs(VectorOf(M, F1) - VectorOf(M, F2))",
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+    {
+      "index": 4285,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4285.png",
+      "answer": "{11, 29}",
+      "fact_expr": "G: Ellipse;k: Number;Expression(G) = (x^2/20 + y^2/k = 1);FocalLength(G) = 6",
+      "coords": {}
+    },
+    {
+      "index": 4286,
+      "success": true,
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4286.png",
+      "answer": "2*sqrt(2)",
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+      "index": 4287,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "error": "Unsupported or unparseable expression"
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+      "success": true,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4291.png",
+      "answer": "x^2/16+y^2/9=1",
+      "fact_expr": "H: Circle;Expression(H) = (x^2 + y^2 = 1);L1: Line;IsTangent(L1, H) = True;G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);A: Point;B: Point;Intersection(L1, G) = {A, B};Intersection(TangentOnPoint(A, G), TangentOnPoint(B, G)) = P;P: Point",
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+    {
+      "index": 4292,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4292.png",
+      "answer": "2*sqrt(5)/5",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;H: Parabola;Expression(H) = (y^2 = 4*x);Length(InterceptChord(Directrix(H), G)) = b",
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+    {
+      "index": 4293,
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4293.png",
+      "answer": "x^2=-(20/3)*y",
+      "fact_expr": "C: Parabola;G: Line;O:Origin;Expression(G) = (2*x - 3*y - 5 = 0);Vertex(C)=O;SymmetryAxis(C)=yAxis;PointOnCurve(Focus(C),G)",
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+    {
+      "index": 4294,
+      "success": true,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4294.png",
+      "answer": "4*sqrt(3)",
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+    {
+      "index": 4295,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "success": true,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4296.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;H: Line;Expression(H) = (y = x);O: Origin;A: Point;Intersection(H, G) = {O, A};B: Point;Intersection(H, Directrix(G)) = B;C: Point;OneOf(Intersection(LineOf(A,F),G))=C",
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+    {
+      "index": 4297,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 4298,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4298.png",
+      "answer": "10",
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+    {
+      "index": 4299,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 4300,
+      "success": true,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4300.png",
+      "answer": "1",
+      "fact_expr": "G: Parabola;H: Line;A: Point;F: Point;B: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B}",
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+    {
+      "index": 4301,
+      "success": true,
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+        "iterations": 89
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4301.png",
+      "answer": "(2,4]",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;O: Origin;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = 2;PointOnCurve(P, G);LeftFocus(G)=F1;RightFocus(G)=F2",
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+      "index": 4302,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 4303,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 4304,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4305,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 2.0,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4305.png",
+      "answer": "x=pm*(\\sqrt{5}/5)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/4 = 1)",
+      "coords": {}
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+    {
+      "index": 4306,
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+        "y_coef": 1.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4306.png",
+      "answer": "{(-oo,-sqrt(17))+(sqrt(17),+oo)}",
+      "fact_expr": "G: Ellipse;H: Line;b: Number;Expression(G) = (x^2/4 + y^2 = 1);Expression(H) = (y = b + 2*x);NumIntersection(H,G)=0",
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+    {
+      "index": 4307,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4307.png",
+      "answer": "x^2/2 - y^2/2 = 1",
+      "fact_expr": "C: Hyperbola;G: Parabola;O: Origin;Expression(G) = (y^2 = 8*x);Center(C) = O;SymmetryAxis(C)=axis;OneOf(Focus(C) )= Focus(G);Expression(OneOf(Asymptote(C))) = (x + y = 0)",
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+    {
+      "index": 4308,
+      "success": true,
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4308.png",
+      "answer": "3",
+      "fact_expr": "l: Line;C: Parabola;B: Point;A: Point;Expression(C) = (y^2 = 6*x);PointOnCurve(Focus(C), l);Intersection(l, C) = {A, B};Abs(LineSegmentOf(A, B)) = 9",
+      "coords": {}
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+    {
+      "index": 4309,
+      "success": true,
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+      "error": null,
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+        "y_coef": 3.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4309.png",
+      "answer": "sqrt(2) - 1",
+      "fact_expr": "G: Ellipse;a: Number;b: Number;H: Circle;I: Line;M: Point;N: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(RightFocus(G), I);IsPerpendicular(I, xAxis);Intersection(I, G) = {M, N};IsDiameter(LineSegmentOf(M, N), H);PointOnCurve(LeftFocus(G), H)",
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+    {
+      "index": 4310,
+      "success": false,
+      "conic_type": "parabola",
+      "error": "validation: parabola_point_off_curve",
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+        "iterations": 44
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4310.png",
+      "answer": "5",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;P: Point;Coordinate(P) = (4, 4);PointOnCurve(P, G)",
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+          4.0
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+        "parabola_point_off_curve"
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+    {
+      "index": 4311,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 4312,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "iterations": 89
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4312.png",
+      "answer": "x^2 - y^2 / 3 = 1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/n + x^2/m = 1);m: Number;n: Number;Eccentricity(G) = 2;H: Parabola;Expression(H) = (y^2 = 8*x);OneOf(Focus(G)) = Focus(H)",
+      "coords": {}
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+    {
+      "index": 4313,
+      "success": true,
+      "conic_type": "ellipse",
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+        "b": 2.6457513110645907,
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+        "x_coef": 11.0,
+        "y_coef": 7.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4313.png",
+      "answer": "2*sqrt(11)/11",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/11 + y^2/7 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 4314,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4315,
+      "success": true,
+      "conic_type": "parabola",
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+        "direction": "right"
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+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4315.png",
+      "answer": "16",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = a*x);a: Number;a>0;V: Point;PointOnCurve(V, G) = True;XCoordinate(V) = 6;Distance(V, Focus(G)) = 10",
+      "coords": {}
+    },
+    {
+      "index": 4316,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4317,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+        "final_loss": 0.0,
+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4317.png",
+      "answer": "(1,0)",
+      "fact_expr": "C: Parabola;p: Number;P: Point;F: Point;p>0;Expression(C) = (y^2 = 2*p*x);Focus(C)=F;PointOnCurve(P, C);Abs(LineSegmentOf(P, F)) = 4;Distance(P, yAxis) = 3",
+      "coords": {}
+    },
+    {
+      "index": 4318,
+      "success": true,
+      "conic_type": "ellipse",
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+        "b": 2.8284271247461903,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 8.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4318.png",
+      "answer": "{6}",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/8 = 1);F: Point;RightFocus(G) = F;H: Circle;Expression(H) = (x^2 + y^2 = 8);M: Point;Quadrant(M) = 1;Z: Line;TangentOnPoint(M, H) = Z;P: Point;Q: Point;Intersection(Z, G) = {P, Q}",
+      "coords": {}
+    },
+    {
+      "index": 4319,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4319.png",
+      "answer": "sqrt(2)-1",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = 2*x);HalfFocalLength(G) = c;XCoordinate(OneOf(Intersection(H,G)))=c;c:Number",
+      "coords": {}
+    },
+    {
+      "index": 4320,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4320.png",
+      "answer": "2",
+      "fact_expr": "l1: Line;Expression(l1) = (3*x - 4*y - 9 = 0);l2: Line;Expression(l2) = (y = 1/4);G: Parabola;Expression(G) = (y = x^2);P: Point;PointOnCurve(P,G) = True",
+      "coords": {}
+    },
+    {
+      "index": 4321,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4321.png",
+      "answer": "(0,2*sqrt(5)/5)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4323.png",
+      "answer": "5/2",
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+      "answer": "y = pm*(sqrt(2)/2)*x",
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+      "answer": "x^2/4 + y^2/3 = 1",
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+      "answer": "x^2 - y^2 / 3 = 1",
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+      "answer": "y=pm*x",
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+      "answer": "sqrt(13)/2",
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+      "answer": "y=(-1/2)*x\ny=(1/2)*x",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4333.png",
+      "answer": "-3/4",
+      "fact_expr": "B: Point;Coordinate(B) = (8, 8);C: Parabola;Expression(C) = (y^2 = 2*p*x);p: Number;PointOnCurve(B, C) = True;Intersection(TangentOnPoint(B,C),Directrix(C))=A;A: Point;F: Point;Focus(C) = F",
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+        "direction": "left"
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+        "note": "using explicit params"
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+      "answer": "3",
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+        "b": 2.070901870727539,
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+        "iterations": 106
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4335.png",
+      "answer": "x^2/12 - y^2/24 = 1",
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+      "conic_type": "ellipse",
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+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4336.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
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+          0.0
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 4338,
+      "success": true,
+      "conic_type": "parabola",
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4338.png",
+      "answer": "-4/3",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;P: Point;PointOnCurve(P, G) = True;A: Point;Coordinate(A) = (1, 1);WhenMin(Perimeter(TriangleOf(P,A,F))) = True",
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+        "A": [
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+          1.0
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+      "index": 4339,
+      "success": true,
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+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4339.png",
+      "answer": "b^2/a^2",
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+    {
+      "index": 4340,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4340.png",
+      "answer": "(1,2)",
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+    {
+      "index": 4341,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4341.png",
+      "answer": "0",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/2 - y^2/b^2 = 1);b: Number;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;Expression(OneOf(Asymptote(G))) = (y = x);P: Point;Coordinate(P) = (sqrt(3), y0);y0: Number;PointOnCurve(P, G)",
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+      "index": 4342,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 4343,
+      "success": true,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4343.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(C) = F;Z: Line;PointOnCurve(F, Z);IsPerpendicular(Z, OneOf(Asymptote(C)));M: Point;Intersection(Z, C) = M;N: Point;FootPoint(Z, OneOf(Asymptote(C))) = N;MidPoint(LineSegmentOf(F, N)) = M",
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+    {
+      "index": 4344,
+      "success": true,
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+      "error": null,
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+        "direction": "right"
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4344.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;M: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(M, G);Distance(M, Focus(G)) = 3",
+      "coords": {}
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+      "index": 4345,
+      "success": true,
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+      "error": null,
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+        "b": 4.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4345.png",
+      "answer": "34/3",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);M:Point;PointOnCurve(M,G);XCoordinate(M)=5",
+      "coords": {}
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+    {
+      "index": 4346,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4346.png",
+      "answer": "4",
+      "fact_expr": "G: Ellipse;O: Origin;N: Point;M: Point;F: Point;Expression(G) = (x^2/25 + y^2/9 = 1);PointOnCurve(M,G);OneOf(Focus(G))=F;Distance(M,F)=2;MidPoint(LineSegmentOf(M,F))=N",
+      "coords": {}
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+    {
+      "index": 4347,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4348,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_focus_mismatch",
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+        "b": 2.2412689930799665,
+        "orientation": "horizontal"
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+        "converged": true,
+        "iterations": 95
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4348.png",
+      "answer": "x^2/36 - y^2/64 = 1",
+      "fact_expr": "G: Hyperbola;Center(G) = O;O: Origin;F: Point;OneOf(Focus(G)) = F;Coordinate(F) = (10, 0);Expression(Asymptote(G)) = (y = pm*(4/3)*x)",
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+        "F": [
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+          0.0
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+      "validation_reasons": [
+        "hyperbola_focus_mismatch"
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+    {
+      "index": 4349,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4349.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "C: Parabola;G: Line;m: Number;k: Number;F: Point;M: Point;N: Point;Expression(C) = (y^2 = 4*x);k>0;Expression(G) = (y = k*x + m);Intersection(G,C)=M;Intersection(G, Directrix(C))=N;Focus(C) = F;2*VectorOf(F, M) = VectorOf(M, N)",
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+    {
+      "index": 4350,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 2.23606797749979,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 5.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4350.png",
+      "answer": "x^2 - y^2/3 = 1",
+      "fact_expr": "G: Hyperbola;H: Ellipse;I: Point;Expression(H) = (x^2/9 + y^2/5 = 1);Focus(G) = Focus(H);Coordinate(I) = (sqrt(2), sqrt(3));PointOnCurve(I, G)",
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+    {
+      "index": 4351,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4352,
+      "success": true,
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+      "error": null,
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+        "direction": "right"
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+      "optimization": {
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4352.png",
+      "answer": "pm*sqrt(3)",
+      "fact_expr": "l: Line;C: Parabola;A: Point;F: Point;B: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};Abs(LineSegmentOf(A, F)) = 3*Abs(LineSegmentOf(B, F))",
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+    {
+      "index": 4353,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4353.png",
+      "answer": "5",
+      "fact_expr": "l: Line;G: Parabola;M: Point;O: Origin;N: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {M, N};Area(TriangleOf(M, O, N)) = sqrt(5)",
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+    {
+      "index": 4354,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4355,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4356,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4356.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/5 - y^2/4 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 4357,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4357.png",
+      "answer": "3*sqrt(2)/4",
+      "fact_expr": "G: Hyperbola;a: Number;b: Number;H: Line;P:Point;e: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = x*(b/(3*a)));Eccentricity(G) = e;Intersection(H,LeftPart(G))=P;LeftFocus(G)=F1;IsPerpendicular(LineSegmentOf(P,F1),xAxis);F1:Point",
+      "coords": {}
+    },
+    {
+      "index": 4358,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4358.png",
+      "answer": "x-y-3=0",
+      "fact_expr": "G: Hyperbola;H: Line;B: Point;A: Point;P: Point;Expression(G) = (x^2 - 4*y^2 = 4);Coordinate(P) = (4, 1);Intersection(H, G) = {A, B};MidPoint(LineSegmentOf(A,B))=P",
+      "coords": {
+        "P": [
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+          1.0
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+    {
+      "index": 4359,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4360,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4360.png",
+      "answer": "(\\sqrt{5}-1)/2",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;F2: Point;A: Point;e: Number;c:Number;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;Abs(LineSegmentOf(F1, F2)) = 2*c;PointOnCurve(A, G);DotProduct(VectorOf(A, F1), VectorOf(F1, F2)) = 0;DotProduct(VectorOf(A, F1), VectorOf(A, F2)) = c^2;Eccentricity(G) = e",
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+    },
+    {
+      "index": 4361,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4362,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4363,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
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+          0.0
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+        "radius": 5.0
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+      "optimization": {
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_4363.png",
+      "answer": "{(x^2/9+y^2/5=1)&Negation(x=-3),(y=0)&Negation(x=-3)&Negation(x=-2)&Negation(x=2)}",
+      "fact_expr": "P: Circle;M:Circle;N:Circle;Expression(M)=((x-2)^2+y^2=25);Expression(N)=((x+2)^2+y^2=1);IsTangent(P,M);IsTangent(P,N)",
+      "coords": {}
+    },
+    {
+      "index": 4364,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4365,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "vertical"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4365.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "C: Hyperbola;G: Circle;r: Number;Expression(C) = (-x^2/4 + y^2/5 = 1);r>0;Expression(G) = (y^2 + (x - 3)^2 = r^2);IsTangent(Asymptote(C),G)",
+      "coords": {}
+    },
+    {
+      "index": 4366,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4367,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4367.png",
+      "answer": "1",
+      "fact_expr": "C: Parabola;Expression(C) = (x^2 = 4*y);P: Point;PointOnCurve(P, C);A: Point;Coordinate(A) = (0, 1);Distance(P, A) = 2",
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+          1.0
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+    },
+    {
+      "index": 4368,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4368.png",
+      "answer": "2",
+      "fact_expr": "G: Ellipse;P: Point;Q: Point;H:Line;M: Point;S: Point;Expression(G) = (x^2/4 + y^2/3 = 1);Expression(H) = (x = 4);Coordinate(M) = (1, 0);PointOnCurve(M,LineSegmentOf(P,Q));IsChordOf(LineSegmentOf(P,Q),G);Intersection(LineOf(P,Q), H) = S",
+      "coords": {
+        "M": [
+          1.0,
+          0.0
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+    },
+    {
+      "index": 4369,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4369.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;A: Point;P: Point;Q: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (0, 2*sqrt(2));PointOnCurve(P, G);PointOnCurve(P, LineSegmentOf(P, Q));IsPerpendicular(LineSegmentOf(P, Q), yAxis);FootPoint(LineSegmentOf(P, Q), yAxis) = Q",
+      "coords": {
+        "A": [
+          0.0,
+          2.8284271247461903
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+    },
+    {
+      "index": 4370,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.7320508075688772,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 3.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4370.png",
+      "answer": "1",
+      "fact_expr": "G: Parabola;H: Ellipse;I: Circle;P: Point;Q: Point;O: Origin;Expression(G) = (x^2 = 2*(sqrt(6)*y));Expression(H) = (x^2/3 + y^2 = 1);PointOnCurve(P, H);PointOnCurve(Q, Directrix(G));IsDiameter(LineSegmentOf(P,Q),I);PointOnCurve(O, I)",
+      "coords": {}
+    },
+    {
+      "index": 4371,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 4.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4371.png",
+      "answer": "6",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/4 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 4372,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4372.png",
+      "answer": "-3",
+      "fact_expr": "G: Parabola;H: Line;O: Origin;A: Point;B: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(Focus(G),H);Intersection(G, H) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 4373,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4373.png",
+      "answer": "sqrt(5)/3",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;M: Point;PointOnCurve(M, G);AngleOf(F1, M, F2) = ApplyUnit(90, degree);N: Point;Intersection(LineOf(M, F1), G) = N;4*Abs(LineSegmentOf(N, F2)) = 5*Abs(LineSegmentOf(M, F2))",
+      "coords": {}
+    },
+    {
+      "index": 4374,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4374.png",
+      "answer": "(1, 0)",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;Focus(G) = {F1, F2};e: Number;Eccentricity(G) = e;e = sqrt(2)/2;P: Point;PointOnCurve(P, G) = True;DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0;Area(TriangleOf(P, F1, F2)) = 1;RightFocus(G) = F2",
+      "coords": {}
+    },
+    {
+      "index": 4375,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4376,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4377,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4378,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_focus_mismatch",
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4378.png",
+      "answer": "x^2/3 - y^2/6 = 1",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;N: Point;M: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(MidPoint(LineSegmentOf(M,N))) = (3, 6);Coordinate(F) = (-3, 0);LeftFocus(G)=F;PointOnCurve(F, H);Intersection(H, G) = {M, N};F:Point",
+      "coords": {
+        "F": [
+          -3.0,
+          0.0
+        ]
+      },
+      "validation_reasons": [
+        "hyperbola_focus_mismatch"
+      ]
+    },
+    {
+      "index": 4379,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4379.png",
+      "answer": "3/2",
+      "fact_expr": "G: Parabola;F: Point;P: Point;Q: Point;l: Line;Expression(G) = (y^2 = 4*x);Focus(G) = F;Directrix(G)=l;PointOnCurve(P, l);OneOf(Intersection(LineOf(P,F),G))=Q;VectorOf(F, P) = 4*VectorOf(F, Q)",
+      "coords": {}
+    },
+    {
+      "index": 4380,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4380.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;M: Point;N: Point;H:Line;C:Circle;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(LeftFocus(G),H);IsPerpendicular(H,xAxis);Intersection(H,G) = {M, N};IsDiameter(LineSegmentOf(M,N),C);PointOnCurve(RightVertex(G),C)",
+      "coords": {}
+    },
+    {
+      "index": 4381,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4381.png",
+      "answer": "ApplyUnit(60,degree)",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 + y^2 = 1);PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G) = F2;Abs(LineSegmentOf(P, F1))*Abs(LineSegmentOf(P, F2)) = 4/3",
+      "coords": {}
+    },
+    {
+      "index": 4382,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4382.png",
+      "answer": "-8",
+      "fact_expr": "r: Ellipse;Expression(r) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;Eccentricity(r) = sqrt(3)/2;A: Point;B: Point;C: Point;PointOnCurve(Vertex(TriangleOf(A, B, C)), r);D: Point;E: Point;M: Point;MidPoint(LineSegmentOf(A, B)) = D;MidPoint(LineSegmentOf(B, C)) = E;MidPoint(LineSegmentOf(A, C)) = M;k1: Number;k2: Number;k3: Number;Slope(OverlappingLine(LineSegmentOf(A, B))) = k1;Slope(OverlappingLine(LineSegmentOf(B, C))) = k2;Slope(OverlappingLine(LineSegmentOf(A, C))) = k3;Negation(k1=0);Negation(k2=0);Negation(k3=0);O: Origin;Slope(LineOf(O, D)) + Slope(LineOf(O, E)) + Slope(LineOf(O, M)) = 2",
+      "coords": {}
+    },
+    {
+      "index": 4383,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_focus_mismatch",
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4383.png",
+      "answer": "x^2/18+y^2/9=1",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4384.png",
+      "answer": "4-2*sqrt(5)",
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+      "coords": {}
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+    {
+      "index": 4385,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4385.png",
+      "answer": "3",
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+      "coords": {}
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+    {
+      "index": 4386,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4386.png",
+      "answer": "y^2 = 6*x",
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+    {
+      "index": 4387,
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+      "answer": "sqrt(6)/2",
+      "fact_expr": "D: Hyperbola;C: Ellipse;P: Point;F1: Point;F2: Point;e: Number;Expression(C) = (x^2/4 + y^2 = 1);Focus(D) = {F1,F2};Focus(C) = {F1,F2};OneOf(Intersection(C,D))=P;IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2));Eccentricity(D) = e",
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+    {
+      "index": 4388,
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+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4389,
+      "success": true,
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+      "answer": "1/4, -1/28",
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+    {
+      "index": 4390,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4390.png",
+      "answer": "sqrt(3)/2",
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+    {
+      "index": 4391,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4391.png",
+      "answer": "2",
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+    {
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+      "success": true,
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4392.png",
+      "answer": "x^2 - y^2/4 = 1",
+      "fact_expr": "G: Hyperbola;C: Parabola;P: Point;O: Origin;Expression(C) = (y^2 = 4*x);Coordinate(P) = (sqrt(5), 4);Center(G) = O;PointOnCurve(P, G);RightVertex(G) = Focus(C)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4393.png",
+      "answer": "y^2 = 8*x",
+      "fact_expr": "G: Parabola;F: Point;Coordinate(F) = (2, 0);Focus(G) = F",
+      "coords": {
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_4394.png",
+      "answer": "(x-2)^2+y^2=1",
+      "fact_expr": "H: Line;A: Point;Coordinate(A) = (4, 0);PointOnCurve(A, H);G: Circle;Expression(G) = (x^2 + y^2 = 4);B: Point;Intersection(H, G) = B;P: Point;MidPoint(LineSegmentOf(A, B)) = P",
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+        "y_coef": 9.0
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+        "note": "using explicit params"
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+      "answer": "8",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};H: Line;PointOnCurve(F1, H);A: Point;B: Point;Intersection(H, G) = {A, B};Abs(LineSegmentOf(F2, A)) + Abs(LineSegmentOf(F2, B)) = 12",
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+      "index": 4396,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "success": true,
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4397.png",
+      "answer": "3",
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+    {
+      "index": 4398,
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+        "b": 1.0,
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+        "y_coef": 1.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4398.png",
+      "answer": "(0, 1/2]",
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+    {
+      "index": 4399,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4399.png",
+      "answer": "3",
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+          2.0
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+      "index": 4400,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4400.png",
+      "answer": "2*sqrt(2)",
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+    {
+      "index": 4401,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4401.png",
+      "answer": "1",
+      "fact_expr": "G: Parabola;H: Line;O: Origin;A: Point;B: Point;k1:Number;k2:Number;Expression(G) = (y^2 = 4*x);Expression(H) = (y = x - 1);Intersection(H, G) = {A, B};Slope(LineSegmentOf(O, A)) = k1;Slope(LineSegmentOf(O,B))=k2",
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+    {
+      "index": 4402,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4402.png",
+      "answer": "2",
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+    {
+      "index": 4403,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4403.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 4404,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4404.png",
+      "answer": "3/2+sqrt(2)",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x);F: Point;Focus(G) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, G) = {A, B}",
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+    {
+      "index": 4405,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_focus_mismatch",
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4405.png",
+      "answer": "x^2 - y^2 / 3 = 1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;Coordinate(F) = (2, 0);OneOf(Focus(G)) = F;H: Circle;Expression(H) = (y^2 + (x - 2)^2 = 3);IsTangent(Asymptote(G), H)",
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+          0.0
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+      "validation_reasons": [
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+    {
+      "index": 4406,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 2.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4406.png",
+      "answer": "x^2/6 - y^2/8 = 1",
+      "fact_expr": "G: Hyperbola;H: Point;C:Hyperbola;Expression(G) = (x^2/3 - y^2/4 = 1);Coordinate(H) = (3, 2);Asymptote(C) = Asymptote(G);PointOnCurve(H,C)",
+      "coords": {
+        "H": [
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+          2.0
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+    },
+    {
+      "index": 4407,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "left"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4407.png",
+      "answer": "(-2, \\pm 2\\sqrt{2})",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = -4*x);M: Point;PointOnCurve(M, G);Distance(M, Focus(G)) = 3",
+      "coords": {}
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+    {
+      "index": 4408,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4408.png",
+      "answer": "9*sqrt(13)/13",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;A: Point;RightVertex(G) = A;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;H: Line;PointOnCurve(F2, H);IsPerpendicular(H, xAxis);B: Point;OneOf(Intersection(H, G)) = B;Abs(LineSegmentOf(F1, F2)) = 2;Abs(LineSegmentOf(F2, B)) = 3/2",
+      "coords": {}
+    },
+    {
+      "index": 4409,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4409.png",
+      "answer": "3",
+      "fact_expr": "H: Ellipse;Expression(H) = (x^2/4 + y^2/m = 1);m: Number;G: Hyperbola;Expression(G) = (x^2 - y^2/n = 1);n: Number;F1: Point;F2: Point;Focus(H) = {F1, F2};Focus(G) = {F1, F2};P: Point;OneOf(Intersection(H, G)) = P",
+      "coords": {}
+    },
+    {
+      "index": 4410,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4411,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4411.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;F:Point;Expression(G) = (x^2/9 - y^2/16 = 1);RightFocus(G)=F",
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+      "answer": "y^2/4 + x^2 = 1",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4413.png",
+      "answer": "16/3",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4414.png",
+      "answer": "12",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);F: Point;Focus(G) = F;A: Point;B: Point;PointOnCurve(A, G);PointOnCurve(B, G);VectorOf(A, F) = 2*VectorOf(F, B)",
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+      "answer": "[\\sqrt{3}/2, 1)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4416.png",
+      "answer": "sqrt(2)+1",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;N: Point;M: Point;O: Origin;F: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Vertex(H) = O;Focus(H) = F;LeftFocus(G)=F;Intersection(H,G)={M,N};MidPoint(LineSegmentOf(M,N)) = F",
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+    {
+      "index": 4417,
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+        "b": 2.23606797749979,
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4417.png",
+      "answer": "34",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/5 = 1);F: Point;RightFocus(G) = F;P: Point;PointOnCurve(P, LeftPart(G));A: Point;Coordinate(A) = (0, 6*sqrt(6))",
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+        "direction": "up"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4418.png",
+      "answer": "7/2",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 2*y);P: Point;PointOnCurve(P, G) = True;YCoordinate(P) = 3",
+      "coords": {}
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+    {
+      "index": 4419,
+      "success": true,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4419.png",
+      "answer": "(\\sqrt{5}/5,1)",
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+        "note": "using explicit params"
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+      "answer": "sqrt(3) + 1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;O: Origin;L: Line;PointOnCurve(O, L);Inclination(L) = pi/3;P: Point;Q: Point;Intersection(L, LeftPart(G)) = P;Intersection(L, RightPart(G)) = Q;H: Circle;IsDiameter(LineSegmentOf(P, Q), H);PointOnCurve(F, H);RightFocus(G) = F;F:Point",
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+      "index": 4421,
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+        "note": "using explicit params"
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+      "answer": "3",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Focus(C) = {F1, F2};PointOnCurve(P, C);IsPerpendicular(VectorOf(P, F2),VectorOf(P, F1));Area(TriangleOf(P, F1, F2)) = 9",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4422.png",
+      "answer": "1",
+      "fact_expr": "G: Hyperbola;E: Point;O: Origin;F: Point;Expression(G) = (x^2/2 - y^2/2 = 1);OneOf(Focus(G)) = F;L:Line;PointOnCurve(F,L);IsPerpendicular(OneOf(Asymptote(G)),L);FootPoint(OneOf(Asymptote(G)),L)=E",
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+      "index": 4423,
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+        "orientation": "horizontal"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4423.png",
+      "answer": "(1,\\sqrt{2})",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;H: Line;PointOnCurve(RightFocus(G), H) = True;Inclination(H) = ApplyUnit(45, degree);NumIntersection(H, RightPart(G)) = 2;e: Number;Eccentricity(G) = e",
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+        "x_coef": 16.0,
+        "y_coef": 12.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4424.png",
+      "answer": "(2, -3)",
+      "fact_expr": "G: Ellipse;H: Line;P: Point;Expression(G) = (x^2/16 + y^2/12 = 1);Expression(H) = (x - 2*y - 12 = 0);PointOnCurve(P, G);WhenMin(Distance(P,H))",
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+    {
+      "index": 4425,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 9.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4425.png",
+      "answer": "4",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;x0:Number;y0:Number;Expression(G) = (x^2/16 + y^2/9 = 1);Coordinate(P) = (x0, y0);PointOnCurve(P, G);Focus(G) = {F1, F2}",
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+    {
+      "index": 4426,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4427,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4428,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.4142135623730951,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4428.png",
+      "answer": "3/4",
+      "fact_expr": "F1: Point;F2: Point;C: Hyperbola;Expression(C) = (x^2 - y^2 = 2);LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C) = True;Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))",
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+    {
+      "index": 4429,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4429.png",
+      "answer": "sqrt(10)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1:Point;F2: Point;O: Origin;a>0;b>0;L:Line;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(F2,L);IsPerpendicular(L,Asymptote(G));FootPoint(L,Asymptote(G))=P;LeftFocus(G)=F1;RightFocus(G)=F2;Tan(AngleOf(P, F2, O)) = 1/3",
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+      "index": 4430,
+      "success": true,
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+        "iterations": 89
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4430.png",
+      "answer": "1",
+      "fact_expr": "G: Hyperbola;a: Number;a>0;Expression(G) = (-y^2/3 + x^2/a^2 = 1);Eccentricity(G) = 2",
+      "coords": {}
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+    {
+      "index": 4431,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4431.png",
+      "answer": "x**2/3-y**2=1",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;O: Origin;F: Point;A: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;IsDiameter(LineSegmentOf(O,F),G);Intersection(G,OneOf(Asymptote(C)))=A;sqrt(3)*Abs(LineSegmentOf(O, F)) = 2*Abs(LineSegmentOf(O, A));Distance(F,l) = 1;l: Line;OneOf(Asymptote(C)) = l",
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+    {
+      "index": 4432,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4432.png",
+      "answer": "1",
+      "fact_expr": "C: Ellipse;A: Point;P: Point;F: Point;Expression(C) = (x^2/4 + y^2/3 = 1);RightFocus(C)=F;Coordinate(A) = (2, 4);PointOnCurve(P, C)",
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+        "A": [
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+          4.0
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+    {
+      "index": 4433,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4433.png",
+      "answer": "y=pm*4*x/3\ny^2=20*x",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2/9 - y^2/16 = 1);G: Parabola;Focus(G) = RightFocus(C)",
+      "coords": {}
+    },
+    {
+      "index": 4434,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4434.png",
+      "answer": "(-oo, 0) + [1, +oo)",
+      "fact_expr": "G: Parabola;H: Curve;a: Real;Expression(G) = (y^2 = 4*x);Expression(H) = And(x^2/a + y^2/4 = 1, y>=0);NumIntersection(Directrix(G), H) = 1",
+      "coords": {}
+    },
+    {
+      "index": 4435,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4435.png",
+      "answer": "sqrt(5)/5",
+      "fact_expr": "G: Hyperbola;H: Point;Expression(G) = (x^2/4 - y^2 = 1);Coordinate(H) = (1, 0)",
+      "coords": {
+        "H": [
+          1.0,
+          0.0
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+      }
+    },
+    {
+      "index": 4436,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4437,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4437.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "G: Parabola;H: Line;A: Point;P: Point;d: Number;Expression(G) = (x^2 = 4*y);Expression(H) = (y = -1);Coordinate(A) = (2, 0);PointOnCurve(P, G);Distance(P, H) = d",
+      "coords": {
+        "A": [
+          2.0,
+          0.0
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+      }
+    },
+    {
+      "index": 4438,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4438.png",
+      "answer": "sqrt(2)+1",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;I: Point;lambda:Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, RightPart(G));LeftFocus(G) = F1;RightFocus(G) = F2;Abs(LineSegmentOf(F1, F2)) = b^2/a;Incenter(TriangleOf(P,F1,F2))=I;lambda*Area(TriangleOf(I,P,F1))=lambda*Area(TriangleOf(I,P,F2))+Area(TriangleOf(I,F1,F2))",
+      "coords": {}
+    },
+    {
+      "index": 4439,
+      "success": false,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4451.png",
+      "answer": "sqrt(6)/2",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2/2 - y^2 = 1)",
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+      "answer": "(2,4)",
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+      "answer": "y^2 = 4*x",
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+      "answer": "2",
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+      "index": 4459,
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+      "answer": "17",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4460.png",
+      "answer": "10/3",
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+    {
+      "index": 4461,
+      "success": true,
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4461.png",
+      "answer": "sqrt(2)",
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+      "error": "Unsupported or unparseable expression"
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+      "index": 4463,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 4464,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 2.0,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4464.png",
+      "answer": "8*sqrt(3)/3",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 4);F: Point;RightFocus(G) = F;H: Line;PointOnCurve(F, H);Inclination(H) = ApplyUnit(105, degree);P: Point;Q: Point;Intersection(H, G) = {P, Q}",
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+      "index": 4465,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4465.png",
+      "answer": "25/8",
+      "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;C: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(Focus(G), H);Intersection(H, G) = {A, B};XCoordinate(A) = 4;L:Line;PointOnCurve(A,L);PointOnCurve(Vertex(G),L);Intersection(L,Directrix(G))=C",
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+    {
+      "index": 4466,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 1.0,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4466.png",
+      "answer": "2*x-y-15=0",
+      "fact_expr": "P: Point;G: Hyperbola;A: Point;B: Point;H: Line;Expression(G) = (x^2 - 4*y^2 = 4);Coordinate(P) = (8, 1);PointOnCurve(P, H) = True;Intersection(H, G) = {A , B};MidPoint(LineSegmentOf(A, B)) = P",
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+        "direction": "up"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4467.png",
+      "answer": "5",
+      "fact_expr": "G: Parabola;F: Point;M: Point;A: Point;O:Origin;Coordinate(F) = (0, 1);Coordinate(M) = (-1, 4);Vertex(G)=O;Focus(G)=F;PointOnCurve(A,G)",
+      "coords": {
+        "F": [
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+          1.0
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+        "M": [
+          -1.0,
+          4.0
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+    {
+      "index": 4468,
+      "success": true,
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4468.png",
+      "answer": "3",
+      "fact_expr": "l: Line;C: Parabola;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;p: Number;A: Point;F: Point;D: Point;B: Point;p>0;Intersection(l,C)={A,B};Intersection(l,Directrix(C))=D;PointOnCurve(F, l);Abs(LineSegmentOf(A, F)) = 6;VectorOf(D, B) = 2*VectorOf(B, F)",
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+    {
+      "index": 4469,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "p": 1.0,
+        "direction": "up"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4469.png",
+      "answer": "(\\sqrt{17}/4, +\\infty)",
+      "fact_expr": "t: Number;P: Point;Coordinate(P) = (2, t);G: Parabola;Expression(G) = (x^2 = 4*y);PointOnCurve(P, G);M: Point;N: Point;PointOnCurve(M, G);PointOnCurve(N, G);Negation(M=P);Negation(N=P);Slope(LineOf(P, M))+Slope(LineOf(P, N)) = 3/2;Q: Point;MidPoint(LineSegmentOf(M, N)) = Q;d: Number;O: Origin;Distance(Q, O) = d",
+      "coords": {}
+    },
+    {
+      "index": 4470,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 2.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4470.png",
+      "answer": "{x^2/20-y^2/16=1, y^2/16-x^2/20=1}",
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+      "answer": "{(-\u221e,-\u221a(2)], [\u221a(2),+\u221e)}",
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+      "answer": "(\\sqrt{5}-1)/2",
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+      "answer": "6*sqrt(3)",
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+      "answer": "13/2",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4484.png",
+      "answer": "2*sqrt(2)",
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+    {
+      "index": 4486,
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+      "answer": "2",
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+    {
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+      "answer": "3*sqrt(2)",
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+      "answer": "(2*sqrt(3)/3, +oo)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4489.png",
+      "answer": "2*sqrt(5)",
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+      "answer": "(\\sqrt{6}, 0)",
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+      "answer": "5/4",
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+      "answer": "12/5",
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+      "index": 4493,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4494.png",
+      "answer": "8",
+      "fact_expr": "G: Hyperbola;H: Parabola;p: Number;Expression(G) = (x^2 - y^2 = 8);p>0;Expression(H) = (y^2 = 2*(p*x));Focus(H) = OneOf(Focus(G))",
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+      "error": "Unsupported or unparseable expression"
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+      "answer": "8",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4497.png",
+      "answer": "y=pm*(sqrt(3)/3)*x",
+      "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (y^2/16 - x^2/m = 1);Eccentricity(G) = e;e = 2;e:Number",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4499.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;Expression(H) = (x + 2*y - 1 = 0);IsPerpendicular(OneOf(Asymptote(G)), H);H: Line",
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+      "index": 4500,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4500.png",
+      "answer": "(5/4)*p^2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;K: Point;Intersection(Directrix(G), xAxis) = K;l: Line;PointOnCurve(K, l);A: Point;B: Point;Intersection(l, G) = {A, B};O: Origin",
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+    {
+      "index": 4501,
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+      "answer": "3*sqrt(5)/5",
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+      "answer": "pm*sqrt(3)/2",
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+      "answer": "2*sqrt(2)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4519.png",
+      "answer": "1/2",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4520.png",
+      "answer": "4*sqrt(3)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4521.png",
+      "answer": "sqrt(6)/2",
+      "fact_expr": "l: Line;G: Hyperbola;b: Number;a: Number;A: Point;F: Point;B: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F;PointOnCurve(F, l);Intersection(l, Asymptote(G)) = {A, B};IsPerpendicular(l,OneOf(Asymptote(G)));VectorOf(A, F) = 3*VectorOf(F, B)",
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+        "b": 4.0,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4522.png",
+      "answer": "4",
+      "fact_expr": "C: Hyperbola;F: Point;Expression(C) = (x^2/9 - y^2/16 = 1);OneOf(Focus(C)) = F",
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+      "index": 4523,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "success": true,
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+        "y_coef": 7.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4524.png",
+      "answer": "8+sqrt(5)",
+      "fact_expr": "F1: Point;F2: Point;G: Ellipse;Expression(G) = (x^2/16 + y^2/7 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;E: Point;PointOnCurve(E, G);N: Point;Coordinate(N) = (5, 1)",
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+      "index": 4525,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 4526,
+      "success": true,
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+        "direction": "up"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4526.png",
+      "answer": "2*sqrt(13)",
+      "fact_expr": "G: Parabola;A: Point;F: Point;P: Point;O: Origin;Expression(G) = (x^2 = 8*y);Focus(G) = F;PointOnCurve(P, Directrix(G));PointOnCurve(A, G);Abs(LineSegmentOf(A, F)) = 4",
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+    {
+      "index": 4527,
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+        "orientation": "vertical"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4527.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/12 + y^2/4 = 1)",
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4528.png",
+      "answer": "y^2=8*x",
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4529.png",
+      "answer": "sqrt(17)/2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x);F: Point;Focus(G) = F;P: Point;PointOnCurve(P, G);A: Point;Coordinate(A) = (0, 2)",
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+    {
+      "index": 4530,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4531,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 4.0,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4531.png",
+      "answer": "13",
+      "fact_expr": "F1: Point;F2: Point;Focus(G) = {F1, F2};G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);M: Point;PointOnCurve(M, G);Distance(M, F1) = 7",
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4532.png",
+      "answer": "sqrt(10)-1",
+      "fact_expr": "G: Parabola;A: Point;P:Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (2, 3);PointOnCurve(P, G)",
+      "coords": {
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+      "index": 4533,
+      "success": true,
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+        "b": 2.23606797749979,
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4533.png",
+      "answer": "x^2/16+y^2/25=1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/5 - y^2/4 = -1);H: Ellipse;Focus(G) = Focus(H);Eccentricity(H) = 0.6",
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+    {
+      "index": 4534,
+      "success": true,
+      "conic_type": "ellipse",
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+        "b": 2.23606797749979,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 5.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4534.png",
+      "answer": "x=-2",
+      "fact_expr": "G: Parabola;p: Number;H: Ellipse;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (x^2/9 + y^2/5 = 1);Focus(G) = RightFocus(H)",
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+    {
+      "index": 4535,
+      "success": true,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4535.png",
+      "answer": "y=pm*(4/3)*x",
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+      "answer": "sqrt(5)",
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+      "answer": "3",
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+      "answer": "sqrt(5)/3",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4541.png",
+      "answer": "10/3",
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+      "answer": "27",
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+        "note": "using explicit params"
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+      "answer": "x^2 - y^2 / 3 = 1  \ny = pm * sqrt(3) * x",
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+      "answer": "5",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 10*x)",
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+        "note": "using explicit params"
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+      "answer": "y=pm*x/2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/36 - y^2/9 = 1)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4547.png",
+      "answer": "sqrt(85)/5",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4548.png",
+      "answer": "sqrt(5)+2",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;A: Point;Coordinate(A) = (1, 2);PointOnCurve(A,C) = True",
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4549.png",
+      "answer": "3",
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+      "index": 4550,
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4550.png",
+      "answer": "(7/4, pm*sqrt(7)/2)",
+      "fact_expr": "G: Parabola;P: Point;Expression(G) = (y^2 = x);PointOnCurve(P, G);Distance(P, Focus(G)) = 2",
+      "coords": {}
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+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 4.0,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4551.png",
+      "answer": "pi/2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, RightPart(G));Abs(LineSegmentOf(P, F1))*Abs(LineSegmentOf(P, F2)) = 32",
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+      "index": 4552,
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+        "orientation": "vertical"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4552.png",
+      "answer": "y^2/100 - x^2/25 = 1",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-x^2/b^2 + y^2/a^2 = 1);b: Number;a: Number;FocalLength(C) = 10*sqrt(5);P: Point;Coordinate(P) = (1, 2);PointOnCurve(P, Asymptote(C))",
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+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4553.png",
+      "answer": "[2,+oo)",
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+      "index": 4554,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4555.png",
+      "answer": "x^2/4 + y^2 = 1",
+      "fact_expr": "G: Ellipse;a: Number;b: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Distance(LeftFocus(G), RightDirectrix(G)) = 7*sqrt(3)/3;Distance(Center(G), Directrix(G)) = 4*sqrt(3)/3",
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+    {
+      "index": 4556,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4556.png",
+      "answer": "(1, \\sqrt{2}+1)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;F1: Point;F2: Point;P: Point;a>0;b>0;c:Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);LeftFocus(G) =F1;RightFocus(G)=F2;PointOnCurve(P,G);Sin(AngleOf(P,F1,F2))/Sin(AngleOf(P,F2,F1))=a/c",
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+    {
+      "index": 4557,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "answer": "3",
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+    {
+      "index": 4558,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 2.8284271247461903,
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4558.png",
+      "answer": "-1",
+      "fact_expr": "G: Hyperbola;A: Point;B: Point;C: Point;O: Origin;D: Point;E: Point;F: Point;Expression(G) = (x^2 - y^2/8 = 1);PointOnCurve(A, G);PointOnCurve(B, G);PointOnCurve(C, G);MidPoint(LineSegmentOf(A, B)) = D;MidPoint(LineSegmentOf(B, C)) = E;MidPoint(LineSegmentOf(A, C)) = F;k1: Number;k2: Number;k3: Number;Slope(LineSegmentOf(O, F)) = k3;Slope(LineSegmentOf(O, D)) = k1;Slope(LineSegmentOf(O, E)) = k2;k1 + k2 + k3 = -8;Negation(k1 = 0);Negation(k2 = 0);Negation(k3 = 0);k4: Number;k5: Number;k6: Number;Slope(LineSegmentOf(A, B)) = k4;Slope(LineSegmentOf(B, C)) = k5;Slope(LineSegmentOf(A, C)) = k6",
+      "coords": {}
+    },
+    {
+      "index": 4559,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4560,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "iterations": 98
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4560.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;PointOnCurve(Focus(G), xAxis);Expression(Asymptote(G)) = (y = pm*(3/2)*x);Length(ImageinaryAxis(G)) = 6",
+      "coords": {}
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+    {
+      "index": 4561,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4561.png",
+      "answer": "sqrt(5)-1",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);l1:Line;l2:Line;O:Origin;PointOnCurve(O,l1);PointOnCurve(O,l2);Slope(l1)>0;Slope(l2)>0;Inclination(l2)=2*Inclination(l1);OneOf(Asymptote(C))=l2;RightFocus(C)=F;L:Line;PointOnCurve(F,L);IsPerpendicular(l1,L);FootPoint(l1,L)=M;PointOnCurve(M,C);M:Point;F:Point",
+      "coords": {}
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+    {
+      "index": 4562,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4562.png",
+      "answer": "{(y^2=16*x), (x^2=-12*y)}",
+      "fact_expr": "G: Parabola;Vertex(G) = O;O: Origin;SymmetryAxis(G) = axis;H: Line;Expression(H) = (3*x - 4*y - 12 = 0);PointOnCurve(Focus(G), H)",
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+    {
+      "index": 4563,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4564,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4565,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4565.png",
+      "answer": "[3/8, 3/4]",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2/3 = 1);A1: Point;A2: Point;LeftVertex(C) = A1;RightVertex(C) = A2;P: Point;PointOnCurve(P, C);Range(Slope(LineOf(P, A2))) = [-2, -1]",
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+    {
+      "index": 4566,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4566.png",
+      "answer": "3",
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+    {
+      "index": 4567,
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+      "error": "Unsupported or unparseable expression"
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+    {
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+        "iterations": 118
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4568.png",
+      "answer": "2*sqrt(3)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/3 + x^2/a^2 = 1);a: Number;Eccentricity(G) = sqrt(6)/2;H: Circle;Expression(H) = (y^2 + (x - 6)^2 = r^2);r: Number;r>0;IsTangent(Asymptote(G), H)",
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+        "y_coef": 2.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4569.png",
+      "answer": "y^2 = 8*x",
+      "fact_expr": "G: Parabola;H: Ellipse;Expression(H) = (x^2/6 + y^2/2 = 1);Focus(G) = RightFocus(H)",
+      "coords": {}
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+    {
+      "index": 4570,
+      "success": true,
+      "conic_type": "circle",
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+        "radius": 1.0
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_4570.png",
+      "answer": "((x - 1/2)^2 + y^2 = 1/4)&(0 < x <= 1)",
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+      "coords": {}
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+    {
+      "index": 4571,
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+      "error": null,
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+        "radius": 2.0
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_4571.png",
+      "answer": "{(y^2+x^2+3*y=0)&(5/3<x<=3)}",
+      "fact_expr": "G: Circle;H: Line;A: Point;B: Point;O:Origin;Expression(G) = (6*y + x^2 + y^2 + 5 = 0);PointOnCurve(O,H);Intersection(H,G)={A,B};IsChordOf(LineSegmentOf(A,B),G);MidPoint(LineSegmentOf(A,B))=M;M:Point",
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+    {
+      "index": 4572,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 4573,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 4574,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4574.png",
+      "answer": "16",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/25 + y^2/16 = 1);PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G) = F2",
+      "coords": {}
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+    {
+      "index": 4575,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_point_off_curve",
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+        "b": 1.6536897822836507,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "iterations": 97
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4575.png",
+      "answer": "2",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;P: Point;Coordinate(P) = (0, sqrt(3));PointOnCurve(P,C);Length(MajorAxis(C)) = 2 * FocalLength(C)",
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+      "index": 4576,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4577,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4577.png",
+      "answer": "y=pm*4*x",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;M: Point;O: Origin;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(M, C);Abs(LineSegmentOf(M, O)) = Abs(LineSegmentOf(O, F1));Area(TriangleOf(M, O, F1)) = 2*a*b",
+      "coords": {}
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+    {
+      "index": 4578,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4578.png",
+      "answer": "3",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);F: Point;LeftFocus(G) = F;H: Line;Expression(H) = (x = m);m: Number;Intersection(H, G) = {A, B};A: Point;B: Point;WhenMax(Perimeter(TriangleOf(F, A, B)))",
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+    {
+      "index": 4579,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4580,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 3.0,
+        "major_axis": "y",
+        "x_coef": 9.0,
+        "y_coef": 16.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4580.png",
+      "answer": "y^2 = pm*20*x",
+      "fact_expr": "G: Parabola;H: Ellipse;O: Origin;Expression(H) = (x^2/9 + y^2/16 = 1);Vertex(G) = O;SymmetryAxis(G)=OverlappingLine(MinorAxis(H));Distance(Focus(G),Vertex(G))=5",
+      "coords": {}
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+    {
+      "index": 4581,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4581.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;M: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(M, G);Distance(M, Focus(G)) = 3;x1: Number;XCoordinate(M) = x1",
+      "coords": {}
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+    {
+      "index": 4582,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4582.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Curve;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2/b^2 + y^2/a^2 = 1);L:Line;PointOnCurve(OneOf(Focus(G)),L);IsPerpendicular(OneOf(Asymptote(G)),L);PointOnCurve(FootPoint(OneOf(Asymptote(G)),L),H)",
+      "coords": {}
+    },
+    {
+      "index": 4583,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4584,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.125,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4584.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;Expression(G) = (y = 2*x^2);F: Point;Focus(G) = F;M: Point;N: Point;PointOnCurve(M, G);PointOnCurve(N, G);Abs(LineSegmentOf(M, F)) + Abs(LineSegmentOf(N, F)) = 17/4",
+      "coords": {}
+    },
+    {
+      "index": 4585,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4585.png",
+      "answer": "1/2",
+      "fact_expr": "G: Ellipse;m: Number;F: Point;P: Point;m>3;Expression(G) = (y^2/3 + x^2/m = 1);OneOf(Focus(G)) = F;PointOnCurve(P, G);Max(Distance(P, F)) = 3",
+      "coords": {}
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+    {
+      "index": 4586,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4587,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4587.png",
+      "answer": "x+y-1=0",
+      "fact_expr": "l: Line;C: Ellipse;M: Point;A: Point;B: Point;F1: Point;Expression(C) = (x^2/4 + y^2/3 = 1);Coordinate(M) = (0, 1);PointOnCurve(M, l);Intersection(l, C) = {A, B};LeftFocus(C)=F1;WhenMax(Perimeter(TriangleOf(A,B,F1)))",
+      "coords": {
+        "M": [
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+          1.0
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+    {
+      "index": 4588,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 2.23606797749979,
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+        "x_coef": 9.0,
+        "y_coef": 5.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4588.png",
+      "answer": "sqrt(15)",
+      "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;O: Origin;M: Point;Expression(G) = (x^2/9 + y^2/5 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);MidPoint(LineSegmentOf(P,F1))=M;Abs(LineSegmentOf(O,F2))=Abs(LineSegmentOf(O,M))",
+      "coords": {}
+    },
+    {
+      "index": 4589,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4589.png",
+      "answer": "2*sqrt(5)",
+      "fact_expr": "P: Point;G: Parabola;Expression(G) = (x^2 = 8*y);PointOnCurve(P, G) = True;A: Point;Coordinate(A) = (4, 0);H: Line;Expression(H) = (y = -2)",
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+        "A": [
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+          0.0
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+    {
+      "index": 4590,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4591,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4591.png",
+      "answer": "2",
+      "fact_expr": "C: Parabola;Expression(C) = (x^2 = 4*y);F: Point;Focus(C) = F;l: Line;PointOnCurve(F, l);Intersection(l, C) = {A, B};A: Point;B: Point;Intersection(TangentOnPoint(A,C),xAxis) = M;Intersection(TangentOnPoint(A,C),yAxis) = N;M: Point;O: Origin;N: Point;Area(TriangleOf(M, O, N)) = 1/2",
+      "coords": {}
+    },
+    {
+      "index": 4592,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4593,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4594,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4594.png",
+      "answer": "5",
+      "fact_expr": "G: Ellipse;m: Number;Expression(G) = (y^2/4 + x^2/m = 1);PointOnCurve(Focus(G), xAxis);FocalLength(G) = 2",
+      "coords": {}
+    },
+    {
+      "index": 4595,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4595.png",
+      "answer": "8",
+      "fact_expr": "G: Parabola;A: Point;B: Point;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;IsChordOf(LineSegmentOf(A, B), G);Inclination(LineSegmentOf(A, B)) = ApplyUnit(45, degree);PointOnCurve(F, LineSegmentOf(A, B))",
+      "coords": {}
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+    {
+      "index": 4596,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 9.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4596.png",
+      "answer": "17",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/81 = 1);P: Point;PointOnCurve(P, G);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;Abs(LineSegmentOf(P, F1)) = 9",
+      "coords": {}
+    },
+    {
+      "index": 4597,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4597.png",
+      "answer": "2*sqrt(3)/3",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(P) = (1, 0);Distance(P, OneOf(Asymptote(C))) = 1/2",
+      "coords": {
+        "P": [
+          1.0,
+          0.0
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+      }
+    },
+    {
+      "index": 4598,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4599,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4599.png",
+      "answer": "[2*sqrt(2), 4)",
+      "fact_expr": "T: Parabola;C: Circle;l1: Line;l2: Line;P: Point;Q: Point;M: Point;Expression(T) = (y^2 = 4*x);Expression(C) = (y^2 + (x - 3)^2 = 4);PointOnCurve(M, T);TangentOfPoint(M, C) = {l1, l2};TangentPoint(l1, C) = P;TangentPoint(l2, C) = Q",
+      "coords": {}
+    },
+    {
+      "index": 4600,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4600.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "H: Line;Expression(H) = (y = x - 1);G: Parabola;Expression(G) = (y^2 = 4*x);Intersection(H,G) = {A,B};A: Point;B: Point;O: Origin",
+      "coords": {}
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+    {
+      "index": 4601,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 4602,
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+      "answer": "3*sqrt(3)",
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+      "answer": "5",
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4604.png",
+      "answer": "2",
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4605.png",
+      "answer": "16/5",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);P: Point;PointOnCurve(P, G);d1: Number;d2: Number;H: Line;Expression(H) = (4*x + 3*y + 8 = 0);Distance(P, Directrix(G)) = d1;Distance(P, H) = d2",
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+      "index": 4606,
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+        "iterations": 89
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4606.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Eccentricity(C) = 2;G: Parabola;Expression(G) = (y^2 = 16*x);RightFocus(C) = Focus(G)",
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+      "index": 4607,
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+      "error": null,
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+        "b": 1.0,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4607.png",
+      "answer": "8",
+      "fact_expr": "H: Parabola;Expression(H) = (y^2 = 2*(p*x));p: Number;p>0;l: Line;Directrix(H) = l;G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1);Z1: Line;Z2: Line;Asymptote(G) = {Z1, Z2};A: Point;Intersection(l, Z1) = A;B: Point;Intersection(l, Z2) = B;Abs(LineSegmentOf(A, B)) = 4",
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+      "index": 4608,
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+        "direction": "up"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4608.png",
+      "answer": "pm*1",
+      "fact_expr": "C: Parabola;P: Point;A: Point;F: Point;t:Number;Expression(C) = (x^2 = 8*y);Intersection(Directrix(C), yAxis) = A;Focus(C) = F;PointOnCurve(P, C);t=Abs(LineSegmentOf(P,A))/Abs(LineSegmentOf(P,F));WhenMax(t)",
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+    {
+      "index": 4609,
+      "success": true,
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+      "error": null,
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+        "b": 3.0,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4609.png",
+      "answer": "16.5",
+      "fact_expr": "G: Hyperbola;H: Point;P: Point;D:Point;Expression(G) = (x^2/16 - y^2/9 = 1);Coordinate(H) = (5, 0);Coordinate(D)=(-5,0);PointOnCurve(P, G);Distance(P, H) = 8.5",
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+      "index": 4610,
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4610.png",
+      "answer": "pm*sqrt(3)",
+      "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;k:Number;Expression(G) = (y^2 = 4*x);Expression(H) = (y = k*(x - 1));Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = 16/3",
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+    {
+      "index": 4611,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4611.png",
+      "answer": "sqrt(3)/2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);Intersection(Directrix(G), xAxis) = P;P: Point;PointOnCurve(P, H);Slope(H) = k;k: Number;k>0;H: Line;Intersection(H, G) = {A, B};B: Point;A: Point;Focus(G) = F;F: Point;Abs(LineSegmentOf(F, A)) = 3*Abs(LineSegmentOf(F, B));Slope(LineOf(A, B)) = k",
+      "coords": {}
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+    {
+      "index": 4612,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4612.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "G: Hyperbola;b: Number;H: Parabola;b>0;Expression(G) = (x^2 - y^2/b^2 = 1);Expression(H) = (y^2 = 8*x);Focus(H) = RightFocus(G)",
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+    {
+      "index": 4613,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4613.png",
+      "answer": "y^2 = - 20*x",
+      "fact_expr": "G: Hyperbola;H: Parabola;Expression(G) = (x^2/16 - y^2/9 = 1);Center(G) = Vertex(H);LeftFocus(G) = Focus(H)",
+      "coords": {}
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+    {
+      "index": 4614,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4614.png",
+      "answer": "sqrt(85)/5",
+      "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;A: Point;B: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F2;PointOnCurve(F2, l);Intersection(l, RightPart(C)) = {A, B};IsChordOf(LineSegmentOf(A, B), C);Abs(LineSegmentOf(A, B)) = 5*a;2*Abs(LineSegmentOf(B, F2)) = 3*Abs(LineSegmentOf(A, F2))",
+      "coords": {}
+    },
+    {
+      "index": 4615,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 2.8284271247461903,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4615.png",
+      "answer": "64*pi",
+      "fact_expr": "G: Hyperbola;H: Circle;O: Origin;N: Point;F1: Point;P: Point;M: Point;F2: Point;lambda:Number;lambda>0;mu:Number;Expression(G) = (x^2/16 - y^2/8 = 1);PointOnCurve(P, RightPart(G));LeftFocus(G) = F1;RightFocus(G) = F2;VectorOf(F1, P) = lambda*VectorOf(P, M);VectorOf(P,N)=mu*(VectorOf(P,M)/Abs(VectorOf(P,M))+VectorOf(P,F2)/Abs(VectorOf(P,F2)));Abs(VectorOf(P, F2)) = 4;DotProduct(VectorOf(P,N),VectorOf(F2,N))=0;Center(H)=O;Radius(H)=LineSegmentOf(O,N)",
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+    },
+    {
+      "index": 4616,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4616.png",
+      "answer": "x^2/2 - y^2/4 = 1",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Point;a>0;b>0;l:Line;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(H) = (1, 0);FocalLength(G) = 2*sqrt(6);OneOf(Asymptote(G)) = l;Distance(H, l) = sqrt(6)/3",
+      "coords": {
+        "H": [
+          1.0,
+          0.0
+        ]
+      }
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+    {
+      "index": 4617,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 2.23606797749979,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 5.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4617.png",
+      "answer": "8/3",
+      "fact_expr": "G: Ellipse;M: Point;N: Point;P: Point;Expression(G) = (x^2/9 + y^2/5 = 1);Coordinate(M) = (2, 0);Coordinate(N) = (-2, 0);PointOnCurve(P, G)",
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+        "M": [
+          2.0,
+          0.0
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+        "N": [
+          -2.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 4618,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4618.png",
+      "answer": "9*x - 32*y + 73 = 0",
+      "fact_expr": "G: Ellipse;H: LineSegment;M: Point;Expression(G) = (x^2/16 + y^2/9 = 1);Coordinate(M) = (-1, 2);IsChordOf(H, G);MidPoint(H) = M",
+      "coords": {
+        "M": [
+          -1.0,
+          2.0
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+    },
+    {
+      "index": 4619,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 3.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4619.png",
+      "answer": "pi/3, 2*pi/3",
+      "fact_expr": "l: Line;C: Parabola;A: Point;B: Point;F: Point;Expression(C) = (y^2 = 12*x);Focus(C) = F;PointOnCurve(F, l);Intersection(l,C)={A,B};IsChordOf(LineSegmentOf(A,B),C);Length(LineSegmentOf(A, B)) = 16",
+      "coords": {}
+    },
+    {
+      "index": 4620,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4620.png",
+      "answer": "(\\sqrt{5} - 1)/2",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;A: Point;F: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(A, G);OneOf(Focus(G)) = F;IsPerpendicular(LineSegmentOf(A,F), xAxis);c: Number;Abs(LineSegmentOf(A, F)) = c;HalfFocalLength(G) = c",
+      "coords": {}
+    },
+    {
+      "index": 4621,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 3.0,
+        "direction": "left"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4621.png",
+      "answer": "y=2*x+1",
+      "fact_expr": "G: Parabola;H: LineSegment;M: Point;Expression(G) = (y^2 = -12*x);Coordinate(M) = (-2, -3);MidPoint(H)=M;IsChordOf(H,G)",
+      "coords": {
+        "M": [
+          -2.0,
+          -3.0
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+    },
+    {
+      "index": 4622,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4623,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4623.png",
+      "answer": "y = pm*sqrt(3)*x",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;B: Point;F: Point;A: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftVertex(C) = A;RightFocus(C) = F;PointOnCurve(B, C);IsPerpendicular(LineSegmentOf(B, F), LineSegmentOf(A, F));Abs(LineSegmentOf(A, F)) = Abs(LineSegmentOf(B, F))",
+      "coords": {}
+    },
+    {
+      "index": 4624,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4624.png",
+      "answer": "4/9",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;G: Line;P: Point;F1: Point;F2: Point;A: Point;I: Point;B: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);Incenter(TriangleOf(P,F1,F2))=I;PointOnCurve(I,G);IsParallel(G,xAxis);Intersection(G,LineSegmentOf(P,F1))=A;Intersection(G,LineSegmentOf(P,F2))=B;Eccentricity(C)=e;e:Number;e=1/2",
+      "coords": {}
+    },
+    {
+      "index": 4625,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4625.png",
+      "answer": "(\\sqrt{5} - 1)/2",
+      "fact_expr": "A: Point;B: Point;G: Ellipse;b: Number;a: Number;e: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (a, 0);Coordinate(B) = (0, b);Eccentricity(G) = e;O: Origin;Distance(O, LineOf(A, B)) = c ;c: Number;HalfFocalLength(G) = c",
+      "coords": {}
+    },
+    {
+      "index": 4626,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4626.png",
+      "answer": "y = (pm*sqrt(3)/3)*x",
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+      "answer": "y^2=2*x-1",
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+      "answer": "{x^2/36-y^2/64=1,y^2/64-x^2/36=1}",
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+      "answer": "0",
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+      "answer": "3",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4642.png",
+      "answer": "48*x-y-95=0",
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+      "answer": "3*sqrt(3)",
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+      "answer": "x=-1",
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+      "answer": "[1/8,1/4)",
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+      "index": 4653,
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+      "answer": "y=x",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4655.png",
+      "answer": "(2,2)",
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+      "index": 4656,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4658.png",
+      "answer": "x^2/6 - y^2/3 = 1",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4659.png",
+      "answer": "sqrt(2)",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4660.png",
+      "answer": "x = pm*sqrt(5)/5",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/4 = 1)",
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+    {
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+        "b": 1.5,
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4661.png",
+      "answer": "(1, 2)",
+      "fact_expr": "G: Hyperbola;a: Number;Expression(G) = (-y^2/3 + x^2/a^2 = 1);a > 1",
+      "coords": {}
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+    {
+      "index": 4662,
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4662.png",
+      "answer": "x = -1",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4663.png",
+      "answer": "sqrt(5)+2",
+      "fact_expr": "l: Line;E: Parabola;A: Point;F: Point;B: Point;C: Point;Expression(E) = (y^2 = 4*x);Focus(E) = F;Expression(l)=(y = k*(x - 1));Intersection(l, E) = {A, B};Intersection(l,yAxis)=C;Abs(LineSegmentOf(A, F)) = Abs(LineSegmentOf(B, C));k:Number",
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+      "index": 4664,
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+      "answer": "x^2+(3/2)*y^2=1",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4666.png",
+      "answer": "4/7",
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+    {
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+        "note": "using explicit params"
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+      "answer": "24",
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+      "index": 4668,
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+      "index": 4669,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 4670,
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4670.png",
+      "answer": "sqrt(5)/2",
+      "fact_expr": "C: Hyperbola;a: Number;G: Parabola;l: Line;e: Number;Expression(C) = (-y^2 + x^2/a^2 = 1);Expression(G) = (y^2 = 8*x);Directrix(G) = l;IsTangent(l, C);Eccentricity(C) = e",
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+      "index": 4671,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 4672,
+      "success": true,
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+        "iterations": 89
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4672.png",
+      "answer": "x^2 - y^2/3 = 1",
+      "fact_expr": "C: Hyperbola;m: Number;n: Number;Negation(m*n=0);Expression(C) = (x^2/m^2 - y^2/n^2 = 1);E: Parabola;Expression(E) = (y^2 = 8*x);RightFocus(C) = Focus(E);Eccentricity(C) = 2",
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+    {
+      "index": 4673,
+      "success": true,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4673.png",
+      "answer": "y^2=12*(x+3)",
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+      "index": 4674,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.0,
+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4674.png",
+      "answer": "11",
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+    {
+      "index": 4675,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4675.png",
+      "answer": "2*sqrt(2)",
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+    {
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+      "success": true,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4676.png",
+      "answer": "4*sqrt(3)/3",
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+      "index": 4677,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4677.png",
+      "answer": "-4",
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+      "index": 4679,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4680,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4680.png",
+      "answer": "sqrt(3)",
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+      "index": 4681,
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+        "b": 1.5,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4681.png",
+      "answer": "2",
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+      "index": 4682,
+      "success": true,
+      "conic_type": "ellipse",
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4682.png",
+      "answer": "(1,3)+(3,+oo)",
+      "fact_expr": "G: Ellipse;m: Number;H: Line;Expression(G) = (y^2/3 + x^2/m = 1);Expression(H) = (y = x + 2);NumIntersection(H, G) = 2",
+      "coords": {}
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+    {
+      "index": 4683,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 4.0,
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+        "x_coef": 25.0,
+        "y_coef": 16.0
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4683.png",
+      "answer": "(3/2,-6/5)",
+      "fact_expr": "J: Point;Coordinate(J) = (3, 0);PointOnCurve(J,L) = True;L: Line;Slope(L) = 4/5;G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1)",
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+          0.0
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+    {
+      "index": 4684,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4685,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4685.png",
+      "answer": "x - 2*y = 0",
+      "fact_expr": "G: Parabola;H: LineSegment;A: Point;Expression(G) = (y^2 = 2*x);Coordinate(A) = (4, 2);MidPoint(H) = A;IsChordOf(H, G)",
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+        "A": [
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+          2.0
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+    {
+      "index": 4686,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4686.png",
+      "answer": "(1,3]",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;Focus(G)={F1,F2};P: Point;PointOnCurve(P,G);Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 4687,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4688,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4688.png",
+      "answer": "sqrt(2)/6",
+      "fact_expr": "G: Parabola;p: Number;H: Circle;p>0;Z: Point;X: Point;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (x^2 + (y - 1)^2 = 1);Intersection(G, H) = {Z, X};Distance(Z, X) = 2*sqrt(3)/3",
+      "coords": {}
+    },
+    {
+      "index": 4689,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4689.png",
+      "answer": "sqrt(7)/2",
+      "fact_expr": "G: Hyperbola;e: Number;Expression(G) = (x^2/4 - y^2/3 = 1);Eccentricity(G) = e",
+      "coords": {}
+    },
+    {
+      "index": 4690,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4690.png",
+      "answer": "(0, 1/2]",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;G: Line;A: Point;F: Point;B: Point;O:Origin;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F;PointOnCurve(O,G);Intersection(G,C) = {A, B};AngleOf(A,F,B)>=ApplyUnit(120,degree)",
+      "coords": {}
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+    {
+      "index": 4691,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
+      },
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4691.png",
+      "answer": "6",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 4692,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4693,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4693.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;H: Line;O: Origin;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, F)) = 2",
+      "coords": {}
+    },
+    {
+      "index": 4694,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4694.png",
+      "answer": "y=pm*(4/3)*x",
+      "fact_expr": "G: Hyperbola;a: Number;H: Parabola;a>0;Expression(G) = (-y^2/9 + x^2/a^2 = 1);Expression(H) = (y^2 = 20*x);Focus(H) = RightFocus(G)",
+      "coords": {}
+    },
+    {
+      "index": 4695,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4696,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
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+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4696.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
+      "fact_expr": "C: Parabola;p: Number;A: Point;O: Origin;B: Point;F: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;G: Line;PointOnCurve(F, G);Intersection(G, C) = {A, B};Abs(LineSegmentOf(A, F)) = 2*Abs(LineSegmentOf(B, F));Area(TriangleOf(A, O, B)) = 6*sqrt(2)",
+      "coords": {}
+    },
+    {
+      "index": 4697,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
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+      "optimization": {
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+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4697.png",
+      "answer": "y = -1",
+      "fact_expr": "G: Parabola;p: Number;Expression(G) = (x^2 = 2*p*y);Coordinate(Focus(G)) = (0, 1)",
+      "coords": {}
+    },
+    {
+      "index": 4698,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4698.png",
+      "answer": "x^2=(16/5)*y",
+      "fact_expr": "C: Parabola;p: Number;A: Point;P: Point;H: Point;0 < p;p < 4;Expression(C) = (x^2 = 2*(p*y));Coordinate(A) = (0, 4);Directrix(C)=l;PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(P,H),l);FootPoint(LineSegmentOf(P,H),l)=H;Abs(LineSegmentOf(P, H)) = Abs(LineSegmentOf(P, A));AngleOf(A, P, H) = ApplyUnit(120, degree);l:Line",
+      "coords": {
+        "A": [
+          0.0,
+          4.0
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+      }
+    },
+    {
+      "index": 4699,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4699.png",
+      "answer": "(1/2)*(a-p)",
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+      "coords": {}
+    },
+    {
+      "index": 4700,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4700.png",
+      "answer": "3",
+      "fact_expr": "C: Ellipse;a: Number;b: Number;F1: Point;P: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, C);LeftFocus(C) =F1;RightFocus(C)=F2;AngleOf(F1, P, F2) = ApplyUnit(90, degree);Area(TriangleOf(P,F1,F2)) = 9",
+      "coords": {}
+    },
+    {
+      "index": 4701,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4702,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
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+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4702.png",
+      "answer": "(0, 6)",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 2*(p*y));p: Number;p>0;H: Circle;Expression(H) = (6*y + x^2 + y^2 - 16 = 0);A: Point;B: Point;Intersection(Directrix(G), H) = {A, B};LineSegmentOf(A, B) = 8",
+      "coords": {}
+    },
+    {
+      "index": 4703,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4703.png",
+      "answer": "sqrt(3)/3",
+      "fact_expr": "A: Point;T: Point;B: Point;F: Point;C: Ellipse;Coordinate(A) = (0, b);Coordinate(B) = (0, -b);Coordinate(F) = (c, 0);Coordinate(T) = (3*c, 0);Expression(C) = (y^2/b^2 + x^2/a^2 = 1);a:Number;b:Number;c:Number;a>b;b>0;c = sqrt(a^2 - b^2);PointOnCurve(Intersection(LineOf(A,T),LineOf(B,F)), C)",
+      "coords": {}
+    },
+    {
+      "index": 4704,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4704.png",
+      "answer": "(1, 2)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/12 = 1);F: Point;RightFocus(G) = F;E: Hyperbola;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;Distance(F, Asymptote(E))<sqrt(3)",
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+    },
+    {
+      "index": 4705,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4705.png",
+      "answer": "11",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;A: Point;PointOnCurve(A, LeftPart(G));Abs(LineSegmentOf(A, F1)) = 5",
+      "coords": {}
+    },
+    {
+      "index": 4706,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.581065485469099,
+        "b": 1.581065485469099,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 1.6812139285133018e-06,
+        "converged": true,
+        "iterations": 95
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4706.png",
+      "answer": "x^2/4-y^2=1",
+      "fact_expr": "G: Hyperbola;F1: Point;F2: Point;P: Point;Coordinate(F1) = (-sqrt(5), 0);Coordinate(F2) = (sqrt(5), 0);Focus(G) = {F1,F2};PointOnCurve(P, G);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0;Abs(LineSegmentOf(P, F1))*Abs(LineSegmentOf(P, F2)) = 2",
+      "coords": {
+        "F1": [
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+          0.0
+        ],
+        "F2": [
+          2.23606797749979,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 4707,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.2175992727279663,
+        "b": 2.434081792831421,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 3.452532837400213e-06,
+        "converged": true,
+        "iterations": 124
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4707.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(OneOf(Asymptote(G)))=(y=2*x)",
+      "coords": {}
+    },
+    {
+      "index": 4708,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "y",
+        "x_coef": 1.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4708.png",
+      "answer": "0",
+      "fact_expr": "G: Ellipse;M: Point;F1: Point;F2: Point;Expression(G) = (x^2 + y^2/2 = 1);Focus(G) = {F1, F2};PointOnCurve(M, G);DotProduct(VectorOf(M, F1), VectorOf(M, F2)) = 0",
+      "coords": {}
+    },
+    {
+      "index": 4709,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4709.png",
+      "answer": "sqrt(15)/3",
+      "fact_expr": "F: Point;M: Hyperbola;Expression(M) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;RightFocus(M) = F;P: Point;PointOnCurve(P, M);O: Origin;Abs(LineSegmentOf(O, P)) = 2*b;AngleOf(P, O, F) = pi/6",
+      "coords": {}
+    },
+    {
+      "index": 4710,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4710.png",
+      "answer": "x^2/5+y^2=1\n2*sqrt(5)/5",
+      "fact_expr": "H: Line;Expression(H) = (x - 2*y + 2 = 0);G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;PointOnCurve(OneOf(Vertex(G)), H) = True;PointOnCurve(OneOf(Focus(G)), H) = True",
+      "coords": {}
+    },
+    {
+      "index": 4711,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4711.png",
+      "answer": "sqrt(3)/2",
+      "fact_expr": "P: Point;B1: Point;G: Ellipse;a: Number;b: Number;B2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);UpperVertex(G) = B2;LowerVertex(G) = B1;PointOnCurve(P, G);Slope(LineOf(P, B1)) = 1/4;Slope(LineOf(P, B2)) = -1",
+      "coords": {}
+    },
+    {
+      "index": 4712,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4712.png",
+      "answer": "3*sqrt(5) - 1",
+      "fact_expr": "G: Parabola;A: Point;P: Point;M: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (4, 6);PointOnCurve(P, G);Projection(P, yAxis) = M",
+      "coords": {
+        "A": [
+          4.0,
+          6.0
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+    },
+    {
+      "index": 4713,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4713.png",
+      "answer": "8",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);F: Point;Focus(G) = F;l: Line;Directrix(G) = l;P: Point;PointOnCurve(P,G) = True;IsPerpendicular(LineSegmentOf(P,A),l) = True;A: Point;FootPoint(LineSegmentOf(P,A),l) = A;Slope(LineOf(A,F)) = -sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 4714,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4714.png",
+      "answer": "6",
+      "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2/5 + y^2/m = 1);Coordinate(OneOf(Focus(G))) = (0, 1)",
+      "coords": {}
+    },
+    {
+      "index": 4715,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4715.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;a: Number;M: Point;l: Line;a>0;m:Number;Expression(G) = (x^2 = a*y);Coordinate(M) = (m, 1);PointOnCurve(M, G);Directrix(G) = l;Distance(M, l) = 2",
+      "coords": {}
+    },
+    {
+      "index": 4716,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4716.png",
+      "answer": "sqrt(5)/2",
+      "fact_expr": "E: Hyperbola;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;F: Point;RightFocus(E) = F;l1: Line;l2: Line;OneOf(Asymptote(E)) = l1;OneOf(Asymptote(E)) = l2;Negation(l1 = l2);H: Line;PointOnCurve(F, H);IsPerpendicular(l1,H);FootPoint(l1,H) = P;P: Point;Intersection(H,l2) = Q;Q: Point;5*VectorOf(P, F) = 3*VectorOf(F, Q)",
+      "coords": {}
+    },
+    {
+      "index": 4717,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4717.png",
+      "answer": "2*p^2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;C: Point;Intersection(Directrix(G), xAxis) = C;H: Line;PointOnCurve(F, H);A: Point;B: Point;Intersection(H, G) = {A, B};DotProduct(VectorOf(B, C), VectorOf(B, F)) = 0",
+      "coords": {}
+    },
+    {
+      "index": 4718,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.7320508075688772,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4718.png",
+      "answer": "1",
+      "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/3 - y^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) + Abs(LineSegmentOf(P, F2)) = 2*sqrt(5)",
+      "coords": {}
+    },
+    {
+      "index": 4719,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "answer": "y=pm*sqrt(2)*x/2",
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+      "answer": "1/2",
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+      "answer": "3",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4727.png",
+      "answer": "{-sqrt(5)/2, -1, 1, sqrt(5)/2}",
+      "fact_expr": "G: Hyperbola;H: Line;k: Real;Expression(G) = (x^2 - y^2 = 4);Expression(H) = (y = k*x - 1);NumIntersection(H, G)=1",
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+        "note": "using explicit params"
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+      "answer": "1",
+      "fact_expr": "C: Parabola;A: Point;Expression(C) = (x^2 = 4*y);PointOnCurve(A, C);Distance(A, Focus(C)) = 2;yA: Number;YCoordinate(A) = yA",
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4729.png",
+      "answer": "sqrt(17)",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 8*x);F: Point;Focus(C) = F;G: Line;M: Point;Coordinate(M) = (-2, 0);PointOnCurve(M, G);A: Point;B: Point;Intersection(G, C) = {A, B};O: Origin;IsParallel(LineSegmentOf(O, A), LineSegmentOf(B, F))",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4730.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1);P: Point;PointOnCurve(P, LeftPart(G)) = True;Distance(P, RightFocus(G)) = 13",
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+      "index": 4731,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 4732,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 4733,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 4734,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4735,
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+      "conic_type": "hyperbola",
+      "error": null,
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4735.png",
+      "answer": "sqrt(2)",
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+    {
+      "index": 4736,
+      "success": true,
+      "conic_type": "parabola",
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4736.png",
+      "answer": "20",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);H: Line;PointOnCurve(Focus(G), H);A: Point;B: Point;x1: Number;x2: Number;y1: Number;y2: Number;Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);Intersection(H, G) = {A, B};x1 + x2 = 16",
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+    {
+      "index": 4737,
+      "success": true,
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+        "x_coef": 3.0,
+        "y_coef": 1.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4737.png",
+      "answer": "(0,pm*1)",
+      "fact_expr": "G: Ellipse;F1: Point;A: Point;F2: Point;B: Point;Expression(G) = (x^2/3 + y^2 = 1);Focus(G) = {F1, F2};PointOnCurve(A, G);PointOnCurve(B, G);VectorOf(F1, A) = 5*VectorOf(F2, B)",
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+    {
+      "index": 4738,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 4739,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4739.png",
+      "answer": "2*sqrt(3)/3",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;M: Point;F: Point;N: Point;l1: Line;l2: Line;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;OneOf(Asymptote(G))=l1;OneOf(Asymptote(G))=l2;Negation(l1=l2);L:Line;PointOnCurve(F, L);IsPerpendicular(l1, L);FootPoint(l1,L)=M;Intersection(l2,L)=N;2*VectorOf(M, F) = VectorOf(F, N)",
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+    {
+      "index": 4740,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4740.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;H: Point;Expression(G) = (y^2 = 4*x);Coordinate(H) = (m, 2);PointOnCurve(H,G);m:Number",
+      "coords": {}
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+    {
+      "index": 4741,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4742,
+      "success": true,
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+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4742.png",
+      "answer": "48/5",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);A: Point;B: Point;C: Point;F: Point;Coordinate(F) = (3, 0);PointOnCurve(A, G);PointOnCurve(B, G);PointOnCurve(C, G);VectorOf(F, A) + VectorOf(F, B) + VectorOf(F, C) = 0",
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+          0.0
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+      "index": 4743,
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+        "direction": "up"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4743.png",
+      "answer": "y=-5",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 20*y)",
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+    {
+      "index": 4744,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4745,
+      "success": true,
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+      "error": null,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4745.png",
+      "answer": "3",
+      "fact_expr": "C: Ellipse;a: Number;b: Number;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);E: Parabola;p: Number;p>0;Expression(E) = (x^2 = 2*(p*y));F: Point;Focus(E) = F;UpperVertex(C) = F;P: Point;OneOf(Intersection(C, E)) = P;Eccentricity(C) = sqrt(3)/3;Abs(LineSegmentOf(P, F)) = 2",
+      "coords": {}
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+    {
+      "index": 4746,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 4.0,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4746.png",
+      "answer": "(x-5)^2+y^2=16",
+      "fact_expr": "H: Ellipse;Expression(H) = (x^2/169 + y^2/144 = 1);Z: Circle;Center(Z) = RightFocus(H);G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);IsTangent(Asymptote(G), Z)",
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+    },
+    {
+      "index": 4747,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4748,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4748.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;A: Point;O: Origin;B: Point;l1: Line;l2: Line;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2 + y^2 = a^2);TangentOfPoint(OneOf(Focus(C)), G) = {l1, l2};TangentPoint(l1, G) = A;TangentPoint(l2, G) = B;AngleOf(A, O, B) = ApplyUnit(90, degree)",
+      "coords": {}
+    },
+    {
+      "index": 4749,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.23606797749979,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4749.png",
+      "answer": "2*sqrt(14)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/5 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 4750,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4751,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4752,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4752.png",
+      "answer": "3",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;Focus(C) = {F1, F2};P: Point;PointOnCurve(P, C);IsPerpendicular(VectorOf(P, F1), VectorOf(P, F2));Area(TriangleOf(P, F1, F2)) = 9",
+      "coords": {}
+    },
+    {
+      "index": 4753,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4754,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4754.png",
+      "answer": "(2,2)",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x);F: Point;Focus(G) = F;P: Point;PointOnCurve(P, G) = True;A: Point;Coordinate(A) = (3, 2);WhenMin(Abs(LineSegmentOf(P, A)) + Abs(LineSegmentOf(P, F)))",
+      "coords": {
+        "A": [
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+      "answer": "x^2/8 + y^2/2 = 1",
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+      "answer": "4*x-3*y-4=0",
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+      "answer": "-4/3",
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+      "answer": "4",
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+        "iterations": 2
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+      "answer": "y=pm*(\\sqrt{3})*x",
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+        "direction": "left"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4777.png",
+      "answer": "(-1/4, 1)",
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+        "direction": "up"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4778.png",
+      "answer": "x^2+(y-4)^2=64",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 16*y);H: Circle;Center(H) = Focus(G);IsTangent(Directrix(G), H)",
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+      "index": 4779,
+      "success": true,
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+        "x_coef": 16.0,
+        "y_coef": 12.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4779.png",
+      "answer": "6",
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+    {
+      "index": 4780,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4781,
+      "success": true,
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+      "error": null,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4781.png",
+      "answer": "12*x-8*y-7=0",
+      "fact_expr": "l: Line;C: Parabola;A: Point;F: Point;B: Point;Expression(C) = (y^2 = 3*x);Focus(C) = F;Slope(l)=3/2;Intersection(l, C) = {A, B};Intersection(l,xAxis)=P;Abs(LineSegmentOf(A,F))+Abs(LineSegmentOf(B,F))=4;P:Point",
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+    {
+      "index": 4782,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4783,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4783.png",
+      "answer": "sqrt(5)-sqrt(3)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2/5 = 1);F: Point;LeftFocus(G) = F;H: Circle;Expression(H) = (x^2 + y^2 = 3);P: Point;T: Point;TangentOfPoint(F, H) = LineOf(F,P);Intersection(LineSegmentOf(F, P), RightPart(G)) = P;TangentPoint(LineOf(F, P), H) = T;M: Point;MidPoint(LineSegmentOf(F, P)) = M;O: Origin",
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+    {
+      "index": 4784,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4785,
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+      "answer": "-1/2",
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+      "answer": "sqrt(6)/3",
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+      "index": 4800,
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+      "index": 4801,
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+      "error": "Unsupported or unparseable expression"
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+      "answer": "(1/3)*sqrt(3)",
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+      "answer": "16",
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+      "answer": "6",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4805.png",
+      "answer": "16/3",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, C) = {A, B};Distance(A, Directrix(C)) = 4",
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+        "iterations": 2
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4806.png",
+      "answer": "x^2/16 + y^2/12 = 1",
+      "fact_expr": "G: Ellipse;m: Number;n: Number;F: Point;m>0;n>0;Expression(G) = (y^2/n^2 + x^2/m^2 = 1);Coordinate(F) = (2, 0);RightFocus(G) = F;Eccentricity(G) = 1/2",
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+          0.0
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+        "ellipse_focus_mismatch"
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+      "index": 4807,
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4807.png",
+      "answer": "pm*2*sqrt(2)",
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+      "index": 4808,
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+      "error": null,
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4808.png",
+      "answer": "12",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4809.png",
+      "answer": "Answer missing",
+      "fact_expr": "F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;PointOnCurve(F1,LineOf(A,B)) = True;Intersection(LineOf(A,B),G) = {A,B};A: Point;B: Point;2*Abs(LineSegmentOf(A, F1)) = 3*Abs(LineSegmentOf(F1, B));Abs(LineSegmentOf(B, F2)) = 2*Abs(LineSegmentOf(B, F1))",
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+    {
+      "index": 4810,
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+      "conic_type": "ellipse",
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+        "x_coef": 9.0,
+        "y_coef": 5.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4810.png",
+      "answer": "12",
+      "fact_expr": "E: Ellipse;M: Circle;P: Point;F1: Point;F2: Point;Q: Point;Expression(E) = (x^2/9 + y^2/5 = 1);Expression(M) = (-8*y - 10*x + x^2 + y^2 + 40 = 0);PointOnCurve(Q, M);LeftFocus(E) = F1;RightFocus(E) = F2;PointOnCurve(P, E)",
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+    {
+      "index": 4811,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4811.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
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+      "index": 4812,
+      "success": true,
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+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4812.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(OneOf(Asymptote(G))) = (y = -2*x)",
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+    {
+      "index": 4813,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4813.png",
+      "answer": "x^2/4 - y^2/8 = 1",
+      "fact_expr": "l: Line;Expression(l) = (y = sqrt(2)*(x + 2*sqrt(3)));C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;F: Point;LeftFocus(C) = F;PointOnCurve(F, l) = True;NumIntersection(l, C)=1",
+      "coords": {}
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+    {
+      "index": 4814,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4815,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4816,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "note": "using explicit params"
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+      "answer": "1/2",
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+      "answer": "x = -3/2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 6*x)",
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+        "note": "using explicit params"
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+      "answer": "sqrt(2)-1",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4819.png",
+      "answer": "2*sqrt(2)",
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+        "y_coef": 9.0
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+        "note": "using explicit params"
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+      "answer": "12*sqrt(7)/7",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4821.png",
+      "answer": "1/20",
+      "fact_expr": "G: Parabola;Expression(G) = (y = 10*x^2)",
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+        "x_coef": 5.0,
+        "y_coef": 4.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4822.png",
+      "answer": "4",
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+    {
+      "index": 4823,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 4824,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "vertical"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4824.png",
+      "answer": "5/3",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/16 + y^2/9 = 1)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4825.png",
+      "answer": "x=-2",
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+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4826.png",
+      "answer": "[2,+oo)",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;F: Point;O: Origin;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F;PointOnCurve(P, RightPart(C));Abs(LineSegmentOf(P, F)) - Abs(LineSegmentOf(P, O)) = 2*a",
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+    {
+      "index": 4827,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4828,
+      "success": true,
+      "conic_type": "parabola",
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4828.png",
+      "answer": "10",
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+    {
+      "index": 4829,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4830,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4831.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
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+    {
+      "index": 4832,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+        "direction": "right"
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4833.png",
+      "answer": "7/2",
+      "fact_expr": "A: Point;Coordinate(A) = (3, 2);F: Point;Focus(G) = F;G: Parabola;Expression(G) = (y^2 = 2*x);P: Point;PointOnCurve(P, G) = True",
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+          2.0
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+        "b": 2.449489742783178,
+        "major_axis": "y",
+        "x_coef": 6.0,
+        "y_coef": 10.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4834.png",
+      "answer": "x^2/12 + y^2/16 = 1",
+      "fact_expr": "G: Ellipse;Z: Ellipse;P: Point;Expression(G) = (x^2/6 + y^2/10 = 1);Coordinate(P) = (-3, 2);PointOnCurve(P, Z);Focus(G) = Focus(Z)",
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+          2.0
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+      "success": true,
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+      "error": null,
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+        "p": 1.0,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4835.png",
+      "answer": "-3",
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+    {
+      "index": 4836,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 4.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4836.png",
+      "answer": "sqrt(3)/2\nx^2/9-y^2/3=1",
+      "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H)=(x^2+4*y^2=16);Focus(G)=Focus(H);Expression(OneOf(Asymptote(G)))=(x + sqrt(3)*y = 0)",
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+    },
+    {
+      "index": 4837,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 9.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4837.png",
+      "answer": "sqrt(52)",
+      "fact_expr": "G: Curve;P: Point;x1: Number;y1: Number;Expression(G) = (x^2/16 + y^2/9 = 1);Coordinate(P) = (x1, y1);PointOnCurve(P, G)",
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+    },
+    {
+      "index": 4838,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4838.png",
+      "answer": "[(-4-sqrt(7))/12, (-4+sqrt(7))/12]",
+      "fact_expr": "C: Circle;Expression(C) = (-8*y - 8*x + x^2 + y^2 + 28 = 0);P: Point;PointOnCurve(P, C);N: Curve;Expression(N) = (x^2 + 4*y^2 = 4);A: Point;B: Point;Intersection(N, xAxis) = {A, B};O: Origin;M: Point;Intersection(LineOf(O, P), N) = M;k1: Number;k2: Number;k3: Number;Slope(LineOf(M, A)) = k1;Slope(LineOf(M, B)) = k2;Slope(LineOf(O, P)) = k3",
+      "coords": {}
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+    {
+      "index": 4839,
+      "success": true,
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+      "error": null,
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4839.png",
+      "answer": "3",
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+      "index": 4840,
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+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4840.png",
+      "answer": "(1/2, 1)",
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+      "coords": {}
+    },
+    {
+      "index": 4841,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
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+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4841.png",
+      "answer": "2",
+      "fact_expr": "C: Parabola;p: Number;F: Point;Expression(C) = (y^2 = 2*(p*x));Coordinate(F) = (1, 0);Focus(C) = F",
+      "coords": {
+        "F": [
+          1.0,
+          0.0
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+      "index": 4842,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4842.png",
+      "answer": "4+sqrt(2)",
+      "fact_expr": "G: Parabola;N: Point;M: Point;F: Point;Expression(G) = (y^2 = 6*x);Coordinate(N) = (5/2, 1);Focus(G) = F;PointOnCurve(M, G)",
+      "coords": {
+        "N": [
+          2.5,
+          1.0
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+      }
+    },
+    {
+      "index": 4843,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4843.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;Intersection(Directrix(G), xAxis) = K;K: Point;P: Point;PointOnCurve(P, G) = True",
+      "coords": {}
+    },
+    {
+      "index": 4844,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4844.png",
+      "answer": "2",
+      "fact_expr": "l: Line;C: Parabola;p: Number;B: Point;A: Point;F: Point;M:Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;Inclination(l)=ApplyUnit(30,degree);PointOnCurve(F, l);Intersection(l, C) = {A, B};MidPoint(LineSegmentOf(A,B))=M;XCoordinate(M)=7",
+      "coords": {}
+    },
+    {
+      "index": 4845,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4845.png",
+      "answer": "3",
+      "fact_expr": "G: Hyperbola;F: Point;Expression(G) = (x^2/4 - y^2/9 = 1);RightFocus(G) = F",
+      "coords": {}
+    },
+    {
+      "index": 4846,
+      "success": true,
+      "conic_type": "ellipse",
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+      "answer": "-4/9",
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+      "answer": "y=x+pm*1",
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+      "answer": "(0,2)+(8,+oo)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4862.png",
+      "answer": "4*sqrt(6)",
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+      "index": 4863,
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+      "answer": "2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x)",
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+      "success": true,
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+      "answer": "6*sqrt(2)/7",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4867.png",
+      "answer": "2*sqrt(5)-1",
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+      "answer": "6",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4870.png",
+      "answer": "4*sqrt(3)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/10 - y^2/2 = 1)",
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+    {
+      "index": 4871,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 4872,
+      "success": true,
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+      "answer": "2*sqrt(3)",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 4874,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4874.png",
+      "answer": "(\\sqrt{5}+1)/2",
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+      "index": 4875,
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+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4875.png",
+      "answer": "sqrt(5)",
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+    {
+      "index": 4876,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.0,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4876.png",
+      "answer": "(5/4,-3/4)",
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+      "index": 4877,
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+        "y_coef": 1.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4877.png",
+      "answer": "sqrt(3)/3",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 + y^2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(60, degree)",
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+    {
+      "index": 4878,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4879,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.6666666666666667,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 2.7777777777777777
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4879.png",
+      "answer": "10\n10/3\n(pm*(10/3)*sqrt(2),0)\n{(pm*5,0),(0,pm*5/3)}",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2 + 9*y^2 = 25)",
+      "coords": {}
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+    {
+      "index": 4880,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4880.png",
+      "answer": "y^2/16 + x^2/9 = 1",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);H: Ellipse;Vertex(H) = {Focus(G), Endpoint(MinorAxis(G))}",
+      "coords": {}
+    },
+    {
+      "index": 4881,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4881.png",
+      "answer": "x^2/16 - y^2/9 = 1",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(OneOf(Asymptote(G)))=(y=(3/4)*x);PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G) = F2;IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2));Abs(LineSegmentOf(P, F1))*Abs(LineSegmentOf(P, F2)) = 18",
+      "coords": {}
+    },
+    {
+      "index": 4882,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4882.png",
+      "answer": "8",
+      "fact_expr": "C: Parabola;p: Number;O: Origin;F: Point;M: Point;p>0;Expression(C) = (y^2 = 2*p*x);Focus(C)=F;PointOnCurve(M,C);Area(CircumCircle(TriangleOf(O,F,M)))=36*pi;IsTangent(CircumCircle(TriangleOf(O,F,M)),Directrix(C))",
+      "coords": {}
+    },
+    {
+      "index": 4883,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4883.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/4 + x^2/a^2 = 1);a: Number;a>2;Eccentricity(G) = sqrt(5)/5",
+      "coords": {}
+    },
+    {
+      "index": 4884,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4884.png",
+      "answer": "(-oo,0)",
+      "fact_expr": "G: ConicSection;H: ConicSection;b: Number;a: Number;a>b;b>0;e1:Number;e2:Number;Expression(H) = (y^2/b^2 + x^2/a^2 = 1);Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(H)=e1;Eccentricity(G)=e2;m:Number;m=lg(e1)+lg(e2)",
+      "coords": {}
+    },
+    {
+      "index": 4885,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4886,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4887,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4887.png",
+      "answer": "(3, 4)",
+      "fact_expr": "G: Ellipse;t: Number;Expression(G) = (x^2/4 + y^2/t = 1);PointOnCurve(Focus(G), xAxis);P: Point;PointOnCurve(P, G);Distance(P, OneOf(Focus(G))) > 1",
+      "coords": {}
+    },
+    {
+      "index": 4888,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4888.png",
+      "answer": "5",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;M: Point;F1: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Focus(C) = {F1, F2};PointOnCurve(M, C);Max(Abs(LineSegmentOf(M, F1))*Abs(LineSegmentOf(M, F2))) = 25",
+      "coords": {}
+    },
+    {
+      "index": 4889,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4889.png",
+      "answer": "(0, \\sqrt{3}/2]",
+      "fact_expr": "l: Line;E: Ellipse;b: Number;F: Point;M: Point;0 < b;b < 2;Expression(E) = (x^2/4 + y^2/b^2 = 1);Expression(l)=(3*x-4*y=0);RightFocus(E) = F;OneOf(Endpoint(MinorAxis(E)))=M;Negation(Distance(M, l) < 4/5)",
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+    },
+    {
+      "index": 4890,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4890.png",
+      "answer": "pm*(3/2)",
+      "fact_expr": "H: Line;Expression(H) = (y = k*x);k: Number;G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);A: Point;B: Point;Intersection(H, G) = {A, B};L1: Line;L2: Line;PointOnCurve(A, L1);PointOnCurve(B, L2);IsPerpendicular(L1, xAxis);IsPerpendicular(L2, xAxis);F1: Point;F2: Point;Focus(G) = {F1, F2};FootPoint(L1, xAxis) = F1;FootPoint(L2, xAxis) = F2",
+      "coords": {}
+    },
+    {
+      "index": 4891,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4892,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4893,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4893.png",
+      "answer": "sqrt(13)/2",
+      "fact_expr": "G: Hyperbola;e: Number;Expression(G) = (x^2/4 - y^2/9 = 1);Eccentricity(G) = e",
+      "coords": {}
+    },
+    {
+      "index": 4894,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.8284271247461903,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 8.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4894.png",
+      "answer": "x=4",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/8 + y^2/4 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 4895,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4895.png",
+      "answer": "(-1,-3/2)",
+      "fact_expr": "G: Ellipse;A: Point;F: Point;O: Origin;B: Point;C: Point;Expression(G) = (x^2/4 + y^2/3 = 1);RightFocus(G) = F;PointOnCurve(A,G);Quadrant(A) = 1;Intersection(OverlappingLine(LineSegmentOf(A, F)), G) = B;Intersection(OverlappingLine(LineSegmentOf(A, O)), G) = C;Area(TriangleOf(A, B, C)) = 3",
+      "coords": {}
+    },
+    {
+      "index": 4896,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4896.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;H: Line;k: Number;A: Point;B: Point;Expression(G) = (y^2 = 8*x);Expression(H) = (y = k*x - 2);Intersection(H, G) = {A, B};Negation(A=B);YCoordinate(MidPoint(LineSegmentOf(A, B))) = 2",
+      "coords": {}
+    },
+    {
+      "index": 4897,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4897.png",
+      "answer": "1",
+      "fact_expr": "C: Hyperbola;F: Point;Expression(C) = (x^2 - y^2 = 1);RightFocus(C) = F",
+      "coords": {}
+    },
+    {
+      "index": 4898,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4898.png",
+      "answer": "16/9",
+      "fact_expr": "G: Parabola;A: Point;B: Point;F: Point;M: Point;Q: Point;P: Point;l: Line;Expression(G) = (y^2 = 2*x);Focus(G) = F;Directrix(G) = l;IsChordOf(LineSegmentOf(A,B),G);PointOnCurve(F,LineSegmentOf(A,B));MidPoint(LineSegmentOf(A,B))=M;L1:Line;L2:Line;PointOnCurve(F,L1);PointOnCurve(M,L2);IsPerpendicular(L1,LineSegmentOf(A,B));IsPerpendicular(L2,LineSegmentOf(A,B));Intersection(L1,l)=P;Intersection(L2,l)=Q;Abs(LineSegmentOf(A,F))=3*Abs(LineSegmentOf(B,F))",
+      "coords": {}
+    },
+    {
+      "index": 4899,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4899.png",
+      "answer": "sqrt(7)/2",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;l: Line;OneOf(Asymptote(C)) = l;G: Circle;Expression(G) = (y^2 + (-c + x)^2 = a^2);c: Number;A: Point;B: Point;Intersection(l, G) = {A, B};c^2 = a^2 + b^2;Abs(LineSegmentOf(A, B)) = a",
+      "coords": {}
+    },
+    {
+      "index": 4900,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4900.png",
+      "answer": "3*sqrt(3)",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 6*x);F: Point;Focus(C) = F;G: Line;A: Point;B: Point;PointOnCurve(F, G);Inclination(G) = ApplyUnit(30, degree);Intersection(G, C) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 4901,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4902,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4902.png",
+      "answer": "(1,0)",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x)",
+      "coords": {}
+    },
+    {
+      "index": 4903,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4904,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4904.png",
+      "answer": "3*sqrt(2)/4",
+      "fact_expr": "H: Line;Expression(H) = (y = x*b/(3*a));G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Intersection(H,LeftPart(G)) = P;P: Point;F1: Point;LeftFocus(G) = F1;IsPerpendicular(LineSegmentOf(P,F1),xAxis) = True;e: Number;Eccentricity(G) = e",
+      "coords": {}
+    },
+    {
+      "index": 4905,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4906,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.4142135623730951,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4906.png",
+      "answer": "x^2/2-y^2=1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/2 = 1);Z: Hyperbola;Asymptote(Z) = Asymptote(G);P: Point;Coordinate(P) = (2, 1);PointOnCurve(P, Z)",
+      "coords": {
+        "P": [
+          2.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 4907,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4907.png",
+      "answer": "y = x",
+      "fact_expr": "G: Parabola;H: Line;P: Point;A: Point;B: Point;Expression(G) = (y^2 = 4*x);Coordinate(P) = (2, 2);PointOnCurve(P,H);Intersection(H, G) = {A, B};VectorOf(P, A) + VectorOf(P, B) = 0",
+      "coords": {
+        "P": [
+          2.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 4908,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4909,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4909.png",
+      "answer": "[2, 3]",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);x1: Number;y1: Number;P: Point;Coordinate(P) = (x1, y1);PointOnCurve(P, G);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2",
+      "coords": {}
+    },
+    {
+      "index": 4910,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4910.png",
+      "answer": "(x-5)^2+y^2=16",
+      "fact_expr": "G: Hyperbola;H: Circle;Expression(G) = (x^2/9 - y^2/16 = 1);RightFocus(G) = Center(H);IsTangent(Asymptote(G),H)",
+      "coords": {}
+    },
+    {
+      "index": 4911,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4911.png",
+      "answer": "pm*(sqrt(3)/3)",
+      "fact_expr": "l: Line;G: Parabola;M: Circle;B:Point;A:Point;Expression(G) = (y^2 = 2*x);Intersection(l,G)={A,B};IsDiameter(LineSegmentOf(A,B),M);IsTangent(Directrix(G),M);Area(M)=16*pi",
+      "coords": {}
+    },
+    {
+      "index": 4912,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
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+      "answer": "-2*p",
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+      "answer": "56/5",
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+      "answer": "(0, 5)",
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+      "answer": "x^2/5+y^2/4=1",
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+      "answer": "x^2/12 - y^2/4 = 1",
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+        "note": "using explicit params"
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+      "answer": "sqrt(2)",
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+      "answer": "2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/12 = 1)",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4936.png",
+      "answer": "1/2",
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+      "answer": "pi/6",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4940.png",
+      "answer": "y^2 = 8*x",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4941.png",
+      "answer": "-2",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2/2 + y^2/m = 1);m: Number;e: Number;Eccentricity(C) = e;M: Circle;Expression(M) = (y^2 + (x - 2)^2 = e^2);IsTangent(Asymptote(C), M) = True",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4942.png",
+      "answer": "7",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);P: Point;Q: Point;PointOnCurve(P, G);PointOnCurve(Q, G);Negation(P=Endpoint(MajorAxis(G)));Negation(Q=Endpoint(MajorAxis(G)));O: Origin;K1: Number;K2: Number;Slope(LineOf(O, P)) = K1;Slope(LineOf(O, Q)) = K2;K1*K2 = -3/4",
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+    {
+      "index": 4943,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 4944,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+        "iterations": 2
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+      "answer": "2*sqrt(3)",
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+      "index": 4946,
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+      "index": 4947,
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+      "answer": "sqrt(10)/3",
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+      "answer": "{(y^2 = 8*x)&(x > 0), (y = 0)&(x < 0)}",
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+        "y_coef": 3.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4949.png",
+      "answer": "2*sqrt(3+2*sqrt(3))",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/3 + x^2/a^2 = 1);a: Number;a>sqrt(3);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;O: Origin;G: Line;PointOnCurve(O, G);Inclination(G) = ApplyUnit(60, degree);M: Point;OneOf(Intersection(G, C)) = M;Abs(VectorOf(M, F1) + VectorOf(M, F2)) = Abs(VectorOf(M, F1) - VectorOf(M, F2))",
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+      "index": 4950,
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+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4951,
+      "success": true,
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+        "b": 4.47213595499958,
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+      "answer": "17",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_4952.png",
+      "answer": "x^2/25+y^2/9=1",
+      "fact_expr": "G: Circle;C:Circle;A: Point;Expression(G) = (-8*x + x^2 + y^2 - 84 = 0);Coordinate(A) = (-4, 0);PointOnCurve(A, C);IsTangent(G,C);P:Point;Center(C)=P",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4953.png",
+      "answer": "(p*k^2 + p, -p*k)",
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+      "success": true,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4954.png",
+      "answer": "x \\pm y - 1 = 0",
+      "fact_expr": "l: Line;G: Parabola;a: Number;F: Point;A: Point;B: Point;Expression(G) = (y^2 = a*x);Coordinate(F) = (1, 0);Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B};LineSegmentOf(A, B) = 8",
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+      "index": 4955,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4955.png",
+      "answer": "18",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4956.png",
+      "answer": "8",
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+    {
+      "index": 4957,
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+      "error": null,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4957.png",
+      "answer": "3/2",
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+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
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+    {
+      "index": 4959,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4960,
+      "success": false,
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+      "error": "validation: hyperbola_point_off_curve",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4960.png",
+      "answer": "x^2/4-y^2=1",
+      "fact_expr": "G: Hyperbola;H: Point;Coordinate(H) = (4, sqrt(3));PointOnCurve(H,G) = True;Expression(Asymptote(G)) = (y=pm*(1/2)*x)",
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+      "index": 4961,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4962,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "a": 2.0,
+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4962.png",
+      "answer": "1/4",
+      "fact_expr": "G: Ellipse;a: Number;b: Number;F: Point;A: Point;O:Origin;H: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Center(G) = O;RightFocus(G)=F;RightVertex(G)=A;Intersection(RightDirectrix(G),xAxis)=H",
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+    },
+    {
+      "index": 4963,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4963.png",
+      "answer": "1",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x);P: Point;Coordinate(P) = (2, 2);l: Line;TangentOfPoint(P, G) = l",
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+        "P": [
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+          2.0
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+      "index": 4964,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 4965,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4965.png",
+      "answer": "1",
+      "fact_expr": "G: Hyperbola;b: Number;b>0;Expression(G) = (x^2/4 - y^2/b^2 = 1);Expression(Asymptote(G)) = (y = pm*(x/2))",
+      "coords": {}
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+    {
+      "index": 4966,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4966.png",
+      "answer": "7",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;G: Point;Coordinate(G) = (-2, 0);l: Line;Slope(l) = 2/3;PointOnCurve(G, l);A: Point;B: Point;Intersection(l, C) = {A, B}",
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+        "G": [
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+          0.0
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+    {
+      "index": 4967,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4967.png",
+      "answer": "5",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);P: Point;PointOnCurve(P, G);F: Point;Focus(G) = F;B: Point;Coordinate(B) = (4, 2)",
+      "coords": {
+        "B": [
+          4.0,
+          2.0
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+    {
+      "index": 4968,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4969,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4970,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 2.23606797749979,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4970.png",
+      "answer": "y^2=12*x",
+      "fact_expr": "G: Hyperbola;H: Parabola;Expression(G) = (x^2/4 - y^2/5 = 1);Vertex(H) = Center(G);RightFocus(G)=Focus(H)",
+      "coords": {}
+    },
+    {
+      "index": 4971,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 4972,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_4972.png",
+      "answer": "2*sqrt(3)/3",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y^2 + (x - 2)^2 = 4);Length(InterceptChord(OneOf(Asymptote(C)),G))=2*sqrt(3)",
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+    {
+      "index": 4973,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4973.png",
+      "answer": "2",
+      "fact_expr": "C: Ellipse;A: Point;B: Point;P: Point;b:Real;b>0;b<5;Coordinate(A) = (-4, 0);Coordinate(B) = (-1, 0);Expression(C) = (x^2/25 + y^2/b^2 = 1);PointOnCurve(P,C);Abs(LineSegmentOf(P,A))=2*Abs(LineSegmentOf(P,B))",
+      "coords": {
+        "A": [
+          -4.0,
+          0.0
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+        "B": [
+          -1.0,
+          0.0
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+      }
+    },
+    {
+      "index": 4974,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4974.png",
+      "answer": "1",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = x);F: Point;Focus(C) = F;A: Point;x0: Number;y0: Number;Coordinate(A) = (x0, y0);PointOnCurve(A, C);Abs(LineSegmentOf(A, F)) = (5/4)*x0",
+      "coords": {}
+    },
+    {
+      "index": 4975,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "down"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_4975.png",
+      "answer": "y=2",
+      "fact_expr": "G: Parabola;Expression(G) = (y = -x^2/8)",
+      "coords": {}
+    },
+    {
+      "index": 4976,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "y",
+        "x_coef": 1.0,
+        "y_coef": 4.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_4976.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "P: Point;Coordinate(P) = (sqrt(2)/2, 1);M: Point;N: Point;PointOnCurve(M, G);PointOnCurve(N, G);G: Ellipse;Expression(G) = (x^2 + y^2/2 = 1);k1: Number;k2: Number;k: Number;Slope(LineOf(P, M)) = k1;Slope(LineOf(P, N)) = k2;Slope(LineOf(M, N)) = k;k1 + k2 = 0",
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+      "answer": "y=pm*(sqrt(7)/3)*x",
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+      "answer": "1/24",
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+      "answer": "x^2 - y^2/(1/2) = 1",
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+      "answer": "13",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5001.png",
+      "answer": "2*sqrt(2)",
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+      "answer": "x^2+(y+2)^2=20",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5004.png",
+      "answer": "x+pm*y=0;pm*3",
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+      "answer": "x^2/16 - y^2/48 = 1",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5022.png",
+      "answer": "5/4",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1)",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "answer": "32",
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+          0.0
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5025.png",
+      "answer": "12",
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+      "index": 5026,
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+      "conic_type": "ellipse",
+      "error": null,
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+        "major_axis": "x",
+        "x_coef": 6.0,
+        "y_coef": 2.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5026.png",
+      "answer": "4",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2 + 3*y^2 = 6)",
+      "coords": {}
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+      "index": 5027,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 5028,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5028.png",
+      "answer": "2",
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+      "index": 5029,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 5030,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5030.png",
+      "answer": "2*sqrt(3)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2/4 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 5031,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_focus_mismatch",
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5031.png",
+      "answer": "x^2/8-y^2=1",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;RightFocus(C) = F;F: Point;Coordinate(F) = (3, 0);Distance(F,OneOf(Asymptote(C))) = 1",
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+          0.0
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+        "b": 8.0,
+        "major_axis": "x",
+        "x_coef": 100.0,
+        "y_coef": 64.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5032.png",
+      "answer": "13",
+      "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/100 + y^2/64 = 1);PointOnCurve(P, G);Distance(P, LeftFocus(G)) = 7",
+      "coords": {}
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+    {
+      "index": 5033,
+      "success": true,
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+      "error": null,
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5033.png",
+      "answer": "(\\sqrt{5} - 1)/2",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;H: LineSegment;PointOnCurve(Focus(G),H);a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);IsPerpendicular(MajorAxis(G), H);IsChordOf(H, G) ;FocalLength(G) = Length(H)",
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+    {
+      "index": 5034,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "vertical"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5034.png",
+      "answer": "2*sqrt(3)/3",
+      "fact_expr": "E: Hyperbola;a: Number;b: Number;C: Circle;a>0;b>0;Expression(E) = (-x^2/b^2 + y^2/a^2 = 1);Expression(C) = (-16*y + x^2 + y^2 + 48 = 0);IsTangent(C, Asymptote(E))",
+      "coords": {}
+    },
+    {
+      "index": 5035,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5035.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (sqrt(3)*x - y + sqrt(3) = 0);IsParallel(OneOf(Asymptote(G)),H)",
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+    {
+      "index": 5036,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5036.png",
+      "answer": "sqrt(21)/3",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, Asymptote(G));Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2));AngleOf(F1, P, F2) = ApplyUnit(60, degree)",
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+    {
+      "index": 5037,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5037.png",
+      "answer": "sqrt(5)/3",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;P: Point;F2: Point;F1: Point;O: Origin;N: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) =F1;RightFocus(C)=F2;PointOnCurve(P,C);Quadrant(P)=2;Intersection(LineSegmentOf(P, F2), yAxis) = N;DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0;Abs(LineSegmentOf(F1, F2)) = 4*Abs(LineSegmentOf(O, N))",
+      "coords": {}
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+    {
+      "index": 5038,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5038.png",
+      "answer": "y^2=8*x",
+      "fact_expr": "G: Parabola;p: Number;P: Point;m:Number;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(P) = (2, m);PointOnCurve(P,G);Distance(P,yAxis)=Distance(P,Focus(G))/2",
+      "coords": {}
+    },
+    {
+      "index": 5039,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5039.png",
+      "answer": "2",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;O: Origin;M: Point;F: Point;N: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;PointOnCurve(F,G);L1:Line;L2:Line;Asymptote(C)={L1,L2};Intersection(G,L1)=M;Intersection(G,L2)=N;DotProduct(VectorOf(O, M), VectorOf(M, F)) = 0;Abs(LineSegmentOf(M, N)) = b",
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+      "index": 5040,
+      "success": true,
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+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5040.png",
+      "answer": "2*sqrt(5)/5",
+      "fact_expr": "F: Point;C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;RightFocus(C) = F;G: Line;Expression(G) = (y = x*b/a);Intersection(G, C) = {A, B};A: Point;B: Point;Cos(AngleOf(A, F, B)) = 1/3",
+      "coords": {}
+    },
+    {
+      "index": 5041,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.0,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5041.png",
+      "answer": "(x-\\sqrt{5})^2+y^2=1",
+      "fact_expr": "G: Hyperbola;H:Circle;Expression(G) = (x^2/4 - y^2 = 1);RightFocus(G)=Center(H);IsTangent(Asymptote(G),H)",
+      "coords": {}
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+    {
+      "index": 5042,
+      "success": true,
+      "conic_type": "parabola",
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+        "direction": "up"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5042.png",
+      "answer": "8",
+      "fact_expr": "G: Parabola;H: Line;Expression(G) = (y = x^2/4);Slope(H)=1;PointOnCurve(Focus(G),H)",
+      "coords": {}
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+    {
+      "index": 5043,
+      "success": true,
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5043.png",
+      "answer": "6",
+      "fact_expr": "Gamma: Ellipse;O: Origin;P: Point;F: Point;Expression(Gamma) = (x^2/4 + y^2/3 = 1);LeftFocus(Gamma) = F;PointOnCurve(P, Gamma)",
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+    {
+      "index": 5044,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5044.png",
+      "answer": "2",
+      "fact_expr": "C: Parabola;p: Number;G: Line;A: Point;E: Point;B: Point;D: Point;Q: Point;F: Point;l: Line;p>0;Expression(C) = (y^2 = 2*(p*x));Coordinate(Q) = (-p/2, 0);Focus(C) = F;Directrix(C) = l;PointOnCurve(Q, G);Intersection(G, C) = {A, B};L1:Line;L2:Line;PointOnCurve(A, L1);PointOnCurve(B,L2);Intersection(l,L1)=D;Intersection(l, L2)=E;Slope(LineSegmentOf(A, E))=k1;Slope(LineSegmentOf(B, D)) = k2;k1:Number;k2:Number;IsPerpendicular(l,L1);IsPerpendicular(l,L2)",
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+    {
+      "index": 5045,
+      "success": true,
+      "conic_type": "parabola",
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5045.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;B: Point;P: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(B) = (3, 2);PointOnCurve(P, G);Focus(G) = F",
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+          2.0
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+      "index": 5046,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5046.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;a: Number;b: Number;c: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = (2*sqrt(3))/3;FocalLength(G) = 2*c;2*a^2 = 3*c;PointOnCurve(P, G);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 2;LeftFocus(G) = F1;RightFocus(G) = F2",
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+    {
+      "index": 5047,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5048,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5048.png",
+      "answer": "2*sqrt(5)",
+      "fact_expr": "C: Parabola;G: Circle;A: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(A, C);Abs(LineSegmentOf(A, F)) = 3;Center(G)=A;Radius(G)=LineSegmentOf(A,F)",
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+    {
+      "index": 5049,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5049.png",
+      "answer": "x^2 - y^2 / 3 = 1",
+      "fact_expr": "G: Hyperbola;H: Parabola;O: Origin;A: Point;B: Point;Expression(H) = (y^2 = 8*x);Focus(H)=F;OneOf(Focus(G))=F;Intersection(Directrix(H), G) = {A, B};Area(TriangleOf(O,A,B))=6;F:Point",
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+    {
+      "index": 5050,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "up"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5050.png",
+      "answer": "4/3",
+      "fact_expr": "G: Parabola;P: Point;F: Point;A: Point;O: Origin;l: Line;Expression(G) = (x^2 = 4*y);Focus(G) = F;Directrix(G)=l;PointOnCurve(P,G);PointOnCurve(P,LineSegmentOf(P,A));IsPerpendicular(LineSegmentOf(P,A),l);FootPoint(LineSegmentOf(P,A),l) = A;AngleOf(A, F, O) = ApplyUnit(30, degree)",
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+    {
+      "index": 5051,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5051.png",
+      "answer": "x^2/12 - y^2/4 = 1",
+      "fact_expr": "G: Hyperbola;H: Parabola;Expression(OneOf(Asymptote(G)))=(x + sqrt(3)*y = 0);Expression(H)=(y^2=16*x);OneOf(Focus(G))=Focus(H)",
+      "coords": {}
+    },
+    {
+      "index": 5052,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 3.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5052.png",
+      "answer": "6*sqrt(2)",
+      "fact_expr": "G: Parabola;M: Point;Expression(G) = (y^2 = 12*x);PointOnCurve(M, G);Distance(M, Focus(G)) = 9",
+      "coords": {}
+    },
+    {
+      "index": 5053,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.4142135623730951,
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+        "x_coef": 4.0,
+        "y_coef": 2.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5053.png",
+      "answer": "-1/2",
+      "fact_expr": "C: Ellipse;A: Point;M: Point;B: Point;Expression(C) = (x^2/4 + y^2/2 = 1);Intersection(C, xAxis) = {A, B};PointOnCurve(M, C);Negation(M = A);Negation(M = B);Slope(LineOf(M,A))=k1;Slope(LineOf(M,B))=k2;k1:Number;k2:Number",
+      "coords": {}
+    },
+    {
+      "index": 5054,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5054.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;H: Parabola;A: Point;B: Point;e: Number;l:Line;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (4*x^2=y);Directrix(H)=l;Intersection(l,Asymptote(C))={A, B};Abs(LineSegmentOf(A, B)) = 1/8;Eccentricity(C)=e",
+      "coords": {}
+    },
+    {
+      "index": 5055,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5055.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;H: Line;PointOnCurve(F, H);A: Point;B: Point;Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, F)) = 2",
+      "coords": {}
+    },
+    {
+      "index": 5056,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "left"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5056.png",
+      "answer": "(-1/4,1)",
+      "fact_expr": "G: Parabola;A: Point;P: Point;Expression(G) = (y^2 = -4*x);Coordinate(A) = (-2, 1);PointOnCurve(P, G);WhenMin(Distance(P,F)+Distance(P,A));Focus(G)=F;F:Point",
+      "coords": {
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+          1.0
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+    {
+      "index": 5057,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5057.png",
+      "answer": "1",
+      "fact_expr": "F1: Point;F2: Point;G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1);Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G) = True;AngleOf(F1, P, F2) = ApplyUnit(90, degree)",
+      "coords": {}
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+    {
+      "index": 5058,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5059,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5059.png",
+      "answer": "4/5",
+      "fact_expr": "F2: Point;F1: Point;LeftFocus(G) = F1;RightFocus(G) = F2;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;P: Point;PointOnCurve(P,RightPart(G)) = True;Abs(LineSegmentOf(P, F2)) = Abs(LineSegmentOf(F1, F2));Expression(Asymptote(G)) = (4*x+pm*3*y = 0)",
+      "coords": {}
+    },
+    {
+      "index": 5060,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5060.png",
+      "answer": "6",
+      "fact_expr": "G: Ellipse;H: Line;F2: Point;A: Point;B: Point;F1: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Focus(G) = {F1, F2};PointOnCurve(F1, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(F2, A)) + Abs(LineSegmentOf(F2, B)) = 14",
+      "coords": {}
+    },
+    {
+      "index": 5061,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5061.png",
+      "answer": "2*sqrt(3)",
+      "fact_expr": "H: Circle;Expression(H) = (4*y + x^2 + y^2 = 0);G: Parabola;Expression(G) = (x^2 = 4*y)",
+      "coords": {}
+    },
+    {
+      "index": 5062,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5062.png",
+      "answer": "ApplyUnit(60,degree)",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/16 + y^2/9 = 1);PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G) = F2;Abs(LineSegmentOf(P, F1))*Abs(LineSegmentOf(P, F2)) = 12",
+      "coords": {}
+    },
+    {
+      "index": 5063,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5063.png",
+      "answer": "8",
+      "fact_expr": "l: Line;G: Parabola;B: Point;A: Point;Expression(G) = (y^2 = 4*x);Intersection(l, G) = {A, B};Coordinate(MidPoint(LineSegmentOf(A,B)))=(3,2)",
+      "coords": {}
+    },
+    {
+      "index": 5064,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5064.png",
+      "answer": "sqrt(3)/3",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;B: Point;F: Point;D: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;OneOf(Endpoint(MinorAxis(C)))=B;Intersection(OverlappingLine(LineSegmentOf(B,F)), C) = D;VectorOf(B, F) = 2*VectorOf(F, D)",
+      "coords": {}
+    },
+    {
+      "index": 5065,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 3.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5065.png",
+      "answer": "{(3,6),(3,-6)}",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 12*x);P:Point;Distance(P,Focus(G))=6;PointOnCurve(P,G)",
+      "coords": {}
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+      "answer": "2/5",
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+      "answer": "sqrt(3)-1",
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+      "answer": "x + 2*y-2 = 0",
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+      "answer": "sqrt(34)-1",
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+      "answer": "16/3",
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+      "answer": "25",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5090.png",
+      "answer": "sqrt(3)/3",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;P: Point;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P,F2),xAxis);Slope(LineOf(P,F1)) = sqrt(3)/3",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5091.png",
+      "answer": "(pm*sqrt(5),0)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/4 = 1)",
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+    {
+      "index": 5092,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 5093,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "success": true,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5094.png",
+      "answer": "sqrt(2)/2",
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+      "answer": "2*sqrt(3)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2 + y^2/4 = 1)",
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+    {
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+      "answer": "8",
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+      "answer": "pm*(sqrt(3)/2)",
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+        "note": "using explicit params"
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+      "answer": "y=pm*((sqrt(5)+1)/2)*x",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5099.png",
+      "answer": "y = pm*x*sqrt(5)/2",
+      "fact_expr": "G: Hyperbola;b: Number;H: Point;Expression(G) = (x^2/4 - y^2/b = 1);Coordinate(H) = (3, 0);RightFocus(G) = H",
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_5100.png",
+      "answer": "(x^2/(1/4)-y^2/(15/4)=1)&(x<0)",
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+    {
+      "index": 5101,
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+        "y_coef": 3.0
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+        "note": "using explicit params"
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+      "answer": "sqrt(5)/5",
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+    {
+      "index": 5102,
+      "success": true,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5102.png",
+      "answer": "y^2=12*x",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(C) = F;G: Ellipse;Expression(G) = (x^2/9 + y^2/4 = 1);Distance(F, Directrix(C)) = Length(MajorAxis(G))",
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5103.png",
+      "answer": "2",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5104.png",
+      "answer": "x^2=-24*y",
+      "fact_expr": "G: Parabola;O: Origin;Vertex(G) = O;SymmetryAxis(G) = yAxis;H: Line;Expression(H) = (3*x - 4*y - 24 = 0);PointOnCurve(Focus(G), H)",
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+      "index": 5105,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 5106,
+      "success": true,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5106.png",
+      "answer": "2*sqrt(3)/3",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 5108,
+      "success": true,
+      "conic_type": "parabola",
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5108.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Parabola;p: Number;Q: Point;O: Origin;F: Point;P: Point;p>0;Expression(G) = (y^2 = 2*(p*x));PointOnCurve(P, G);Focus(G)=F;PointOnCurve(Q, LineSegmentOf(P,F));2*Abs(VectorOf(P, Q)) = Abs(VectorOf(Q, F))",
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+    {
+      "index": 5109,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 5110,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "up"
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+      "optimization": {
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5110.png",
+      "answer": "24",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 2*(p*y));p: Number;p>0;H: Point;Coordinate(H) = (m, 3);m: Number;PointOnCurve(H, G);Distance(H, Focus(G)) = 5",
+      "coords": {}
+    },
+    {
+      "index": 5111,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "up"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5111.png",
+      "answer": "3*sqrt(2)/4",
+      "fact_expr": "G: Line;Expression(G) = (y = k*x - 2);k: Number;k>0;C: Parabola;Expression(C) = (x^2 = 8*y);A: Point;B: Point;Intersection(G, C) = {A, B};F: Point;Focus(C) = F;Abs(LineSegmentOf(F, A)) = 2*Abs(LineSegmentOf(F, B))",
+      "coords": {}
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+    {
+      "index": 5112,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
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+      "optimization": {
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_5112.png",
+      "answer": "3/4",
+      "fact_expr": "C1: Circle;C:Hyperbola;a: Number;a>0;Expression(C) = (-y^2 + x^2/a^2 = 1);Expression(C1) = ((x - 1)^2 + (y + 2)^2 = 4);IsTangent(OneOf(Asymptote(C)),C1);Quadrant(TangentPoint(OneOf(Asymptote(C)),C1))=3",
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+    {
+      "index": 5113,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5113.png",
+      "answer": "y^2/2 - x^2/4 = 1",
+      "fact_expr": "G: Hyperbola;C: Hyperbola;A: Point;Expression(G) = (x^2/2 - y^2 = 1);Coordinate(A) = (-2, 2);PointOnCurve(A, C);Asymptote(G) = Asymptote(C)",
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+          2.0
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+    {
+      "index": 5114,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "major_axis": "x",
+        "x_coef": 5.0,
+        "y_coef": 4.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5114.png",
+      "answer": "2",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/5 + y^2/4 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 5115,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5115.png",
+      "answer": "8/3",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;A: Point;B: Point;PointOnCurve(A, G);PointOnCurve(B, G);VectorOf(A, F) = 3*VectorOf(F, B);IsChordOf(LineSegmentOf(A, B), G)",
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+    {
+      "index": 5116,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 5117,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 5118,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5118.png",
+      "answer": "pm*p/2",
+      "fact_expr": "G: Parabola;p: Number;A: Point;B: Point;M: Point;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(M) = (xm, ym);xm:Number;ym:Number;Focus(G) = F;IsChordOf(LineSegmentOf(A, B), G);PointOnCurve(F, LineSegmentOf(A, B));MidPoint(LineSegmentOf(A, B)) = M;xm = (3/4)*p",
+      "coords": {}
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+    {
+      "index": 5119,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5119.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;Asymptote(G) = {l1, l2};l1: Line;l2: Line;PointOnCurve(F2, V) = True;IsPerpendicular(V, l1) = True;V: Line;Intersection(V, l1) = P;Intersection(V, l2) = Q;P: Point;Q: Point;O: Origin;VectorOf(O, F2) + VectorOf(O, Q) = 2*VectorOf(O, P)",
+      "coords": {}
+    },
+    {
+      "index": 5120,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5120.png",
+      "answer": "{1/3, 3}",
+      "fact_expr": "C: Ellipse;G: Line;A: Point;B: Point;F1: Point;F2: Point;Expression(C) = (x^2/2 + y^2 = 1);LeftFocus(C) =F1;RightFocus(C)=F2;PointOnCurve(F2, G);Intersection(G, C) = {A, B};Area(TriangleOf(A, B, F1)) = 4/3",
+      "coords": {}
+    },
+    {
+      "index": 5121,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5121.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "G: Hyperbola;F1: Point;F2: Point;P: Point;Expression(G) = (x^2/2 - y^2 = 1);PointOnCurve(P, G);Focus(G) = {F1, F2};Abs(LineSegmentOf(P, F1)) + Abs(LineSegmentOf(P, F2)) = 4*sqrt(2)",
+      "coords": {}
+    },
+    {
+      "index": 5122,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5122.png",
+      "answer": "sqrt(17)/3",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;A: Point;B: Point;F: Point;C: Point;O:Origin;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(A, G);PointOnCurve(B, G);PointOnCurve(C, G);PointOnCurve(O,LineSegmentOf(A,B));RightFocus(G)=F;PointOnCurve(F,LineSegmentOf(A,C));IsPerpendicular(LineSegmentOf(B, F), LineSegmentOf(A, C));3*Abs(LineSegmentOf(A, F)) = Abs(LineSegmentOf(A, C))",
+      "coords": {}
+    },
+    {
+      "index": 5123,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5124,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5124.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "C: Hyperbola;a: Number;a>0;Expression(C) = (-y^2/5 + x^2/a^2 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 5125,
+      "success": true,
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+        "b": 1.7320508075688772,
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+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5125.png",
+      "answer": "sqrt(2)/2",
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+      "coords": {}
+    },
+    {
+      "index": 5126,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.4142135623730951,
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5126.png",
+      "answer": "2*sqrt(3)",
+      "fact_expr": "C: Hyperbola;F1: Point;P: Point;F2: Point;Expression(C) = (x^2 - y^2 = 2);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);AngleOf(F1, P, F2) = ApplyUnit(60, degree)",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 5128,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 5129,
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5129.png",
+      "answer": "8",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);F: Point;Focus(G) = F;C: Point;Intersection(Directrix(G), xAxis) = C;H: Line;PointOnCurve(F, H);A: Point;B: Point;Intersection(H, G) = {A, B};AngleOf(C, B, F) = ApplyUnit(90, degree)",
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+    },
+    {
+      "index": 5130,
+      "success": true,
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+      "error": null,
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+        "b": 1.7320508075688772,
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+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5130.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/a^2+y^2/b^2=1);a: Number;b: Number;a>b;b>0;Eccentricity(C) = sqrt(3)/2;F: Point;RightFocus(C) = F;G: Line;PointOnCurve(F, G);k: Number;k>0;Slope(G) = k;A: Point;B: Point;Intersection(G, C) = {A, B};VectorOf(A, F) = 3*VectorOf(F, B)",
+      "coords": {}
+    },
+    {
+      "index": 5131,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5131.png",
+      "answer": "5/4",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);F: Point;Focus(G) = F;A: Point;Quadrant(A) = 1;PointOnCurve(A, G);Abs(LineSegmentOf(A, F)) = 2;P: Point;PointOnCurve(P, Directrix(G));WhenMin(Abs(LineSegmentOf(P, A))+ Abs(LineSegmentOf(P, F)))",
+      "coords": {}
+    },
+    {
+      "index": 5132,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5133,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5134,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5134.png",
+      "answer": "4",
+      "fact_expr": "C: Parabola;A: Point;F: Point;B: Point;l: Line;Expression(C) = (y^2 = 4*x);Focus(C) = F;Directrix(C) = l;In(A, l);Intersection(LineSegmentOf(A,F), C) = B;VectorOf(F, A) = 3*VectorOf(F, B)",
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+    },
+    {
+      "index": 5135,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5136,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5137,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5137.png",
+      "answer": "(1,\\sqrt{5}/2]",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;P:Point;PointOnCurve(P, RightPart(C)) = True;L1: Line;L2: Line;PointOnCurve(P, L1) = True;PointOnCurve(P, L2) = True;J1: Line;J2: Line;Asymptote(C) = {J1, J2};IsParallel(L1, J1) = True;IsParallel(L2, J2) = True;Intersection(L1, J2) = M;Intersection(L2, J1) = N;M: Point;N: Point;O: Origin;Area(TriangleOf(O, M, N)) = S;S: Number;S >= b^2/2",
+      "coords": {}
+    },
+    {
+      "index": 5138,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5139,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 5140,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5141,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5141.png",
+      "answer": "4",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);A: Point;LeftVertex(G) = A;I: Point;Coordinate(I) = (-6/5, 0);M: Point;N: Point;PointOnCurve(I, LineOf(M, N));Intersection(LineOf(M, N), G) = {M, N};Negation(A=M);Negation(A=N);k1: Number;Slope(LineOf(A, M)) = k1;k2: Number;Slope(LineOf(A, N)) = k2",
+      "coords": {
+        "I": [
+          -1.2,
+          0.0
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+    {
+      "index": 5142,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 5143,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5143.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "l: Line;C: Parabola;G: Circle;B: Point;A: Point;F: Point;Expression(C) = (y^2 = 2*x);Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};Abs(LineSegmentOf(A, F)) = Abs(LineSegmentOf(B, F));IsDiameter(LineSegmentOf(A,B),G)",
+      "coords": {}
+    },
+    {
+      "index": 5144,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 5145,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5145.png",
+      "answer": "{y^2=12*x, y^2=-4*x}",
+      "fact_expr": "G: Parabola;O: Origin;Vertex(G) = O;PointOnCurve(Focus(G), xAxis);H: Line;Expression(H) = (2*x - y + 1 = 0);Length(InterceptChord(H, G)) = sqrt(15)",
+      "coords": {}
+    },
+    {
+      "index": 5146,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 4.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5146.png",
+      "answer": "8*x+9*y-25=0",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/9 + y^2/4 = 1);P: Point;Coordinate(P) = (2, 1);PointOnCurve(P, l) = True;l: Line;Intersection(l, C) = {A, B};A: Point;B: Point;MidPoint(LineSegmentOf(A, B)) = P",
+      "coords": {
+        "P": [
+          2.0,
+          1.0
+        ]
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+    },
+    {
+      "index": 5147,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5147.png",
+      "answer": "5",
+      "fact_expr": "G: Parabola;M: Point;N: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(M, G);IsPerpendicular(LineSegmentOf(M, N), xAxis);Intersection(LineSegmentOf(M, N), xAxis) = N;Abs(LineSegmentOf(M, F)) = 6",
+      "coords": {}
+    },
+    {
+      "index": 5148,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5148.png",
+      "answer": "-2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1);A1: Point;LeftVertex(G) = A1;F2: Point;RightFocus(G) = F2;P: Point;PointOnCurve(P, RightPart(G))",
+      "coords": {}
+    },
+    {
+      "index": 5149,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
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+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5149.png",
+      "answer": "{x^2 = 2*y, x^2 = 4*y}",
+      "fact_expr": "G: Parabola;p: Number;A: Point;B: Point;M: Point;p>0;Expression(G) = (x^2 = 2*(p*y));Coordinate(M) = (2, -2*p);l1: Line;l2: Line;TangentOfPoint(M, G) = {l1, l2};TangentPoint(l1, G) = A;TangentPoint(l2, G) = B;YCoordinate(MidPoint(LineSegmentOf(A, B))) = 6",
+      "coords": {}
+    },
+    {
+      "index": 5150,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5150.png",
+      "answer": "y^2 = -6x",
+      "fact_expr": "G: Parabola;C:Curve;O:Origin;Expression(C)=(2*x-4*y+3=0);SymmetryAxis(G)=xAxis;Vertex(G)=O;PointOnCurve(Focus(G),C)",
+      "coords": {}
+    },
+    {
+      "index": 5151,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5151.png",
+      "answer": "2*sqrt(3)/3",
+      "fact_expr": "H: Circle;Expression(H) = ((x - sqrt(3))^2 + (y - 1)^2 = 9);G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Intersection(H,Asymptote(G)) = {A,B};Quadrant(A) = 2;Quadrant(B) = 4;A: Point;B: Point;Abs(LineSegmentOf(A, B)) = 2*sqrt(6)",
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+    },
+    {
+      "index": 5152,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5152.png",
+      "answer": "25",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G)",
+      "coords": {}
+    },
+    {
+      "index": 5153,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5153.png",
+      "answer": "8*sqrt(2)/7",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/2 + y^2 = 1);F1: Point;LeftFocus(G) = F1;l: Line;PointOnCurve(F1, l) = True;Inclination(l) = ApplyUnit(60, degree);Intersection(l, G) = {A, B};A: Point;B: Point",
+      "coords": {}
+    },
+    {
+      "index": 5154,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 4.47213595499958,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 20.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5154.png",
+      "answer": "4",
+      "fact_expr": "F: Point;C: Ellipse;Expression(C) = (x^2/20 + y^2/4 = 1);RightFocus(C) = F;P: Point;PointOnCurve(P, C) = True;A: Point;Coordinate(A) = (-2, 1);WhenMin(Perimeter(TriangleOf(A, P, F))) = True",
+      "coords": {
+        "A": [
+          -2.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 5155,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5156,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5157,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5157.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;H: Point;x0:Number;y0:Number;Expression(G) = (y^2 = 8*x);Coordinate(H) = (x0, y0);PointOnCurve(H, G);Distance(H, Focus(G)) = x0^2",
+      "coords": {}
+    },
+    {
+      "index": 5158,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5158.png",
+      "answer": "1",
+      "fact_expr": "F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);M: Point;PointOnCurve(M, G) = True",
+      "coords": {}
+    },
+    {
+      "index": 5159,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5160,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5161,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5161.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "C: Parabola;Expression(C) = (x^2 = 4*y);F: Point;Focus(C) = F;P: Point;G: Line;Expression(G) = (x - y - 2 = 0);PointOnCurve(P, G);TangentOfPoint(P, C) = {LineOf(P, A), LineOf(P, B)};TangentPoint(LineOf(P, A), C) = A;TangentPoint(LineOf(P, B), C) = B;B: Point;A: Point;O: Origin",
+      "coords": {}
+    },
+    {
+      "index": 5162,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5163,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5164,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
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+      "answer": "x^2/11 - y^2/(99/16) = 1",
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+      "answer": "(\\sqrt{5}-1)/2",
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+      "answer": "2*sqrt(3)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5187.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/4 + y^2/2 = 1)",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5188.png",
+      "answer": "4*sqrt(10)-8",
+      "fact_expr": "G: Circle;H:Curve;M: Point;P: Point;Q: Point;Expression(G) = (x^2 + (y - 2)^2 = 1);Expression(H) = (x^2 = 8*y);Coordinate(M) = (0, 5);PointOnCurve(M,H);PointOnCurve(Q, G);PointOnCurve(P,H)",
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+      "index": 5189,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5189.png",
+      "answer": "x^2/25 + y^2/16 = 1",
+      "fact_expr": "G: Hyperbola;C: Ellipse;H: Point;Expression(G) = (x^2/4 - y^2/5 = 1);Coordinate(H) = (5, 0);Center(G) = Center(C);Focus(G) = Focus(C);PointOnCurve(H,C)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_5190.png",
+      "answer": "(-\\infty,-1/3)+(1/3,+\\infty)",
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+    {
+      "index": 5191,
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+        "direction": "left"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5191.png",
+      "answer": "(-2, 0)  \nx = 2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = -8*x)",
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+    {
+      "index": 5192,
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+      "answer": "[2,2*sqrt(2)]",
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+      "answer": "sqrt(15)/5",
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+      "answer": "{0,1/2}",
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+      "answer": "(1/2,1)",
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+      "answer": "y=pm*sqrt(3)*x/3\n4",
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+      "answer": "sqrt(10)/3",
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+      "answer": "(-sqrt(6)/2,0)",
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+      "answer": "y=pm*sqrt(3)*x",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "answer": "x=-1",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5213.png",
+      "answer": "x^2/25 + y^2/16 = 1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/5 - y^2/4 = 1);Focus(G) = Focus(H);H: Ellipse;Eccentricity(H) = 3/5",
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+      "answer": "5/3",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5215.png",
+      "answer": "3/5",
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+      "error": "Unsupported or unparseable expression"
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+      "index": 5217,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "success": true,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5218.png",
+      "answer": "sqrt(41)/5",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(OneOf(Asymptote(G))) = (y = (4/5)*x)",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5219.png",
+      "answer": "2*sqrt(5) + 1",
+      "fact_expr": "G: Ellipse;H: Circle;M: Point;N: Point;Expression(G) = (x^2/18 + y^2/9 = 1);Expression(H) = (x^2 + (y - 1)^2 = 1);PointOnCurve(M, G);PointOnCurve(N, H)",
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+    {
+      "index": 5220,
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+        "y_coef": 1.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5220.png",
+      "answer": "5/4",
+      "fact_expr": "G: Ellipse;a: Number;a>0;Expression(G) = (y^2 + x^2/a^2 = 1);Distance(LeftFocus(G),RightVertex(G))=2",
+      "coords": {}
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+    {
+      "index": 5221,
+      "success": true,
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+        "orientation": "vertical"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5221.png",
+      "answer": "2",
+      "fact_expr": "G: Circle;Expression(G) = (y^2 + (x - 4)^2 = 4);C: Hyperbola;Expression(C) = (-x^2/b^2 + y^2/a^2 = 1);b: Number;a: Number;a>0;b>0;IsTangent(G, Asymptote(C))",
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+    {
+      "index": 5222,
+      "success": true,
+      "conic_type": "ellipse",
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+        "y_coef": 5.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5222.png",
+      "answer": "10*x+9*y-29=0",
+      "fact_expr": "l: Line;C: Ellipse;A: Point;B: Point;M: Point;Expression(C) = (x^2/9 + y^2/5 = 1);Coordinate(M) = (2, 1);PointOnCurve(M, l);Intersection(l, C) = {A, B};MidPoint(LineSegmentOf(A, B)) = M",
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+    {
+      "index": 5223,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5223.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/b^2 = 1);b: Number;b>0;Eccentricity(G) = sqrt(3)",
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+    {
+      "index": 5224,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 5225,
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+        "direction": "right"
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5225.png",
+      "answer": "2*sqrt(3)-1",
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+    {
+      "index": 5226,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 5227,
+      "success": true,
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+      "error": null,
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5227.png",
+      "answer": "(3, 2*sqrt(6))",
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+    {
+      "index": 5228,
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+      "error": null,
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+        "b": 3.0,
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+        "x_coef": 16.0,
+        "y_coef": 9.0
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5228.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
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+          -5.0
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+    {
+      "index": 5229,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 5230,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5230.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "l: Line;G: Hyperbola;a: Number;C: Circle;a>0;Expression(G) = (-y^2/2 + x^2/a^2 = 1);Center(C)=RightFocus(G);IsTangent(C, Asymptote(G));Expression(l)=(x-sqrt(3)*y=0);Length(InterceptChord(l,C))=2",
+      "coords": {}
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+    {
+      "index": 5231,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 5232,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "p": 1.0,
+        "direction": "up"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5232.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);L: Line;Slope(L) = -1/2;A: Point;B: Point;Intersection(L, G) = {A, B};H: Circle;IsDiameter(LineSegmentOf(A, B), H);P: Point;TangentPoint(H, Directrix(G)) = P",
+      "coords": {}
+    },
+    {
+      "index": 5233,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 2.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5233.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;F: Point;Expression(G) = (x^2 - y^2/4 = 1);RightFocus(G) = F",
+      "coords": {}
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+    {
+      "index": 5234,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 5235,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5235.png",
+      "answer": "18",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - 4*y^2 = 4);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;L: Line;PointOnCurve(F1,L) = True;A: Point;B: Point;Intersection(L, LeftPart(G)) = {A, B};Abs(LineSegmentOf(A, B)) = 5",
+      "coords": {}
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+    {
+      "index": 5236,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 2.0,
+        "major_axis": "y",
+        "x_coef": 4.0,
+        "y_coef": 16.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5236.png",
+      "answer": "3",
+      "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/4 + y^2/16 = 1);PointOnCurve(P, G);F1:Point;F2:Point;OneOf(Focus(G))=F1;OneOf(Focus(G))=F2;Negation(F1=F2);Distance(P, F1) = 5",
+      "coords": {}
+    },
+    {
+      "index": 5237,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5237.png",
+      "answer": "sqrt(3)/3",
+      "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 + y^2 = 1);PointOnCurve(P, G);Focus(G) = {F1, F2};AngleOf(F1, P, F2) = ApplyUnit(60, degree)",
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+    {
+      "index": 5238,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5239,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5240,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5240.png",
+      "answer": "sqrt(6)/3",
+      "fact_expr": "H: Line;Expression(H) = (x - y - 4 = 0);G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Intersection(H, G) = {A, B};A: Point;B: Point;XCoordinate(MidPoint(LineSegmentOf(A, B))) = 3",
+      "coords": {}
+    },
+    {
+      "index": 5241,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5242,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "left"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5242.png",
+      "answer": "sqrt(17)/2",
+      "fact_expr": "P: Point;G: Parabola;Expression(G) = (y^2 = -2*x);PointOnCurve(P, G) = True;M: Point;Coordinate(M) = (0, 2)",
+      "coords": {
+        "M": [
+          0.0,
+          2.0
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+    },
+    {
+      "index": 5243,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5243.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;p: Number;H: Ellipse;Expression(G) = (y^2 = 2*p*x);Expression(H) = (x^2/5 - y^2/p^2 = 1);PointOnCurve(LeftFocus(H),Directrix(G))",
+      "coords": {}
+    },
+    {
+      "index": 5244,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
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+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5244.png",
+      "answer": "1",
+      "fact_expr": "G: Parabola;p: Number;A: Point;B: Point;C: Point;D: Point;F: Point;p>0;Expression(G) = (x^2 = 2*(p*y));Focus(G) = F;PointOnCurve(A,G);PointOnCurve(B,G);AngleOf(A, F, B) = ApplyUnit(60, degree);IsChordOf(LineSegmentOf(A,B),G);MidPoint(LineSegmentOf(A,B))=C;PointOnCurve(C,LineOf(C,D));IsPerpendicular(Directrix(G),LineOf(C,D));FootPoint(Directrix(G),LineOf(C,D))=D",
+      "coords": {}
+    },
+    {
+      "index": 5245,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.4970548748970032,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.00010081290383823216,
+        "converged": true,
+        "iterations": 80
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5245.png",
+      "answer": "3/2",
+      "fact_expr": "G: Parabola;p: Number;A: Point;p>0;Expression(G) = (x^2 = 2*(p*y));Coordinate(A) = (sqrt(2), 1);PointOnCurve(A, G);F: Point;Focus(G) = F",
+      "coords": {
+        "A": [
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+          1.0
+        ]
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+    },
+    {
+      "index": 5246,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5247,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5247.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "G: Parabola;p: Number;H: Line;P: Point;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (y = x + 3);Expression(Directrix(G)) = (x = -2);PointOnCurve(P, G)",
+      "coords": {}
+    },
+    {
+      "index": 5248,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5248.png",
+      "answer": "6;3",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/a^2+y^2/b^2 = 1);a: Number;b: Number;a > b;b > 0;Eccentricity(G) = sqrt(3)/2;P: Point;PointOnCurve(P, G);F1: Point;F2: Point;Focus(G) = {F1, F2};Distance(P, F1) + Distance(P, F2) = 12",
+      "coords": {}
+    },
+    {
+      "index": 5249,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5249.png",
+      "answer": "x^2/4 - y^2/12 = 1",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;l: Line;Expression(l) = (x + sqrt(3)*y = 0);IsPerpendicular(OneOf(Asymptote(C)), l);Distance(OneOf(Focus(C)), l) = 2",
+      "coords": {}
+    },
+    {
+      "index": 5250,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5250.png",
+      "answer": "8*sqrt(3)/5",
+      "fact_expr": "A: Point;F: Point;LowerVertex(C) = A;LeftFocus(C) = F;C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;l: Line;PointOnCurve(A,l) = True;Inclination(l) = ApplyUnit(60,degree);Intersection(l,xAxis) = M;Intersection(l,C) = N;M: Point;N: Point;YCoordinate(N) = 3*b/5;Perimeter(TriangleOf(F, M, N)) = 6",
+      "coords": {}
+    },
+    {
+      "index": 5251,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.449489742783178,
+        "major_axis": "y",
+        "x_coef": 6.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5251.png",
+      "answer": "2*sqrt(6)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/6 + y^2/9 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 5252,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5252.png",
+      "answer": "8*sqrt(2)/5",
+      "fact_expr": "H: Line;Expression(H) = (y = x - 1);G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 5253,
+      "success": false,
+      "error": "Circle from diameter: cannot parse diameter pattern"
+    },
+    {
+      "index": 5254,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5255,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.4142135623730951,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5255.png",
+      "answer": "Answer missing",
+      "fact_expr": "H: Ellipse;Expression(H) = (x^2/6 + y^2/2 = 1);G: Hyperbola;Expression(G) = (x^2/2 - y^2/2 = 1);F1: Point;F2: Point;Focus(H) = {F1, F2};P: Point;OneOf(Intersection(H, G)) = P;Focus(G) = {F1, F2}",
+      "coords": {}
+    },
+    {
+      "index": 5256,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5257,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0010501146316528,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 3.455598402979376e-07,
+        "converged": true,
+        "iterations": 81
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5257.png",
+      "answer": "2\n(1,0)",
+      "fact_expr": "G: Parabola;p: Number;H:Point;F:Point;p>0;Focus(G)=F;Expression(G) = (y^2 = 2*(p*x));Coordinate(H) = (1, 2);PointOnCurve(H, G)",
+      "coords": {
+        "H": [
+          1.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 5258,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
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+        "major_axis": "y",
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+      "answer": "(-17/8, 15/8)",
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+      "coords": {}
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+      "answer": "x^{-y^2/3}=1",
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+      "answer": "2",
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+      "answer": "sqrt(5)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5276.png",
+      "answer": "sqrt(7)/4",
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+      "answer": "2",
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+      "answer": "x^2 - y^2/4 = 1",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+        "note": "using explicit params"
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+      "answer": "32",
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+      "answer": "y=pm*3*x",
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+      "success": false,
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+        "y_coef": 2.0
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+        "note": "using explicit params"
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+      "answer": "12",
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+        "y_coef": 1.0
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+        "note": "using explicit params"
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+      "answer": "2*sqrt(3)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1)",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5287.png",
+      "answer": "3",
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+      "coords": {}
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+        "direction": "right"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5288.png",
+      "answer": "4/7",
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+        "b": 1.0,
+        "major_axis": "y",
+        "x_coef": 1.0,
+        "y_coef": 4.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5289.png",
+      "answer": "y=-3",
+      "fact_expr": "G: Parabola;Expression(G) = (y = a*x^2);a: Number;a>0;H: Ellipse;Expression(H) = (x^2 + y^2/10 = 1);Focus(G)=OneOf(Focus(H))",
+      "coords": {}
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+      "index": 5290,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 5291,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 5292,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "answer": "(2,4)",
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+      "answer": "2",
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+      "answer": "2*sqrt(2)",
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+      "answer": "(\\sqrt{13} - 2)/3",
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+      "index": 5310,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "answer": "4*sqrt(5)/5",
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+      "success": true,
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+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "answer": "4",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 5314,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "y_coef": 1.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "answer": "8",
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+      "success": true,
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+      "error": null,
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+        "b": 8.0,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "answer": "15",
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5316.png",
+      "answer": "y^2=x",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;Distance(Focus(G),Directrix(G))=1/2",
+      "coords": {}
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+      "index": 5317,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5317.png",
+      "answer": "5/4",
+      "fact_expr": "G: Hyperbola;a: Number;b: Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(Asymptote(G)) = (y = pm*((3/4)*x))",
+      "coords": {}
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+      "index": 5318,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5318.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;A: Point;F: Point;P: Point;l: Line;Expression(G) = (y^2 = 4*x);Focus(G) = F;Directrix(G)=l;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P,A),l);FootPoint(LineSegmentOf(P,A),l)=A;Inclination(LineSegmentOf(A, F)) = (2*pi)/3",
+      "coords": {}
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+    {
+      "index": 5319,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
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+          -0.0
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+        "radius": 1.0
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_5319.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
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+      "index": 5320,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 4.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5320.png",
+      "answer": "x^2/9 - y^2/3 = 1",
+      "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2 + 4*y^2 = 16);Focus(G)=Focus(H);Expression(OneOf(Asymptote(G))) = (x + sqrt(3)*y = 0)",
+      "coords": {}
+    },
+    {
+      "index": 5321,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.7320508075688772,
+        "b": 1.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5321.png",
+      "answer": "1/3",
+      "fact_expr": "G: Hyperbola;H: Ellipse;P:Point;F1:Point;F2:Point;Expression(G) = (x^2/3 - y^2 = 1);Expression(H) = (x^2/6 + y^2/2 = 1);Focus(G)={F1,F2};Focus(H)={F1,F2};OneOf(Intersection(G,H))=P",
+      "coords": {}
+    },
+    {
+      "index": 5322,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+      "answer": "2",
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+      "answer": "pm*sqrt(2)",
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+      "answer": "sqrt(15)/2",
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+      "answer": "7/8",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5328.png",
+      "answer": "sqrt(2)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5329.png",
+      "answer": "1/2",
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+      "index": 5330,
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+      "answer": "9",
+      "fact_expr": "G: Hyperbola;m: Number;m>0;Expression(G) = (x^2/16 - y^2/m = 1);Eccentricity(G) = 5/4",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5331.png",
+      "answer": "y^2 = -36*(x - 4)",
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+    {
+      "index": 5332,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 5333,
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+        "note": "using explicit params"
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+      "answer": "(18,pm*12)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5334.png",
+      "answer": "(0, 1/12)",
+      "fact_expr": "G: Parabola;Expression(G) = (y = 3*x^2)",
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+    {
+      "index": 5335,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5335.png",
+      "answer": "11/3",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2/4 - y^2/3 = 1);F1: Point;F2: Point;Focus(C) = {F1, F2};P: Point;PointOnCurve(P, C);PointOnCurve(MidPoint(LineSegmentOf(P, F1)), yAxis)",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5336.png",
+      "answer": "(1, 3]",
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+    {
+      "index": 5337,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5337.png",
+      "answer": "(\\sqrt{5}, +\\infty)",
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+    {
+      "index": 5338,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.0,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5338.png",
+      "answer": "2*sqrt(5)/5",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2/4 - y^2 = 1)",
+      "coords": {}
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+    {
+      "index": 5339,
+      "success": true,
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+        "direction": "right"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5339.png",
+      "answer": "10",
+      "fact_expr": "G: Parabola;p: Real;O:Origin;p>0;Expression(G) = (y^2 = 2*(p*x));Distance(Focus(G),O)=5",
+      "coords": {}
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+      "index": 5340,
+      "success": true,
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+      "error": null,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5340.png",
+      "answer": "(x-2)^2+y^2=2",
+      "fact_expr": "G: Hyperbola;E: Circle;Expression(G) = (x^2 - y^2 = 2);Center(E) = RightFocus(G);IsTangent(Asymptote(G), E) = True",
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+    {
+      "index": 5341,
+      "success": true,
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+      "error": null,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5341.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "l: Line;C: Ellipse;b: Number;a: Number;A: Point;F1: Point;B: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(Focus(C),xAxis);LeftFocus(C)=F1;RightFocus(C)=F2;PointOnCurve(F1, l);Intersection(l,C)={A,B};VectorOf(A, F1) = 3*VectorOf(F1, B);3*Abs(VectorOf(B, F2)) = 5*Abs(VectorOf(A, F2))",
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+    {
+      "index": 5342,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 5343,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5344,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5344.png",
+      "answer": "20",
+      "fact_expr": "C: Hyperbola;m: Number;G: Parabola;A: Point;B: Point;m>0;Expression(C) = (2*x^2 - y^2 = m);Expression(G) = (y^2 = 16*x);Intersection(C, Directrix(G)) = {A,B};Abs(LineSegmentOf(A, B)) = 4*sqrt(3)",
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+    {
+      "index": 5345,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 5346,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "up"
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5346.png",
+      "answer": "2*sqrt(5)/5",
+      "fact_expr": "G: Parabola;P: Point;F: Point;A: Point;Expression(G) = (x^2 = 4*y);Coordinate(A) = (3/2, -1);Focus(G) = F;PointOnCurve(P, G)",
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+    {
+      "index": 5347,
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+      "error": null,
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+        "x_coef": 2.0,
+        "y_coef": 1.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5347.png",
+      "answer": "x^2 + y^2 = 2",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/2 + y^2 = 1);M: Point;PointOnCurve(M, C);l: Line;PointOnCurve(M, l);IsPerpendicular(l,xAxis);N: Point;FootPoint(l,xAxis) = N;P: Point;VectorOf(N, P) = sqrt(2)*VectorOf(N, M)",
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+    {
+      "index": 5348,
+      "success": false,
+      "error": "Circle from diameter: missing point coordinates"
+    },
+    {
+      "index": 5349,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 4.5,
+        "direction": "up"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5349.png",
+      "answer": "pi/6",
+      "fact_expr": "P: Point;Coordinate(P) = (3*sqrt(3), 4);l: Line;PointOnCurve(P, l);C: Parabola;Expression(C) = (x^2 = 18*y);A: Point;B: Point;Intersection(l, C) = {A, B};IsChordOf(LineSegmentOf(A, B), C);MidPoint(LineSegmentOf(A, B)) = P",
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+          4.0
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+    {
+      "index": 5350,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5350.png",
+      "answer": "4",
+      "fact_expr": "G: Ellipse;O: Circle;P: Point;F2: Point;Q: Point;M: Point;Expression(G) = (x^2/4 + y^2/3 = 1);RightFocus(G) = F2;Expression(O) = (x^2 + y^2 = 3);PointOnCurve(M, O) = True;Quadrant(M) = 1;Intersection(TangentOfPoint(M,O), G) = {P, Q}",
+      "coords": {}
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+    {
+      "index": 5351,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5351.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2/a^2 - y^2/b^2 = 1);a: Number;b: Number;a>b;b>0;F: Point;RightFocus(C) = F;A1: Point;A2: Point;LeftVertex(C) = A1;RightVertex(C) = A2;L1: Line;L2: Line;OneOf(Asymptote(C)) = L1;OneOf(Asymptote(C)) = L2;Negation(L1 = L2);H: Line;PointOnCurve(F,H);IsParallel(L1,H);P: Point;Intersection(H,L2) = P;G: Circle;IsDiameter(LineSegmentOf(A1,A2),G);PointOnCurve(P,G)",
+      "coords": {}
+    },
+    {
+      "index": 5352,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5352.png",
+      "answer": "(3,0)\ny^2=12*x",
+      "fact_expr": "C: Parabola;G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);RightFocus(G)=F;Vertex(C)=O;O:Origin;Focus(C)=F;F:Point",
+      "coords": {}
+    },
+    {
+      "index": 5353,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
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+      "answer": "6",
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+      "index": 5354,
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+      "index": 5355,
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+      "answer": "3/4",
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+      "answer": "4*sqrt(10)/5",
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+      "answer": "sqrt(3)",
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+      "index": 5358,
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+        "y_coef": 3.0
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+        "note": "using explicit params"
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+      "answer": "sqrt(3) - 1",
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+      "index": 5359,
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+      "answer": "3",
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+    {
+      "index": 5360,
+      "success": true,
+      "conic_type": "parabola",
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+        "direction": "up"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5360.png",
+      "answer": "sqrt(17)",
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+        "B": [
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+        "direction": "right"
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+      "answer": "3",
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+      "answer": "12",
+      "fact_expr": "G: Hyperbola;a: Number;F1: Point;F2: Point;P: Point;a>0;Expression(G) = (x^2 - y^2/a^2 = 1);Coordinate(F1) = (-2, 0);Coordinate(F2) = (2, 0);Focus(G) = {F1,F2};PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(60, degree)",
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+          0.0
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+      "index": 5363,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5363.png",
+      "answer": "2*sqrt(3)/3",
+      "fact_expr": "E: Hyperbola;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;A: Point;Coordinate(A) = (2, 0);RightVertex(E) = A;Distance(A,OneOf(Asymptote(E))) = 1",
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+          0.0
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+    {
+      "index": 5364,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5364.png",
+      "answer": "6",
+      "fact_expr": "G: Parabola;H: Line;F: Point;A: Point;B: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A,B};Abs(LineSegmentOf(F, A))*Abs(LineSegmentOf(F, B)) = 6",
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+    {
+      "index": 5365,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5365.png",
+      "answer": "8",
+      "fact_expr": "A: Point;B: Point;l: Line;G: Parabola;F: Point;Expression(G) = (y^2 = 8*x);Focus(G) = F;PointOnCurve(F, l);Intersection(l,G)={A,B};Distance(MidPoint(LineSegmentOf(A,B)), yAxis) = 2",
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+    {
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+      "success": true,
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+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5366.png",
+      "answer": "5",
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+      "coords": {}
+    },
+    {
+      "index": 5367,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5367.png",
+      "answer": "sqrt(10)/2",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;LeftFocus(C) = F1;RightFocus(C) = F2;F1: Point;F2: Point;PointOnCurve(F2,L) = True;L: Line;Intersection(L,RightPart(C)) = {P,Q};P: Point;Q: Point;IsPerpendicular(LineSegmentOf(P, Q), LineSegmentOf(P, F1)) = True;3*Abs(LineSegmentOf(P, Q)) = 4*Abs(LineSegmentOf(P, F1))",
+      "coords": {}
+    },
+    {
+      "index": 5368,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 4.0,
+        "major_axis": "x",
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+        "y_coef": 16.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5368.png",
+      "answer": "10",
+      "fact_expr": "G: Ellipse;P: Point;F: Point;Expression(G) = (x^2/25 + y^2/16 = 1);PointOnCurve(P, G);LeftFocus(G) = F;Distance(P, F) = 6",
+      "coords": {}
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+    {
+      "index": 5369,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 5370,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5370.png",
+      "answer": "2*sqrt(3)",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;O: Origin;F: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F;PointOnCurve(P, G);Distance(P,O)=HalfFocalLength(G);Area(TriangleOf(P, O, F)) = 6",
+      "coords": {}
+    },
+    {
+      "index": 5371,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 3.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5371.png",
+      "answer": "8",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 12*y);F: Point;Focus(G) = F;P: Point;Coordinate(P) = (2, 1);l: Line;PointOnCurve(P, l)  ;Intersection(l, G) = {A, B};A: Point;B: Point;MidPoint(LineSegmentOf(A, B)) = P",
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+          1.0
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+    {
+      "index": 5372,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5373,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_point_off_curve",
+      "params": {
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+        "b": 1.4970015287399292,
+        "orientation": "horizontal"
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+        "converged": true,
+        "iterations": 2
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5373.png",
+      "answer": "x^2/(8/5)-y^2/4=1",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;P: Point;Coordinate(P) = (sqrt(2), 1);PointOnCurve(P, C);Distance(OneOf(Focus(C)), OneOf(Asymptote(C))) = 2",
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+          1.0
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+      "validation_reasons": [
+        "hyperbola_point_off_curve"
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+    {
+      "index": 5374,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_focus_mismatch",
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5374.png",
+      "answer": "x^2/8+y^2/4=1",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;B: Point;F: Point;M: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (2, 0);Coordinate(M) = (-sqrt(2)*a/2, 0);UpperVertex(C) = B;RightFocus(C) = F;IsPerpendicular(LineSegmentOf(B, F), LineSegmentOf(B, M))",
+      "coords": {
+        "F": [
+          2.0,
+          0.0
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+      "validation_reasons": [
+        "ellipse_focus_mismatch"
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+    {
+      "index": 5375,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5375.png",
+      "answer": "24/7",
+      "fact_expr": "C: Ellipse;G: Line;Expression(C) = (x^2/4 + y^2/3 = 1);PointOnCurve(LeftFocus(C), G);Slope(G)=1",
+      "coords": {}
+    },
+    {
+      "index": 5376,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5376.png",
+      "answer": "2+4*sqrt(2)",
+      "fact_expr": "C: Hyperbola;M: Point;P: Point;F: Point;Expression(C) = (x^2 - y^2/3 = 1);Coordinate(M) = (0, 2);RightFocus(C)=F;PointOnCurve(P,LeftPart(C))",
+      "coords": {
+        "M": [
+          0.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 5377,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5377.png",
+      "answer": "2*pi/3",
+      "fact_expr": "P: Point;Coordinate(P) = (x1, y1);x1: Number;y1: Number;Q: Point;Coordinate(Q) = (x2, y2);x2: Number;y2: Number;G: Parabola;Expression(G) = (y^2 = 4*x);PointOnCurve(P, G) = True;PointOnCurve(Q, G) = True;x1 + x2 + 2 = (2/3)*sqrt(3)*Abs(LineSegmentOf(P, Q));F: Point;Focus(G) = F",
+      "coords": {}
+    },
+    {
+      "index": 5378,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5378.png",
+      "answer": "(1/2,-1)",
+      "fact_expr": "G: Parabola;H: Line;Expression(G) = (y^2 = 2*x);Expression(H) = (x - y + 3 = 0);P:Point;PointOnCurve(P,G);Q:Point;PointOnCurve(Q,H);WhenMin(Distance(P,Q))",
+      "coords": {}
+    },
+    {
+      "index": 5379,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.23606797749979,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5379.png",
+      "answer": "ApplyUnit(60,degree)",
+      "fact_expr": "G: Hyperbola;F1: Point;P: Point;F2: Point;Expression(G) = (3*x^2 - 5*y^2 = 15);PointOnCurve(P, G);Focus(G)={F1,F2};Area(TriangleOf(F1, P, F2)) = 3*sqrt(3)",
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+      "answer": "4/3",
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+      "answer": "4/3",
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+      "answer": "[\\sqrt{2}/2,1)",
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+      "answer": "y = -0.5*x+4",
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+      "answer": "4",
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+      "index": 5400,
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+      "answer": "[0, 2]",
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+      "answer": "2",
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+      "answer": "x^2/18 - y^2/9 = 1",
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+      "answer": "sqrt(5)/2",
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+      "index": 5405,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5405.png",
+      "answer": "y^2/4 - x^2/16 = 1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1);Asymptote(G) = Asymptote(G1);H: Point;Coordinate(H) = (2, sqrt(5));PointOnCurve(H, G1) = True;G1: Hyperbola",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5406.png",
+      "answer": "6*sqrt(2)",
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+    {
+      "index": 5407,
+      "success": true,
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+        "x_coef": 9.0,
+        "y_coef": 4.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5407.png",
+      "answer": "(0,pm*2)",
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5408.png",
+      "answer": "(1, 0)",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);D: Point;PointOnCurve(D, G) = True;l: Line;Expression(l) = (x = -1);Distance(D, l) = Distance(D, F);F: Point",
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+    {
+      "index": 5409,
+      "success": true,
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5409.png",
+      "answer": "3",
+      "fact_expr": "G: Hyperbola;m: Number;H: Parabola;Expression(G) = (x^2 - y^2/m = 1);Expression(H) = (y^2 = 8*x);OneOf(Focus(G)) = Focus(H)",
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+      "index": 5410,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5411,
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+      "error": null,
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+        "note": "using explicit params"
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+      "answer": "sqrt(14)/4",
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+        "b": 3.0,
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+        "x_coef": 16.0,
+        "y_coef": 9.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5412.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
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+      "index": 5414,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 5415,
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+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5415.png",
+      "answer": "4",
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+      "index": 5416,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 5417,
+      "success": true,
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+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5417.png",
+      "answer": "(1,\\sqrt{2})",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(RightFocus(G), H);Inclination(H) = ApplyUnit(45, degree);NumIntersection(H, RightPart(G)) = 2",
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+    {
+      "index": 5418,
+      "success": true,
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+      "error": null,
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+        "direction": "up"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5418.png",
+      "answer": "2+2*sqrt(2)",
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+    {
+      "index": 5419,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 5420,
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5420.png",
+      "answer": "(1, 3/2)",
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+          1.0
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "conic_type": "hyperbola",
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+        "b": 1.0,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5422.png",
+      "answer": "1/2",
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+    {
+      "index": 5423,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5423.png",
+      "answer": "3*sqrt(5)/20",
+      "fact_expr": "G: Parabola;H: Line;P: Point;Expression(G) = (y^2 = 2*x);Expression(H) = (y = 2*x + 1);PointOnCurve(P,G)",
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+      "index": 5424,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 5425,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.4142135623730951,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5425.png",
+      "answer": "3/4",
+      "fact_expr": "C: Hyperbola;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/2 - y^2/2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))",
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+    {
+      "index": 5426,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5426.png",
+      "answer": "4",
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+    {
+      "index": 5427,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5427.png",
+      "answer": "3",
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+    {
+      "index": 5428,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "up"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5428.png",
+      "answer": "1/2",
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+    {
+      "index": 5429,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5429.png",
+      "answer": "3/2",
+      "fact_expr": "G: Parabola;p: Number;H: Line;A: Point;B: Point;F: Point;p>0;Expression(G) = (y^2 = 2*p*x);Focus(G) = F;PointOnCurve(F, H);IsPerpendicular(H,xAxis);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = 3",
+      "coords": {}
+    },
+    {
+      "index": 5430,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5430.png",
+      "answer": "4",
+      "fact_expr": "F1: Point;F2: Point;G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G) = True;O: Origin;M: Point;MidPoint(LineSegmentOf(F1, P)) = M;Abs(LineSegmentOf(O, M)) = 3",
+      "coords": {}
+    },
+    {
+      "index": 5431,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "y",
+        "x_coef": 1.0,
+        "y_coef": 4.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5431.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2 + y^2/b^2 = 1);b: Number;0 < b;b < 1;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;H: Line;Slope(H) = 1;PointOnCurve(F1, H);A: Point;B: Point;Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = 4/3",
+      "coords": {}
+    },
+    {
+      "index": 5432,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5432.png",
+      "answer": "y=pm*(sqrt(10)/4)*x",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;p: Number;F: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);p>0;Expression(H) = (y^2 = 2*p*x);Focus(H)=F;RightFocus(G)=F;P:Point;Coordinate(P)=(3*a^2/p,2*b^2/p);PointOnCurve(P,G)",
+      "coords": {}
+    },
+    {
+      "index": 5433,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5433.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(F1, F2),LineSegmentOf(P, F2));AngleOf(P,F1,F2)=ApplyUnit(30,degree)",
+      "coords": {}
+    },
+    {
+      "index": 5434,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5434.png",
+      "answer": "-1",
+      "fact_expr": "P: Point;Coordinate(P) = (-3, 3);M: Point;Coordinate(M) = (3, 0);L: Line;PointOnCurve(M,L) = True;G: Parabola;Expression(G) = (y^2 = 4*x);Intersection(L,G) = {A,B};A: Point;B: Point;Slope(LineOf(P,A)) = k1;k1: Number;Slope(LineOf(P,B)) = k2;k2: Number",
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+          3.0
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+        "M": [
+          3.0,
+          0.0
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+      "index": 5435,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 2.070901870727539,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 1.4661259228887502e-06,
+        "converged": true,
+        "iterations": 106
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5435.png",
+      "answer": "x \\pm \\sqrt{2} \\cdot y = 0",
+      "fact_expr": "a: Number;b: Number;a > b;b > 0;C1: Ellipse;Expression(C1) = (y^2/b^2 + x^2/a^2 = 1);C2: Hyperbola;Expression(C2) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(C1)*Eccentricity(C2) = sqrt(3)/2",
+      "coords": {}
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+    {
+      "index": 5436,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5436.png",
+      "answer": "8*sqrt(6)/9",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;Coordinate(F) = (1, 0);l: Line;Expression(l) = (y = m + x);A: Point;B: Point;Intersection(l, G) = {A, B};m: Number;m >= 0;m < 1;Negation(A=B)",
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+        "F": [
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+          0.0
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+      "index": 5437,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5437.png",
+      "answer": "3/5",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 5438,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 5439,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5439.png",
+      "answer": "13/6",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;a > b;b > 0;e:Number;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);a <= sqrt(3);Eccentricity(G)=e",
+      "coords": {}
+    },
+    {
+      "index": 5440,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5440.png",
+      "answer": "sqrt(2)+1",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;F: Point;B: Point;A: Point;a>0;b<0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightVertex(C) = A;RightFocus(C) = F;Intersection(LeftDirectrix(C), xAxis) = B;MidPoint(LineSegmentOf(B,F)) = A",
+      "coords": {}
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+    {
+      "index": 5441,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5441.png",
+      "answer": "{2, 1/2}",
+      "fact_expr": "P: Point;a: Number;b: Number;G: Hyperbola;Coordinate(P) = (a, b);Expression(G) = (x^2 - y^2 = 1);PointOnCurve(P, RightPart(G));Distance(P, Asymptote(G)) = sqrt(2)",
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+      "answer": "pm*1",
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+      "answer": "sqrt(10)/2",
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+      "answer": "(x+pm*1)^2+(y-(1/2))^2=1",
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+      "answer": "x^2+(y-1)^2=1/5",
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+      "answer": "{20/9, 36/5}",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5452.png",
+      "answer": "1/2",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5453.png",
+      "answer": "1",
+      "fact_expr": "F1: Point;F2: Point;Focus(G) = {F1, F2};G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1);P: Point;PointOnCurve(P, G);AngleOf(F1,P,F2) = ApplyUnit(90,degree)",
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+      "index": 5454,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5454.png",
+      "answer": "2",
+      "fact_expr": "G: Ellipse;O: Circle;A: Point;B: Point;P: Point;Expression(G) = (x^2/4 + y^2/3 = 1);Expression(O) = (x^2 + y^2 = 1);IsDiameter(LineSegmentOf(A, B), O);PointOnCurve(P, G)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5455.png",
+      "answer": "8",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5456.png",
+      "answer": "4*sqrt(7)/7",
+      "fact_expr": "C: Hyperbola;G:Circle;b: Number;a: Number;A: Point;O: Origin;B: Point;F: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;Center(G)=F;Radius(G)=HalfFocalLength(C);Intersection(G,Asymptote(C))={O,A,B};Perimeter(TriangleOf(A, O, B)) = 7*a",
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+      "index": 5457,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 5458,
+      "success": true,
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+        "direction": "up"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5458.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;A: Point;B: Point;F: Point;Expression(G) = (x^2 = 4*y);Focus(G) = F;PointOnCurve(F, LineSegmentOf(A, B));IsChordOf(LineSegmentOf(A, B), G);Length(LineSegmentOf(A, B)) = 6;MidPoint(LineSegmentOf(A, B)) = M;M: Point",
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+    {
+      "index": 5459,
+      "success": true,
+      "conic_type": "parabola",
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+        "direction": "up"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5459.png",
+      "answer": "9/4",
+      "fact_expr": "C: Parabola;B: Point;A: Point;O: Origin;F: Point;Expression(C) = (x^2 = 4*y);Focus(C) = F;Intersection(LineOf(A, B), C) = {A, B};2*VectorOf(O, A) + VectorOf(O, B) - 3*VectorOf(O, F) = 0;IsChordOf(LineSegmentOf(A, B), C)",
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+    {
+      "index": 5460,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5460.png",
+      "answer": "5",
+      "fact_expr": "G:Hyperbola;Expression(G) = (-y^2/9 + x^2/a^2 = 1);a: Number;P: Point;PointOnCurve(P, RightPart(G));Expression(OneOf(Asymptote(G))) = (3*x - y = 0);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;Abs(LineSegmentOf(P, F2)) = 3",
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+    {
+      "index": 5461,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "x_coef": 2.0,
+        "y_coef": 8.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5461.png",
+      "answer": "[-1, 3]",
+      "fact_expr": "C: Ellipse;A: Point;B: Point;a: Real;Expression(C) = (x^2/2 + y^2/8 = 1);Coordinate(A) = (a, 2*a);Coordinate(B) = (a - 2, 2*a - 4);IsIntersect(LineSegmentOf(A,B), C)",
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+    {
+      "index": 5462,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 5463,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 5464,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5464.png",
+      "answer": "8",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;A: Point;B: Point;C: Point;Negation(A=B);Negation(A=C);Negation(B=C);PointOnCurve(A, G);PointOnCurve(B, G);PointOnCurve(C, G);F: Point;Focus(G) = F;VectorOf(F, A) + VectorOf(F, B) + VectorOf(F, C) = 0;Expression(OverlappingLine(LineSegmentOf(B, C))) = (4*x + y - 20 = 0)",
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+    {
+      "index": 5465,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 5466,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5466.png",
+      "answer": "10 + sqrt(2)",
+      "fact_expr": "G: Ellipse;A: Point;B: Point;P: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Coordinate(A) = (3, 1);Coordinate(B) = (-4, 0);PointOnCurve(P, G)",
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+          1.0
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+        "B": [
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+          0.0
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+    {
+      "index": 5467,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5467.png",
+      "answer": "1/8",
+      "fact_expr": "G: Parabola;Expression(G) = (y = 4*x^2)",
+      "coords": {}
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+    {
+      "index": 5468,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5468.png",
+      "answer": "16",
+      "fact_expr": "G: Parabola;M: Point;O: Origin;N: Point;Expression(G) = (y^2 = 8*x);PointOnCurve(M,G);PointOnCurve(N,G);AngleOf(M, O, N) = ApplyUnit(90, degree)",
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+    {
+      "index": 5469,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5470,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.4142135623730951,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5470.png",
+      "answer": "y^2/18 - x^2/9 = 1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/2 = 1);Z: Hyperbola;H: Point;Coordinate(H) = (3, -6);PointOnCurve(H, Z);Asymptote(G) = Asymptote(Z)",
+      "coords": {
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+    },
+    {
+      "index": 5471,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 5472,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5473,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 5474,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5474.png",
+      "answer": "(1, 0)",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;H: Circle;Expression(H) = (-2*y + x^2 + y^2 - 1 = 0);Length(InterceptChord(Directrix(G), H)) = 2",
+      "coords": {}
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+    {
+      "index": 5475,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5476,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 4.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5476.png",
+      "answer": "(0, pm*2)",
+      "fact_expr": "P: Point;F1: Point;F2: Point;Focus(G) = {F1, F2};G: Ellipse;Expression(G) = (x^2/9 + y^2/4 = 1);PointOnCurve(P, G);Vertex(TriangleOf(P, F1, F2)) = {P, F1, F2};Area(TriangleOf(P, F1, F2)) = 2*sqrt(5)",
+      "coords": {}
+    },
+    {
+      "index": 5477,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5477.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;a: Number;Expression(G) = (-y^2/9 + x^2/a^2 = 1);a>0;Expression(OneOf(Asymptote(G))) = (3*x - 2*y = 0)",
+      "coords": {}
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+    {
+      "index": 5478,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+      "answer": "y = -1/8",
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+      "answer": "(2/3)*sqrt(21)",
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+      "answer": "x = 4",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1)",
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+      "answer": "(\\sqrt{5}-1)/2",
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+      "answer": "x^2/15 - y^2/5 = 1",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5494.png",
+      "answer": "(\\sqrt{5}, +\\infty)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5495.png",
+      "answer": "x^2 - y^2 / 3 = 1",
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+      "success": false,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5496.png",
+      "answer": "4*y^2/31 - 25*x^2/31 = 1",
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+      "answer": "2*sqrt(3)/3",
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+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
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+      "answer": "2",
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+      "answer": "3*x-8*y+19=0",
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+      "success": false,
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+      "index": 5502,
+      "success": true,
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+        "note": "using explicit params"
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+      "answer": "4*sqrt(2)",
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+      "index": 5503,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 5504,
+      "success": true,
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+        "y_coef": 3.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5504.png",
+      "answer": "sqrt(3)/2",
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+    {
+      "index": 5505,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 5506,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 5507,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5507.png",
+      "answer": "1/2",
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+    {
+      "index": 5508,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5508.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;H: Parabola;p: Real;Expression(G) = (x^2 - y^2/3 = 1);p>0;Expression(H) = (y^2 = 2*(p*x));PointOnCurve(LeftFocus(G), Directrix(H))",
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+    {
+      "index": 5509,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5509.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "l: Line;C: Ellipse;b: Number;a: Number;A: Point;F: Point;B: Point;e: Number;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Expression(l)=(y=x+1);Intersection(l,C)={A,B};Intersection(l,xAxis)=F;LeftFocus(C)=F;VectorOf(A, F) = 3*VectorOf(F, B);Eccentricity(C) = e",
+      "coords": {}
+    },
+    {
+      "index": 5510,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 5511,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "answer": "27/4",
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+      "answer": "(-2,0)  \nx=9/2",
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+      "answer": "sqrt(5)/3",
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+      "answer": "8/3",
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+      "answer": "2*p",
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+      "answer": "x^2/16+y^2/12=1",
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+      "answer": "sqrt(5)",
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+      "answer": "8*sqrt(2)/3",
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+      "answer": "20",
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+      "answer": "y=pm*sqrt(3)*x",
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+      "answer": "[\\sqrt{3}/3,1)",
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+      "answer": "Answer is incorrect\nApplyUnit(120, degree)",
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+      "answer": "x^2 - y^2 = 1",
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+      "answer": "16*sqrt(2)",
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+      "answer": "4",
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+      "answer": "2",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5538.png",
+      "answer": "(\\sqrt{3}+1)/2",
+      "fact_expr": "F: Point;RightFocus(C) = F;C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;M: Point;PointOnCurve(M,RightPart(C)) = True;O: Origin;Abs(LineSegmentOf(F, M)) = 2*Abs(LineSegmentOf(O, F));AngleOf(O, F, M) = ApplyUnit(120, degree)",
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+        "iterations": 56
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5539.png",
+      "answer": "5/4",
+      "fact_expr": "G: Parabola;a: Number;A: Point;Expression(G) = (y^2 = a*x);Coordinate(A) = (1/4, 1);PointOnCurve(A, G)",
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+      "answer": "x^2/2 - y^2/6 = 1",
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+      "answer": "9/2",
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+      "answer": "sqrt(3)/2",
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+      "answer": "sqrt(14)/4",
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+      "answer": "y=pm*(sqrt(2)*x)",
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+      "answer": "sqrt(5)",
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+      "answer": "sqrt(2)",
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+      "answer": "x = 1",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = -4*x)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5552.png",
+      "answer": "sqrt(3)*x+pm*2*y=0",
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+      "answer": "14",
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+      "answer": "3",
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+      "index": 5556,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "answer": "-2",
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+        "note": "using explicit params"
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+      "answer": "15/4",
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+        "direction": "up"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5559.png",
+      "answer": "(3*sqrt(3))/2",
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+        "y_coef": 9.0
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+      "answer": "20",
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+        "note": "using explicit params"
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+      "answer": "pm*1",
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+      "answer": "1/2",
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+      "answer": "sqrt(3)+1",
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+      "answer": "2",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5568.png",
+      "answer": "sqrt(3)/2",
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+      "index": 5569,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_5569.png",
+      "answer": "{x^2/25+y^2/21=1,x^2/16+y^2/12=1}",
+      "fact_expr": "M: Circle;C1:Circle;C2:Circle;M1:Point;Expression(C1)=((x+2)^2+y^2=1);Expression(C2)=((x-2)^2+y^2=81);IsTangent(M,C1);IsTangent(M,C2);Center(M)=M1",
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+      "index": 5570,
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+        "direction": "down"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5570.png",
+      "answer": "8",
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+      "coords": {}
+    },
+    {
+      "index": 5571,
+      "success": true,
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+      "error": null,
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5571.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
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+          1.0
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+    {
+      "index": 5572,
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+        "b": 1.7320508075688772,
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+        "x_coef": 12.0,
+        "y_coef": 3.0
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5572.png",
+      "answer": "1/7",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/12 + y^2/3 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G) = True;PointOnCurve(MidPoint(LineSegmentOf(P, F1)), yAxis) = True",
+      "coords": {}
+    },
+    {
+      "index": 5573,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5573.png",
+      "answer": "1/2",
+      "fact_expr": "G: Parabola;H: Line;O: Origin;A: Point;B: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (y = 2*x - 3);Intersection(H, G) = {A, B};Slope(LineOf(O, A)) = k1;Slope(LineOf(O, B)) = k2;k1:Number;k2:Number",
+      "coords": {}
+    },
+    {
+      "index": 5574,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5574.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;P: Point;F: Point;Expression(G) = (x^2 = 4*y);PointOnCurve(P, G);Focus(G) = F;Abs(LineSegmentOf(P, F)) = 5",
+      "coords": {}
+    },
+    {
+      "index": 5575,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5575.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;M: Point;F2: Point;F1: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, H);Inclination(H) = ApplyUnit(30, degree);Intersection(H, RightPart(G)) = M;IsPerpendicular(LineSegmentOf(M, F2), xAxis)",
+      "coords": {}
+    },
+    {
+      "index": 5576,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5576.png",
+      "answer": "(x-1)^2+y^2=36",
+      "fact_expr": "G: Parabola;H: Circle;P: Point;Expression(G) = (y^2 = 4*x);Coordinate(P) = (5, -2*sqrt(5));Focus(G)=Center(H);PointOnCurve(P,H)",
+      "coords": {
+        "P": [
+          5.0,
+          -4.47213595499958
+        ]
+      }
+    },
+    {
+      "index": 5577,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5577.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;P: Point;F2: Point;F1: Point;O:Origin;B: Point;e:Number;b > a;a > 0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2 + y^2 = b^2);LeftFocus(C) = F1;RightFocus(C) = F2;Eccentricity(C)=e;PointOnCurve(P, RightPart(C));IsPerpendicular(LineSegmentOf(P, F2), LineSegmentOf(F1, F2));PointOnCurve(B, G);VectorOf(O, B) = (2/sqrt(21))*VectorOf(O, P)",
+      "coords": {}
+    },
+    {
+      "index": 5578,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5578.png",
+      "answer": "34",
+      "fact_expr": "G: Hyperbola;a: Number;P: Point;F1: Point;F2: Point;a>0;Expression(G) = (-y^2/9 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);Abs(LineSegmentOf(P,F1))=2*Abs(LineSegmentOf(P,F2));Abs(LineSegmentOf(P,F1))=16",
+      "coords": {}
+    },
+    {
+      "index": 5579,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5579.png",
+      "answer": "3*sqrt(2)/4",
+      "fact_expr": "C2: Hyperbola;C1: Ellipse;F1: Point;A: Point;F2: Point;Expression(C1) = (x^2/4 + y^2 = 1);Focus(C1) = {F1, F2};Focus(C2) = {F1, F2};Intersection(C1, C2) = A;Quadrant(A) = 2;AngleOf(F1, A, F2) = pi/3",
+      "coords": {}
+    },
+    {
+      "index": 5580,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.366584062576294,
+        "b": 2.364173412322998,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.00017838670464698225,
+        "converged": true,
+        "iterations": 89
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5580.png",
+      "answer": "y = pm*sqrt(3)*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;Eccentricity(G) = 2",
+      "coords": {}
+    },
+    {
+      "index": 5581,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5582,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5582.png",
+      "answer": "9*sqrt(5)/2",
+      "fact_expr": "G: Parabola;H: Line;I: Point;A: Point;B: Point;C: Point;Expression(G) = (y^2 = 6*x);Coordinate(I) = (3, 0);PointOnCurve(I, H);Intersection(H, G) = {A, B};Intersection(H,yAxis)=C;VectorOf(A, B) = 3*VectorOf(B, C)",
+      "coords": {
+        "I": [
+          3.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 5583,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5583.png",
+      "answer": "8",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l: Line;Inclination(l) = ApplyUnit(45, degree);PointOnCurve(F, l);A: Point;B: Point;Intersection(l, C) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 5584,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5584.png",
+      "answer": "sqrt(5)*2",
+      "fact_expr": "C: Parabola;A: Point;K: Point;F: Point;O: Origin;Expression(C) = (y^2 = 8*x);Focus(C) = F;Intersection(Directrix(C), xAxis) = K;PointOnCurve(A, C);Abs(LineSegmentOf(A, K)) = sqrt(2)*Abs(LineSegmentOf(A, F))",
+      "coords": {}
+    },
+    {
+      "index": 5585,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5585.png",
+      "answer": "(0, \\sqrt{2}/2]",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;p: Point;F1: Point;F2: Point;a>b;b>0;x1:Number;y1:Number;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(p) = (x1, y1);PointOnCurve(p, G);Focus(G) = {F1, F2};AngleOf(F1,p,F2)<=ApplyUnit(90,degree)",
+      "coords": {}
+    },
+    {
+      "index": 5586,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5586.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "H: Line;Expression(H) = (x + 2*y - 3 = 0);G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;A: Point;B: Point;Intersection(H, G) = {A, B};Z: Line;Expression(Z) = (3*x - 4*y + 1 = 0);PointOnCurve(MidPoint(LineSegmentOf(A, B)), Z)",
+      "coords": {}
+    },
+    {
+      "index": 5587,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5588,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5589,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_point_off_curve",
+      "params": {
+        "a": 1.4149149626832767,
+        "b": 2.450642358201144,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 1.0319236343469396e-05,
+        "converged": true,
+        "iterations": 94
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5589.png",
+      "answer": "x^2 - y^2 / 3 = 1",
+      "fact_expr": "O: Origin;Center(G) = O;G: Hyperbola;Expression(Asymptote(G)) = (y = pm*(sqrt(3)*x));H: Point;Coordinate(H) = (sqrt(2), sqrt(3));PointOnCurve(H, G)",
+      "coords": {
+        "H": [
+          1.4142135623730951,
+          1.7320508075688772
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+      },
+      "validation_reasons": [
+        "hyperbola_point_off_curve"
+      ]
+    },
+    {
+      "index": 5590,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5591,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 0.929953396320343,
+        "b": 2.629857301712036,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 4.594453912432073e-06,
+        "converged": true,
+        "iterations": 175
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5591.png",
+      "answer": "8",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Eccentricity(G) = 3;H: Circle;Expression(H) = (m - 6*y + x^2 + y^2 = 0);m: Number;IsTangent(Asymptote(G), H) = True",
+      "coords": {}
+    },
+    {
+      "index": 5592,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5592.png",
+      "answer": "y=pm*sqrt(3)*x",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;D: Ellipse;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(D) = (x^2/4 + y^2/3 = 1);InterReciprocal(Eccentricity(C),Eccentricity(D))",
+      "coords": {}
+    },
+    {
+      "index": 5593,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5593.png",
+      "answer": "(4, 4)",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);Q: Point;PointOnCurve(Q, G);A: Point;Coordinate(A) = (7, 8);WhenMin(Distance(Q, A) + Distance(Q, Directrix(G)))",
+      "coords": {
+        "A": [
+          7.0,
+          8.0
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+      }
+    },
+    {
+      "index": 5594,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5594.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;M: Point;PointOnCurve(M, G);XCoordinate(M) = 4;F: Point;Focus(G) = F;Distance(M, F) = 5",
+      "coords": {}
+    },
+    {
+      "index": 5595,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5595.png",
+      "answer": "[sqrt(2)/2, 1]",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(G) = F;L: Line;PointOnCurve(F, L);IsPerpendicular(L, OneOf(Asymptote(G)));A: Point;FootPoint(L, OneOf(Asymptote(G))) = A;O: Origin;Area(TriangleOf(O, A, F)) = 2;e: Number;Eccentricity(G) = e;In(e, [sqrt(17), sqrt(65)])",
+      "coords": {}
+    },
+    {
+      "index": 5596,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5596.png",
+      "answer": "x^2 - y^2 / 3 = 1",
+      "fact_expr": "H: Parabola;Expression(H) = (y^2 = 4*x);G: Hyperbola;Focus(H) = Vertex(G);Vertex(H) = Center(G);Eccentricity(G) = 2",
+      "coords": {}
+    },
+    {
+      "index": 5597,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5597.png",
+      "answer": "pm*1",
+      "fact_expr": "G: Parabola;B: Point;A: Point;Expression(G) = (x^2 = 4*y);IsChordOf(LineSegmentOf(A, B),G);Length(LineSegmentOf(A, B)) = 8;WhenMin(Distance(MidPoint(LineSegmentOf(A,B)),xAxis))",
+      "coords": {}
+    },
+    {
+      "index": 5598,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5598.png",
+      "answer": "5/3",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Expression(Asymptote(G)) = (y = pm*x*4/3);e: Number;Eccentricity(G) = e",
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+      "answer": "2*sqrt(5)",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5600.png",
+      "answer": "8/3",
+      "fact_expr": "C: Parabola;G: Line;A: Point;B: Point;F: Point;E: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F, G);Intersection(G, C) = {A, B};VectorOf(A, F) = 3*VectorOf(F, B);MidPoint(LineSegmentOf(A, B)) = E",
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+      "index": 5601,
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+      "error": "Unsupported or unparseable expression"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5602.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
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+      "answer": "3*x+pm*4*y=0",
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+        "direction": "up"
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+        "note": "using explicit params"
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+      "answer": "1",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);P:Point;PointOnCurve(P,G)",
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+    {
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+      "error": null,
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+        "a": 1.5030499696731567,
+        "b": 2.00545072555542,
+        "orientation": "horizontal"
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+        "iterations": 109
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5606.png",
+      "answer": "3",
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+      "coords": {}
+    },
+    {
+      "index": 5607,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_point_off_curve",
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+        "b": 1.4970015287399292,
+        "orientation": "horizontal"
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+        "converged": true,
+        "iterations": 2
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5607.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
+      "fact_expr": "G: Hyperbola;a: Number;H: Point;a>0;Expression(G) = (-y^2/16 + x^2/a^2 = 1);Coordinate(H) = (3, 0);PointOnCurve(H, G)",
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+          0.0
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+      "index": 5608,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 5609,
+      "success": true,
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+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5609.png",
+      "answer": "y=pm*sqrt(3)*x",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;O: Origin;F: Point;A: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;RightVertex(C)=A;Abs(LineSegmentOf(O, F)) = 2*Abs(LineSegmentOf(O, A))",
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+    {
+      "index": 5610,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
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+          2.0
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+        "radius": 1.0
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_5610.png",
+      "answer": "(0,1/8)",
+      "fact_expr": "C: Parabola;m: Number;M: Circle;Expression(C) = (m*y + 4*x^2 = 0);Expression(M) = ((x - 1)^2 + (y - 2)^2 = 1);PointOnCurve(Center(M),C)",
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+    {
+      "index": 5611,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5612,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5613,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5613.png",
+      "answer": "sqrt(6)/3",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;G: Circle;F: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2 + y^2 = b^2);RightFocus(C) = F;L1:Line;L2:Line;TangentOfPoint(F,G)={L1,L2};IsPerpendicular(L1,L2)",
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+    {
+      "index": 5614,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5614.png",
+      "answer": "8/5",
+      "fact_expr": "G: Ellipse;H: Line;A: Point;B: Point;Expression(G) = (x^2/4 + y^2 = 1);PointOnCurve(RightFocus(G), H);Slope(H) = 1;Intersection(H, G) = {A, B};IsChordOf(LineSegmentOf(A, B), G)",
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+    },
+    {
+      "index": 5615,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5616,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.125,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5616.png",
+      "answer": "(1/2,1/2)",
+      "fact_expr": "G: Parabola;H: Line;Expression(G) = (y = 2*x^2);Expression(H) = (2*x - y - 4 = 0);PointOnCurve(P, G);WhenMin(Distance(P,H));P:Point",
+      "coords": {}
+    },
+    {
+      "index": 5617,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5617.png",
+      "answer": "(-oo, 0)",
+      "fact_expr": "a: Number;b: Number;e1: Number;e2: Number;m: Number;a>b;b>0;C1: ConicSection;C2: ConicSection;Expression(C1) = (x^2/a^2+y^2/b^2=1);Expression(C2) = (x^2/a^2-y^2/b^2=1);Eccentricity(C1) = e1;Eccentricity(C2) = e2;m = lg(e1) + lg(e2)",
+      "coords": {}
+    },
+    {
+      "index": 5618,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5618.png",
+      "answer": "3",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;Focus(C) = {F1, F2};P: Point;PointOnCurve(P, C);AngleOf(F1, P, F2) = ApplyUnit(60, degree);Area(TriangleOf(P, F1, F2)) = 3*sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 5619,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5619.png",
+      "answer": "5/3",
+      "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;F1: Point;F2: Point;B: Point;A: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F2, l);Intersection(l, RightPart(C)) = {A, B};Abs(LineSegmentOf(F1, F2)) = Abs(LineSegmentOf(F1, B));VectorOf(A, F2) = 2*VectorOf(F2, B)",
+      "coords": {}
+    },
+    {
+      "index": 5620,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5620.png",
+      "answer": "16",
+      "fact_expr": "F2: Point;F1: Point;LeftFocus(G) = F1;RightFocus(G) = F2;G: Hyperbola;Expression(G) = (x^2/4 - y^2/3 = 1);P: Point;PointOnCurve(P, LeftPart(G))",
+      "coords": {}
+    },
+    {
+      "index": 5621,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5622,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5623,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5623.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;l1:Line;l2:Line;Expression(l1)=(3*x - 4*y - 9 = 0);Expression(l2)=(y = -1/4);Expression(G)=(y=x^2);PointOnCurve(P,G);P:Point",
+      "coords": {}
+    },
+    {
+      "index": 5624,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5624.png",
+      "answer": "pi/4",
+      "fact_expr": "E: Parabola;p: Number;B: Point;A: Point;F: Point;p>0;Expression(E) = (y^2 = 2*(p*x));Focus(E) = F;PointOnCurve(A,E);PointOnCurve(B,E);Quadrant(A)=1;Quadrant(B)=1;Abs(LineSegmentOf(F, A)) = 3;Abs(LineSegmentOf(F, B)) = 5;Abs(LineSegmentOf(A, B)) = 2*sqrt(2)",
+      "coords": {}
+    },
+    {
+      "index": 5625,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5625.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "G: Parabola;P: Point;A: Point;F: Point;x1:Number;y1:Number;Expression(G) = (y^2 = 4*x);Coordinate(P) = (x1, y1);Coordinate(A) = (-1, 0);Focus(G) = F;PointOnCurve(P, G)",
+      "coords": {
+        "A": [
+          -1.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 5626,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5627,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5627.png",
+      "answer": "sqrt(2)+1",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;I: Point;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) =F1;RightFocus(C)=F2;PointOnCurve(P, C);Center(InscribedCircle(TriangleOf(P,F1,F2)))=I;DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = DotProduct(VectorOf(P, I), VectorOf(P, F1));IsPerpendicular(LineSegmentOf(P,F2),LineSegmentOf(F1,F2))",
+      "coords": {}
+    },
+    {
+      "index": 5628,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5628.png",
+      "answer": "6",
+      "fact_expr": "C: Parabola;A: Point;B: Point;P: Point;Expression(C) = (y^2 = 4*x);IsChordOf(LineSegmentOf(A,B),C);MidPoint(LineSegmentOf(A,B))=P;Distance(P, yAxis) = 2",
+      "coords": {}
+    },
+    {
+      "index": 5629,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5630,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5630.png",
+      "answer": "15",
+      "fact_expr": "G: Ellipse;P: Point;M: Point;F1: Point;F2: Point;Expression(G) = (x^2/25 + y^2/16 = 1);Coordinate(M) = (6, 4);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G)",
+      "coords": {
+        "M": [
+          6.0,
+          4.0
+        ]
+      }
+    },
+    {
+      "index": 5631,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5632,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5633,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5633.png",
+      "answer": "6",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);F: Point;Focus(G) = F;l: Line;Directrix(G) = l;P: Point;PointOnCurve(P,G) = True;A: Point;IsPerpendicular(LineSegmentOf(P, A), l) = True;FootPoint(LineSegmentOf(P, A), l) = A;Slope(LineOf(A, F)) = -sqrt(2)",
+      "coords": {}
+    },
+    {
+      "index": 5634,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5635,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
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+      "answer": "y^2/64 - x^2/36 = 1",
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+      "answer": "y=-1/16",
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+      "answer": "1/2",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5658.png",
+      "answer": "x^2/16+y^2/7=1",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "answer": "4",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1)",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5663.png",
+      "answer": "2*sqrt(15)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5664.png",
+      "answer": "[sqrt(5)/2,sqrt(5)]",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5665.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "G: Ellipse;H: Line;A: Point;B: Point;Expression(G) = (x^2 + 4*y^2 = 4);Expression(H) = (x - 2*y + 2 = 0);Intersection(H, G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 5666,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
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+      "index": 5667,
+      "success": true,
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+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5667.png",
+      "answer": "sqrt(10)/2",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;l: Line;PointOnCurve(F2, l);Intersection(l, RightPart(C)) = {P, Q};P: Point;Q: Point;Abs(LineSegmentOf(P, F1)) = 3*Abs(LineSegmentOf(P, F2));Abs(LineSegmentOf(P, Q)) = 4*Abs(LineSegmentOf(P, F2))",
+      "coords": {}
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+    {
+      "index": 5668,
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+      },
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5668.png",
+      "answer": "sqrt(3)/2",
+      "fact_expr": "C1: Ellipse;Expression(C1) = (x^2/a1^2+y^2/b1^2=1);a1: Number;b1: Number;a1>b1;b1>0;C2: Hyperbola;Expression(C2) = (x^2/a2^2-y^2/b2^2=1);a2: Number;b2: Number;a2>0;b2>0;F1: Point;F2: Point;Focus(C1) = {F1, F2};Focus(C2) = {F1, F2};P: Point;OneOf(Intersection(C1, C2)) = P;e1: Number;e2: Number;Eccentricity(C1) = e1;Eccentricity(C2) = e2;AngleOf(F1, P, F2) = pi/3",
+      "coords": {}
+    },
+    {
+      "index": 5669,
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+      "error": null,
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+        "direction": "up"
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+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5669.png",
+      "answer": "x^2=-4*y",
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+      "coords": {}
+    },
+    {
+      "index": 5670,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "p": 1.0,
+        "direction": "right"
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+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5670.png",
+      "answer": "y^2=16*x",
+      "fact_expr": "C: Parabola;Expression(C)=(y^2=2*p*x);p:Number;p>0;M: Point;F: Point;Focus(C)=F;m: Number;Coordinate(M) = (2, m);Abs(LineSegmentOf(M,F))=6;PointOnCurve(M,C)",
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+    {
+      "index": 5671,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_focus_mismatch",
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5671.png",
+      "answer": "5",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/16 + x^2/a^2 = 1);a: Number;a>0;F: Point;OneOf(Focus(G)) = F;Coordinate(F) = (3, 0)",
+      "coords": {
+        "F": [
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+          0.0
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+        "ellipse_focus_mismatch"
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+    {
+      "index": 5672,
+      "success": true,
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+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5672.png",
+      "answer": "7*sqrt(2)/8",
+      "fact_expr": "G: Parabola;H: Line;P: Point;Expression(G) = (y = x^2);Expression(H) = (y = x - 2);PointOnCurve(P, G)",
+      "coords": {}
+    },
+    {
+      "index": 5673,
+      "success": true,
+      "conic_type": "ellipse",
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5673.png",
+      "answer": "(0,2]+[8,+oo)",
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+    },
+    {
+      "index": 5674,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "a": 4.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 9.0
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5674.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
+      "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/16 + y^2/9 = 1);LeftFocus(G) =F1;RightFocus(G)=F2;PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(60, degree)",
+      "coords": {}
+    },
+    {
+      "index": 5675,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5675.png",
+      "answer": "10",
+      "fact_expr": "l: Line;C: Parabola;F: Point;A: Point;B: Point;Expression(C) = (x^2 = 8*y);Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};XCoordinate(A)=2",
+      "coords": {}
+    },
+    {
+      "index": 5676,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5677,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5677.png",
+      "answer": "5/4",
+      "fact_expr": "l: Line;G: Parabola;B: Point;A: Point;F: Point;Expression(G) = (y = x^2);Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B};Abs(LineSegmentOf(A, B)) = 3",
+      "coords": {}
+    },
+    {
+      "index": 5678,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 2.6457513110645907,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 7.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5678.png",
+      "answer": "2*sqrt(14)",
+      "fact_expr": "C: Ellipse;G: Line;P: Point;F: Point;Q: Point;O: Origin;Expression(C) = (x^2/16 + y^2/7 = 1);LeftFocus(C) = F;PointOnCurve(O, G);Intersection(G, C) = {P, Q};Abs(LineSegmentOf(P, F))*Abs(LineSegmentOf(Q, F)) = 9",
+      "coords": {}
+    },
+    {
+      "index": 5679,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5679.png",
+      "answer": "2*sqrt(5)/5",
+      "fact_expr": "P: Point;Coordinate(P) = (x0, y0);x0: Number;y0: Number;Negation(x0=pm*a);E: Ellipse;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>b;b>0;PointOnCurve(P,E) = True;M: Point;N: Point;LeftVertex(E) = M;RightVertex(E) = N;Slope(LineOf(P,M)) * Slope(LineOf(P,N)) = -1/5",
+      "coords": {}
+    },
+    {
+      "index": 5680,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5681,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5682,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "y",
+        "x_coef": 1.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5682.png",
+      "answer": "2/3",
+      "fact_expr": "G: Ellipse;E: Point;P: Point;Q: Point;Expression(G) = (x^2 + y^2/4 = 1);PointOnCurve(P,G);PointOnCurve(Q,G);Coordinate(E)=(0, 1);IsPerpendicular(LineSegmentOf(E, P), LineSegmentOf(E, Q))",
+      "coords": {
+        "E": [
+          0.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 5683,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5683.png",
+      "answer": "[1, +\\infty)",
+      "fact_expr": "H: Ellipse;Expression(H) = (y^2 + x^2/a^2 = 1);a: Number;a>0;F1: Point;F2: Point;G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;F: Point;Focus(G) = F;VectorOf(F1, F) = 3*VectorOf(F, F2);z: Number;z>=a^2-p^2;Focus(H)={F1,F2}",
+      "coords": {}
+    },
+    {
+      "index": 5684,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5684.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "l: Line;G: Hyperbola;b: Number;a: Number;P: Point;Q: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Slope(l)=sqrt(2)/2;Intersection(l, G) = {P, Q};F1:Point;F2:Point;Focus(G)={F1,F2};Projection(P,xAxis)=F1;Projection(Q,xAxis)=F2;Negation(P=Q)",
+      "coords": {}
+    },
+    {
+      "index": 5685,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5685.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;p: Number;A: Point;p>0;Expression(G) = (y^2 = 2*p*x);Coordinate(A) = (x0, 2);x0:Number;PointOnCurve(A,G);Distance(A,Focus(G))=2*Distance(A,yAxis)",
+      "coords": {}
+    },
+    {
+      "index": 5686,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.7320508075688772,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5686.png",
+      "answer": "(-2, 0)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 5687,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5688,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5688.png",
+      "answer": "sqrt(5)/2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x + 2*y - 1 = 0);IsParallel(OneOf(Asymptote(G)), H)",
+      "coords": {}
+    },
+    {
+      "index": 5689,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5690,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5690.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(G) = F;M: Point;IsPerpendicular(LineSegmentOf(M, F), xAxis);P: Point;MidPoint(LineSegmentOf(M, F)) = P;A: Point;B: Point;Vertex(G) = {A, B};WhenMax(AngleOf(A, P, B));PointOnCurve(M, G)",
+      "coords": {}
+    },
+    {
+      "index": 5691,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5691.png",
+      "answer": "2",
+      "fact_expr": "C: Hyperbola;a: Number;b: Number;G: Circle;A: Point;O: Origin;B: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2 + y^2 = a^2);l1:Line;l2:Line;PointOnCurve(OneOf(Focus(C)),l1);PointOnCurve(OneOf(Focus(C)),l2);TangentOfPoint(OneOf(Focus(C)), G) = {l1, l2} ;TangentPoint(l1,G)=A;TangentPoint(l2,G)=B;AngleOf(A, O, B) = ApplyUnit(120,degree)",
+      "coords": {}
+    },
+    {
+      "index": 5692,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.7320508075688772,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 3.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5692.png",
+      "answer": "4*sqrt(3)",
+      "fact_expr": "G: Ellipse;H: Line;A: Point;B: Point;F2: Point;F1: Point;Expression(G) = (x^2/3 + y^2 = 1);Focus(G) = {F1, F2};PointOnCurve(F1, H);Intersection(H, G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 5693,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5693.png",
+      "answer": "{5,3}",
+      "fact_expr": "G: Ellipse;m: Number;Expression(G) = (y^2/4 + x^2/m = 1);FocalLength(G) = 2",
+      "coords": {}
+    },
+    {
+      "index": 5694,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          -2.0,
+          0.0
+        ],
+        "radius": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_5694.png",
+      "answer": "{1, 4}",
+      "fact_expr": "C1: Circle;C2: Circle;C3: Circle;Omega: Curve;r: Number;r > 0;Expression(C1) = (y^2 + (x - 2)^2 = 4);Expression(C2) = ((x + r)^2 + y^2 = r^2);IsOutTangent(C3, C1);IsOutTangent(C3, C2);Locus(Center(C3))=Omega;Eccentricity(Omega) = 3",
+      "coords": {}
+    },
+    {
+      "index": 5695,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5695.png",
+      "answer": "pm*sqrt(3)/2",
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+      "answer": "5/6",
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+      "answer": "9/2",
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+      "answer": "15",
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+      "answer": "-8*sqrt(2)/5",
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+      "answer": "x^2/18 - y^2/8 = 1",
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+      "answer": "4*sqrt(3)/3",
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+      "answer": "y=pm*(1/2)*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/8 - y^2/2 = 1)",
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+      "answer": "sqrt(3) - 1",
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+      "success": false,
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+      "answer": "7",
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+      "answer": "sqrt(3)*x+pm*y=0",
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+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5712.png",
+      "answer": "(\\sqrt{5}/2,+\\infty)",
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+        "x_coef": 0.25,
+        "y_coef": 1.0
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+        "final_loss": 0.0,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5713.png",
+      "answer": "[-sqrt(5)/2,sqrt(5)/2]",
+      "fact_expr": "G: Ellipse;H: Line;m: Real;Expression(G) = (4*x^2 + y^2 = 1);Expression(H) = (y = m + x);IsIntersect(G, H)",
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+      "index": 5714,
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+        "direction": "right"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5714.png",
+      "answer": "8",
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+        "note": "using explicit params"
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+      "answer": "4",
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+        "note": "using explicit params"
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+      "answer": "3",
+      "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 - y^2/5 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, LeftPart(G));Abs(LineSegmentOf(P, F1)) + Abs(LineSegmentOf(P, F2)) = 8",
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+      "index": 5717,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5717.png",
+      "answer": "4*sqrt(2)",
+      "fact_expr": "l: Line;C: Parabola;M: Point;O: Origin;A: Point;F: Point;B: Point;S1: Number;S2: Number;Expression(C) = (y^2 = 4*x);Coordinate(M) = (4, 0);Focus(C) = F;PointOnCurve(M, l);Intersection(l, C) = {A, B};Area(TriangleOf(O, A, F)) = S1;Area(TriangleOf(O, B, F)) = S2",
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+          0.0
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+      "index": 5718,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5718.png",
+      "answer": "5",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);G: Circle;A: Point;F: Point;M: Point;Coordinate(M) = (0, 2);PointOnCurve(A, C);Focus(C) = F;IsDiameter(LineSegmentOf(A, F), G) ;NumIntersection(G, yAxis) = 1;Intersection(G, yAxis) = M",
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+        "M": [
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+          2.0
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+      "index": 5719,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5719.png",
+      "answer": "sqrt(2)+1",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;A: Point;B: Point;F1: Point;F2: Point;a>0;b>0;H: Line;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1,H);IsPerpendicular(H,xAxis);Intersection(C,H) = {A,B};Abs(LineSegmentOf(A,B)) = 2*Abs(LineSegmentOf(F1,F2))",
+      "coords": {}
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+    {
+      "index": 5720,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 6.0,
+        "y_coef": 1.0
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5720.png",
+      "answer": "4*sqrt(6)",
+      "fact_expr": "A: Point;B: Point;G: Ellipse;F2: Point;F1: Point;Expression(G) = (x^2/6 + y^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1,LineOf(A, B));Intersection(LineOf(A, B), G) = {A,B}",
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+    {
+      "index": 5721,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+        "final_loss": 0.0,
+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5721.png",
+      "answer": "pm*4",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;M: Point;m: Number;Coordinate(M) = (1, m);PointOnCurve(M, G);Distance(M, Focus(G)) = 5",
+      "coords": {}
+    },
+    {
+      "index": 5722,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 12.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5722.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/12 + y^2/3 = 1);F1: Point;LeftFocus(G) = F1;P: Point;PointOnCurve(P, G);M: Point;MidPoint(LineSegmentOf(P, F1)) = M;PointOnCurve(M, yAxis)",
+      "coords": {}
+    },
+    {
+      "index": 5723,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5724,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5725,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
+      },
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5725.png",
+      "answer": "2",
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+      "coords": {}
+    },
+    {
+      "index": 5726,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5727,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 2.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5727.png",
+      "answer": "{-1, 1, -sqrt(5)/2, sqrt(5)/2}",
+      "fact_expr": "G: Hyperbola;H: Line;k: Number;Expression(G) = (x^2 - y^2 = 4);Expression(H) = (y = k*x - 1);NumIntersection(H, G) = 1",
+      "coords": {}
+    },
+    {
+      "index": 5728,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5729,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5729.png",
+      "answer": "sqrt(2)",
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+      "coords": {}
+    },
+    {
+      "index": 5730,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5731,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5731.png",
+      "answer": "4+sqrt(5)",
+      "fact_expr": "G: Ellipse;A: Point;B: Point;P: Point;Expression(G) = (x^2/4 + y^2/3 = 1);Coordinate(A) = (1, 0);Coordinate(B) = (1, 1);PointOnCurve(P, G)",
+      "coords": {
+        "A": [
+          1.0,
+          0.0
+        ],
+        "B": [
+          1.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 5732,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5732.png",
+      "answer": "sqrt(3)",
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+      "coords": {}
+    },
+    {
+      "index": 5733,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5733.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;Q: Point;F1: Point;F2: Point;P: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) =F1;RightFocus(G)=F2;PointOnCurve(F1,H);L1:Line;L2:Line;Asymptote(G)={L1,L2};Intersection(H,L1)=P;Intersection(H,L2)=Q;MidPoint(LineSegmentOf(F1,Q))=P;IsPerpendicular(LineSegmentOf(Q, F1), LineSegmentOf(Q, F2))",
+      "coords": {}
+    },
+    {
+      "index": 5734,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5734.png",
+      "answer": "(1,2]",
+      "fact_expr": "C: Hyperbola;G: Circle;x0: Number;y0: Number;H: Line;b: Number;a: Number;a>0;b>0;P: Point;Expression(G) = ((x - x0)^2 + (y - y0)^2 = 1);Expression(H) = (2*a - a*y + b*x = 0);Coordinate(P) = (x0, y0);Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, H);NumIntersection(G, RightPart(C))=0",
+      "coords": {}
+    },
+    {
+      "index": 5735,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 2.5275303393962245,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 4.481942266905428e-06,
+        "converged": true,
+        "iterations": 95
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5735.png",
+      "answer": "3",
+      "fact_expr": "G: Hyperbola;a: Number;a>0;Expression(G) = (-y^2/(a + 3)^2 + x^2/a^2 = 1);Expression(OneOf(Asymptote(G))) = (y = 2*x)",
+      "coords": {}
+    },
+    {
+      "index": 5736,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5736.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/6 - y^2/m = 1);m: Number;Length(ImageinaryAxis(G)) = 6*sqrt(2)",
+      "coords": {}
+    },
+    {
+      "index": 5737,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5738,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5738.png",
+      "answer": "sqrt(6)/3",
+      "fact_expr": "C1:Ellipse;C2:Circle;A: Point;O: Origin;B: Point;Expression(C1) = (x^2/36 + y^2/b^2 = 1);Expression(C2)=(x^2+y^2=9);PointOnCurve(P,C2);Intersection(TangentOfPoint(P,C2),C1)={A,B};AngleOf(A, O, B) = pi/2;P:Point;b: Number;0 < b;b < 6",
+      "coords": {}
+    },
+    {
+      "index": 5739,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 5.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5739.png",
+      "answer": "5*x + pm*4*y = 0",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/25 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 5740,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5740.png",
+      "answer": "-8",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 2*(a*y));a: Number;Expression(Directrix(G)) = (y = 4)",
+      "coords": {}
+    },
+    {
+      "index": 5741,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.046616818012331,
+        "b": 3.9644919697671575,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 4.0501961192983754e-07,
+        "converged": true,
+        "iterations": 93
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5741.png",
+      "answer": "13",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);P: Point;PointOnCurve(P, G);F1: Point;Coordinate(F1) = (-5, 0);Distance(P, F1) = 7;F2: Point;Coordinate(F2) = (5, 0)",
+      "coords": {
+        "F1": [
+          -5.0,
+          0.0
+        ],
+        "F2": [
+          5.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 5742,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5742.png",
+      "answer": "6",
+      "fact_expr": "G: Parabola;F: Point;A: Point;B: Point;C: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(A, G);PointOnCurve(B, G);PointOnCurve(C, G);VectorOf(F, A) + VectorOf(F, B) + VectorOf(F, C) = 0",
+      "coords": {}
+    },
+    {
+      "index": 5743,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5744,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5744.png",
+      "answer": "sqrt(3)/3",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;LeftFocus(C) = F;O: Origin;PointOnCurve(O, G) = True;G: Line;Intersection(G, C) = {P, Q};P: Point;Q: Point;AngleOf(P, F, Q) = ApplyUnit(120, degree);Abs(LineSegmentOf(O, F)) = sqrt(3);Abs(LineSegmentOf(O, P)) = sqrt(7)",
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+    },
+    {
+      "index": 5745,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
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+          2.0
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+        "radius": 2.23606797749979
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+      "optimization": {
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+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_5745.png",
+      "answer": "x^2/4 - y^2/12 = 1",
+      "fact_expr": "C: Circle;Expression(C) = (-4*y - 6*x + x^2 + y^2 + 8 = 0);A: Point;B: Point;Intersection(C, axis) = {A, B};G: Hyperbola;OneOf(Focus(G)) = A;OneOf(Vertex(G)) = B",
+      "coords": {}
+    },
+    {
+      "index": 5746,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5746.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F2: Point;F1: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(P,Directrix(G));IsPerpendicular(LineSegmentOf(P,F1),LineSegmentOf(P,F2));DotProduct(VectorOf(P,F1),VectorOf(P,F2))=4*a*b",
+      "coords": {}
+    },
+    {
+      "index": 5747,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5747.png",
+      "answer": "(x-5)^2+y^2=16",
+      "fact_expr": "G: Hyperbola;H: Ellipse;C:Circle;Expression(G) = (x^2/9 - y^2/16 = 1);Expression(H) = (x^2/169 + y^2/144 = 1);Center(C)=RightFocus(H);IsTangent(Asymptote(G),C)",
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+    },
+    {
+      "index": 5748,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 5749,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5749.png",
+      "answer": "[\\sqrt{17}, +\\infty)",
+      "fact_expr": "P: Point;x1: Number;y1: Number;Coordinate(P) = (x1, y1);C: Ellipse;Expression(C) = (x^2/4 + y^2 = 1);PointOnCurve(P, C);a: Number;2*x1 - y1 + a >=0",
+      "coords": {}
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+    {
+      "index": 5750,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 5751,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5752,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5753,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5754,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 3.872983346207417,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5754.png",
+      "answer": "5",
+      "fact_expr": "G: Hyperbola;P: Point;Expression(G) = (x^2 - y^2/15 = 1);PointOnCurve(P, G);F1:Point;F2:Point;OneOf(Focus(G))=F1;OneOf(Focus(G))=F2;Negation(F1=F2);Distance(P, F1) = 3",
+      "coords": {}
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+    {
+      "index": 5755,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5755.png",
+      "answer": "-4/3",
+      "fact_expr": "G: Ellipse;l: Line;A: Point;B: Point;M: Point;m:Number;Expression(G) = (x^2/4 + y^2/3 = 1);Slope(l)=1;Intersection(l,G)={A,B};Coordinate(M) = (m, 1);MidPoint(LineSegmentOf(A, B)) = M",
+      "coords": {}
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+    {
+      "index": 5756,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5756.png",
+      "answer": "Slight",
+      "fact_expr": "G: Parabola;Expression(G) = (y = x^2);P:Point;PointOnCurve(P,G);Inclination(TangentOnPoint(P,G))=pi/4",
+      "coords": {}
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+    {
+      "index": 5757,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5757.png",
+      "answer": "(4,0)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 5758,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5758.png",
+      "answer": "7",
+      "fact_expr": "C: Parabola;p: Number;G: Line;A: Point;F: Point;O: Origin;B: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;PointOnCurve(F, G);Intersection(G, C) = {A, B};Abs(LineSegmentOf(A, F)) = 8*Abs(LineSegmentOf(O, F))",
+      "coords": {}
+    },
+    {
+      "index": 5759,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 2.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5759.png",
+      "answer": "{(0,-sqrt(2)),(0,sqrt(2))}",
+      "fact_expr": "G: Ellipse;P: Point;F1:Point;F2:Point;m:Number;Expression(G) = (x^2/4 + y^2/2 = 1);PointOnCurve(P, G);Distance(P,F1)*Distance(P,F2)=m;WhenMax(m);Focus(G)={F1,F2}",
+      "coords": {}
+    },
+    {
+      "index": 5760,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5760.png",
+      "answer": "pm*sqrt(2)/2",
+      "fact_expr": "C: Parabola;B: Circle;B1:Point;O: Origin;F: Point;A: Point;Expression(C) = (x^2 = 4*y);Focus(C) = F;PointOnCurve(A, C);CircumCircle(TriangleOf(O,F,A))=B;Center(B)=B1;IsTangent(Directrix(C),B);H:Line;PointOnCurve(B,H);IsTangent(H,C)",
+      "coords": {}
+    },
+    {
+      "index": 5761,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5761.png",
+      "answer": "6",
+      "fact_expr": "G: Parabola;F: Point;A: Point;B: Point;C: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(A,G);PointOnCurve(B,G);PointOnCurve(C,G);VectorOf(F, A) + VectorOf(F, B) + VectorOf(F, C) = 0",
+      "coords": {}
+    },
+    {
+      "index": 5762,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.4142135623730951,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5762.png",
+      "answer": "x^2/2 - y^2/4 = 1",
+      "fact_expr": "G:Hyperbola;H: Point;C:Hyperbola;Expression(G)=(x^2-y^2/2=1);Coordinate(H)=(2,2);Asymptote(G)=Asymptote(C);PointOnCurve(H,C)",
+      "coords": {
+        "H": [
+          2.0,
+          2.0
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+    },
+    {
+      "index": 5763,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5763.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;O: Origin;F: Point;L1:Line;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);OneOf(Focus(G))=F;PointOnCurve(F,L1);IsPerpendicular(OneOf(Asymptote(G)),L1);PointOnCurve(FootPoint(OneOf(Asymptote(G)),L1),PerpendicularBisector(LineSegmentOf(O,F)))",
+      "coords": {}
+    },
+    {
+      "index": 5764,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.7320508075688772,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 3.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5764.png",
+      "answer": "{(0, 1), (0, -1)}",
+      "fact_expr": "G: Ellipse;F1: Point;A: Point;F2: Point;B: Point;Expression(G) = (x^2/3 + y^2 = 1);Focus(G) = {F1, F2};PointOnCurve(A, G);PointOnCurve(B, G);VectorOf(F1, A) = 5*VectorOf(F2, B)",
+      "coords": {}
+    },
+    {
+      "index": 5765,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5765.png",
+      "answer": "3/2",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2 - y^2 = 1);F: Point;RightFocus(C) = F;P: Point;PointOnCurve(P, LeftPart(C));A: Point;Coordinate(A) = (0, sqrt(2));WhenMin(Perimeter(TriangleOf(A, P, F)))",
+      "coords": {
+        "A": [
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+          1.4142135623730951
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+    },
+    {
+      "index": 5766,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5766.png",
+      "answer": "x-y=0",
+      "fact_expr": "l: Line;C: Parabola;A: Point;B: Point;Expression(C) = (y^2 = 4*x);Coordinate(MidPoint(LineSegmentOf(A,B))) = (2, 2);Intersection(l, C) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 5767,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5767.png",
+      "answer": "6",
+      "fact_expr": "G: Hyperbola;Expression(G) = (4*x^2 - 9*y^2 = 36)",
+      "coords": {}
+    },
+    {
+      "index": 5768,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5768.png",
+      "answer": "[\\sqrt{2}/2,1)",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C)= F2;l: Line;RightDirectrix(C)=l;P: Point;PointOnCurve(P,l);AngleOf(F1,F2,P)=ApplyUnit(120,degree);ApplyUnit(30,degree)<=AngleOf(F1,P,F2);AngleOf(F1,P,F2)<ApplyUnit(60,degree)",
+      "coords": {}
+    },
+    {
+      "index": 5769,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5769.png",
+      "answer": "x=-3/2",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;O: Origin;F: Point;Focus(C) = F;P: Point;PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(P, F), xAxis);Q: Point;PointOnCurve(Q, xAxis);IsPerpendicular(LineSegmentOf(P, Q), LineSegmentOf(O, P));Abs(LineSegmentOf(F, Q)) = 6",
+      "coords": {}
+    },
+    {
+      "index": 5770,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5771,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5771.png",
+      "answer": "4",
+      "fact_expr": "C: Parabola;A: Point;P: Point;E: Point;l:Line;x0:Number;y0:Number;Negation(x0=0);Expression(C) = (x^2 = 4*y);Coordinate(A) = (0, 3);Coordinate(P) = (x0, y0);PointOnCurve(P,C);PointOnCurve(P, l);TangentOfPoint(P,C)=l;IsPerpendicular(l,LineSegmentOf(P, A));Intersection(l, yAxis) = E",
+      "coords": {
+        "A": [
+          0.0,
+          3.0
+        ]
+      }
+    },
+    {
+      "index": 5772,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5772.png",
+      "answer": "y = x",
+      "fact_expr": "l: Line;C: Parabola;F: Point;A: Point;B: Point;O: Origin;Coordinate(MidPoint(LineSegmentOf(A, B))) = (2, 2);Coordinate(F) = (1, 0);Vertex(C) = O;Intersection(l, C) = {A, B};Focus(C) = F",
+      "coords": {
+        "F": [
+          1.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 5773,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5773.png",
+      "answer": "0",
+      "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;M: Point;l: Line;C:Line;Expression(G) = (y^2 = 4*x);Expression(H) = (x - y - 1 = 0);Expression(l) = (x = -1);Intersection(H,G)={A,B};PointOnCurve(MidPoint(LineSegmentOf(A,B)),C);IsPerpendicular(C,l);FootPoint(C,l)=M",
+      "coords": {}
+    },
+    {
+      "index": 5774,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5775,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5775.png",
+      "answer": "13",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1);P: Point;PointOnCurve(P, RightPart(G));H: Circle;Expression(H) = (y^2 + (x + 5)^2 = 9);Z: Circle;Expression(Z) = (y^2 + (x - 5)^2 = 4);M: Point;PointOnCurve(M, H);N: Point;PointOnCurve(N, Z)",
+      "coords": {}
+    },
+    {
+      "index": 5776,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5776.png",
+      "answer": "4*sqrt(3)",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/9 + y^2 = 1);P: Point;PointOnCurve(P, C);E: Circle;Expression(E) = (x^2 + (y - 4)^2 = 3);Q: Point;PointOnCurve(Q, E)",
+      "coords": {}
+    },
+    {
+      "index": 5777,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5777.png",
+      "answer": "3",
+      "fact_expr": "l: Line;C: Parabola;p: Number;A: Point;F: Point;D: Point;B: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;PointOnCurve(F, l);Intersection(l,C)={A,B};Intersection(l,Directrix(C))=D;Abs(LineSegmentOf(A, F)) = 6;VectorOf(D, B) = 2*VectorOf(B, F)",
+      "coords": {}
+    },
+    {
+      "index": 5778,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5779,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5779.png",
+      "answer": "12",
+      "fact_expr": "l: Line;C: Parabola;p: Number;O: Origin;F: Point;A: Point;B: Point;p>0;Expression(C) = (x^2 = 2*(p*y));Focus(C) = F;Abs(LineSegmentOf(O, F)) = 3/4;PointOnCurve(F, l);Slope(l)=sqrt(3);Intersection(l, C) = {A,B}",
+      "coords": {}
+    },
+    {
+      "index": 5780,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5780.png",
+      "answer": "sqrt(3)/2",
+      "fact_expr": "H: Parabola;Expression(H) = (y^2 = 4*x);G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 5781,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5781.png",
+      "answer": "sqrt(13)/2",
+      "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;G: Circle;F1: Point;P: Point;T: Point;F2: Point;a > b;b > 0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2 + y^2 = a^2);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, l);TangentPoint(l, G) = T;Intersection(l, RightPart(C)) = P;LineSegmentOf(F1, P) = 3*LineSegmentOf(F1, T)",
+      "coords": {}
+    },
+    {
+      "index": 5782,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5782.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 5783,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5783.png",
+      "answer": "ApplyUnit(45,degree)",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);M: Point;PointOnCurve(M, C) = True;F: Point;Focus(C) = F;O: Origin;Abs(LineSegmentOf(M,F)) = 2;K: Point;Intersection(Directrix(C),xAxis) = K",
+      "coords": {}
+    },
+    {
+      "index": 5784,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5785,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5785.png",
+      "answer": "y=(1/2)*x^2+1",
+      "fact_expr": "G: Parabola;Expression(G) = (y = x^2/4);F: Point;Focus(G) = F;H: Line;PointOnCurve(F, H);A: Point;B: Point;Intersection(H, G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 5786,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.7320508075688772,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 3.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5786.png",
+      "answer": "1",
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+      "index": 5787,
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+      "error": null,
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5787.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "G: Parabola;P: Point;F: Point;A: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (-1, 0);PointOnCurve(P, G);Focus(G) = F",
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+        "converged": true,
+        "note": "using explicit params"
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+      "answer": "(1/2,2/3)",
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5789.png",
+      "answer": "sqrt(6)/2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (-6*x + x^2 + y^2 + 5 = 0);Length(InterceptChord(Asymptote(G),H))=2",
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+    {
+      "index": 5790,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 4.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5790.png",
+      "answer": "16",
+      "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/36 - y^2/16 = 1);PointOnCurve(P,G);Focus(G)={F1,F2};IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2))",
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+        "b": 4.0,
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+        "y_coef": 16.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5791.png",
+      "answer": "(5, 0)",
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+    {
+      "index": 5792,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
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+          0.0
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_5792.png",
+      "answer": "4*x^2/25+4*y^2/21=1",
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+      "index": 5793,
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "answer": "1",
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+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5794.png",
+      "answer": "9/2",
+      "fact_expr": "G: Parabola;A: Point;P: Point;Expression(G) = (y^2 = 2*x);Coordinate(A) = (7/2, 4);PointOnCurve(P, G);Distance(P, yAxis) = d1;Distance(P, A) = d2;d1:Number;d2:Number",
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+      "index": 5795,
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+        "b": 1.0,
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+        "y_coef": 1.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5795.png",
+      "answer": "0",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2 = 1);F1: Point;F2: Point;Focus(C) = {F1,F2};M: Point;PointOnCurve(M,C);I: Point;Incenter(TriangleOf(M,F1,F2)) = I;YCoordinate(I) = 2-sqrt(3)",
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+    {
+      "index": 5796,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5796.png",
+      "answer": "[\\pi/4, 3\\pi/4]",
+      "fact_expr": "l: Line;C: Parabola;A: Point;F: Point;B: Point;Expression(C) = (y^2 = 8*x);Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};Abs(LineSegmentOf(A, F))*Abs(LineSegmentOf(B, F)) <= 32",
+      "coords": {}
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+    {
+      "index": 5797,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5797.png",
+      "answer": "9",
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+    {
+      "index": 5798,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 9.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5798.png",
+      "answer": "16",
+      "fact_expr": "C: Ellipse;M: Point;F1: Point;F2: Point;Expression(C) = (x^2/16 + y^2/9 = 1);Focus(C) = {F1, F2};PointOnCurve(M, C)",
+      "coords": {}
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+    {
+      "index": 5799,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5799.png",
+      "answer": "8",
+      "fact_expr": "C: Parabola;G: Line;H: Point;F: Point;M: Point;N: Point;Expression(C) = (y^2 = 4*x);Coordinate(H) = (-2, 0);Focus(C)=F;PointOnCurve(H, G);Slope(G) = 2/3;Intersection(G, C) = {M, N}",
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+          0.0
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+      "index": 5800,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 5801,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5802,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5802.png",
+      "answer": "10",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);l: Line;PointOnCurve(Focus(G), l);A: Point;B: Point;Intersection(l, G) = {A, B};XCoordinate(MidPoint(LineSegmentOf(A, B))) = 4",
+      "coords": {}
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+    {
+      "index": 5803,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 4.0,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5803.png",
+      "answer": "x-y-5=0",
+      "fact_expr": "G: Hyperbola;H: Line;A: Point;Expression(G) = (x^2/9 - y^2/16 = 1);Coordinate(A) = (4, -1);PointOnCurve(A,H);PointOnCurve(RightFocus(G),H)",
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+          -1.0
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+      "index": 5804,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5804.png",
+      "answer": "x^2-4*y+4=0",
+      "fact_expr": "G: Parabola;P: Point;F: Point;Expression(G) = (y = x^2/8);Focus(G) = F;PointOnCurve(P, G)",
+      "coords": {}
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+    {
+      "index": 5805,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_focus_mismatch",
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5805.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;F1: Point;A: Point;P: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (0, 1);Coordinate(F1) = (2*sqrt(2), 0);RightFocus(G) = F1;PointOnCurve(P, LeftPart(G));Min(Perimeter(TriangleOf(A, P, F1))) = 8",
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+          1.0
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+        "F1": [
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+      "index": 5806,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 5807,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5807.png",
+      "answer": "20",
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+      "index": 5808,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 5809,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "up"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5809.png",
+      "answer": "x^2 = 2*y - 1",
+      "fact_expr": "C: Parabola;F: Point;Coordinate(F) = (0, 1);Focus(C) = F;Directrix(C) = xAxis",
+      "coords": {
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+          1.0
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+      "index": 5810,
+      "success": true,
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5810.png",
+      "answer": "-1",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Eccentricity(C) = sqrt(3)/3;l: Line;Intersection(l, C) = {A, B};A: Point;B: Point;M: Point;Coordinate(M) = (3, 2);MidPoint(LineSegmentOf(A, B)) = M",
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+          2.0
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+      "index": 5811,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5811.png",
+      "answer": "(0, 2]+[8, +\\infty)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/m = 1);m: Number;Eccentricity(G) = e;In(e, [sqrt(2)/2, 1));e: Number",
+      "coords": {}
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+    {
+      "index": 5812,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5812.png",
+      "answer": "{1,-1}",
+      "fact_expr": "H: Ellipse;Expression(H) = (x^2/4 + y^2/a^2 = 1);G: Hyperbola;Expression(G) = (-y^2/2 + x^2/a^2 = 1);a: Number;Focus(H) = Focus(G)",
+      "coords": {}
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+    {
+      "index": 5813,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5814,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
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+          0.0
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+        "radius": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_5814.png",
+      "answer": "(x-2)^2+y^2=4",
+      "fact_expr": "G: Circle;A: Point;P: Point;M: Point;Expression(G) = (x^2 + y^2 = 9);Coordinate(A) = (6, 0);PointOnCurve(M, G);VectorOf(A, P) = 2*VectorOf(P, M)",
+      "coords": {}
+    },
+    {
+      "index": 5815,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5815.png",
+      "answer": "(0,\\sqrt{6})",
+      "fact_expr": "l: Line;G: Hyperbola;F1: Point;P: Point;Q: Point;F2: Point;Expression(G) = (x^2/2 - y^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2, l);Slope(l) = k;k:Number;k<-1;Intersection(l, RightPart(G)) = {P, Q};Area(TriangleOf(F1, P, Q)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 5816,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_point_off_curve",
+      "params": {
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+        "iterations": 106
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5816.png",
+      "answer": "(\\sqrt{5}, +\\infty)",
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+          1.0
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+      "index": 5817,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
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+      "coords": {}
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+    {
+      "index": 5818,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5818.png",
+      "answer": "pi/3",
+      "fact_expr": "G: Hyperbola;F1: Point;P: Point;F2: Point;Expression(G) = (3*x^2 - 5*y^2 = 15);PointOnCurve(P, G);Focus(G) = {F1, F2};Area(TriangleOf(F1, P, F2)) = 3*sqrt(2)",
+      "coords": {}
+    },
+    {
+      "index": 5819,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 9.0
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+        "final_loss": 0.0,
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5819.png",
+      "answer": "9",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/9 = 1);P: Point;PointOnCurve(P, G);F1: Point;F2: Point;Focus(G) = {F1, F2};AngleOf(F1, P, F2) = ApplyUnit(90, degree)",
+      "coords": {}
+    },
+    {
+      "index": 5820,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5820.png",
+      "answer": "5",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;H: Point;Coordinate(H) = (p/2, 1);PointOnCurve(H, G);K: Point;PointOnCurve(K, G);YCoordinate(K) = 3",
+      "coords": {}
+    },
+    {
+      "index": 5821,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5821.png",
+      "answer": "(\\sqrt{6}, 4)",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;F1: Point;F2: Point;P: Point;c: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);RightFocus(C) = F2;LeftFocus(C) = F1;Expression(Asymptote(C)) = (y = pm*(2*x));FocalLength(C) = 2*sqrt(10);PointOnCurve(P, C);Quadrant(P) = 1;Area(TriangleOf(P, F1, F2)) = 4*sqrt(10)",
+      "coords": {}
+    },
+    {
+      "index": 5822,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5823,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5823.png",
+      "answer": "8",
+      "fact_expr": "l: Line;G: Parabola;M: Point;O: Origin;A: Point;B: Point;F: Point;t:Number;Expression(G) = (y^2 = 4*x);Coordinate(M) = (t, 2);Quadrant(M) = 1;Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B};VectorOf(O, M) = (VectorOf(O, A) + VectorOf(O, B))/2",
+      "coords": {}
+    },
+    {
+      "index": 5824,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5824.png",
+      "answer": "(1,2)",
+      "fact_expr": "G: Hyperbola;a: Number;b: Number;H: Circle;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2 + (y + 2)^2 = 1);NumIntersection(Asymptote(G), H) = 0",
+      "coords": {}
+    },
+    {
+      "index": 5825,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5825.png",
+      "answer": "3",
+      "fact_expr": "G: Curve;B: Point;A: Point;C: Point;a:Number;b:Number;Expression(G) = (x^2 = 4*y);Coordinate(C) = (-1, -1);L1:Line;L2:Line;TangentOfPoint(C,G)={L1,L2};TangentPoint(L1,G)=A;TangentPoint(L2,G)=B;Expression(LineOf(A,B)) = (y/b + x/a = 1)",
+      "coords": {
+        "C": [
+          -1.0,
+          -1.0
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+      }
+    },
+    {
+      "index": 5826,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5826.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "G: Parabola;A: Point;B: Point;P: Point;Expression(G) = (y^2 = 2*x);Coordinate(A) = (-1, 0);Coordinate(B) = (1, 0);PointOnCurve(P,G);Abs(LineSegmentOf(P,A))=m*Abs(LineSegmentOf(P,B));m:Number",
+      "coords": {
+        "A": [
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+          0.0
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+        "B": [
+          1.0,
+          0.0
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+    },
+    {
+      "index": 5827,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5828,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5829,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5830,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5830.png",
+      "answer": "-2",
+      "fact_expr": "G: Hyperbola;P: Point;A1: Point;F2: Point;Expression(G) = (x^2 - y^2/3 = 1);LeftVertex(G) = A1;RightFocus(G)=F2;PointOnCurve(P,RightPart(G))",
+      "coords": {}
+    },
+    {
+      "index": 5831,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5832,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 6.0,
+        "y_coef": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5832.png",
+      "answer": "-4",
+      "fact_expr": "G: Parabola;p: Number;H: Ellipse;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (x^2/6 + y^2/2 = 1);Focus(G) = LeftFocus(H)",
+      "coords": {}
+    },
+    {
+      "index": 5833,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5833.png",
+      "answer": "m-a",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Ellipse;m: Number;n: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b + x^2/a = 1);m > n;n > 0;Expression(H) = (y^2/n + x^2/m = 1);Focus(G)={F1,F2};Focus(H)={F1,F2};Intersection(H,G)=P",
+      "coords": {}
+    },
+    {
+      "index": 5834,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5834.png",
+      "answer": "{y^2=-8*x,x^2=8*y}",
+      "fact_expr": "G: Parabola;H: Line;O: Origin;Expression(H) = (x - y + 2 = 0);Vertex(G) = O;SymmetryAxis(G)=axis;PointOnCurve(Focus(G), H)",
+      "coords": {}
+    },
+    {
+      "index": 5835,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5836,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5836.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "C: Ellipse;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;G: Line;PointOnCurve(F2, G);A: Point;B: Point;Intersection(G, C) = {A, B};AngleOf(A, F1, B) = ApplyUnit(90, degree);M: Circle;IsTangent(M, OverlappingLine(LineSegmentOf(F1, A)));IsTangent(M, OverlappingLine(LineSegmentOf(F1, B)));IsTangent(M, LineOf(A, B));Expression(C)=(x^2/2+y^2=1)",
+      "coords": {}
+    },
+    {
+      "index": 5837,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5838,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5838.png",
+      "answer": "9",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Focus(G) = {F1, F2};Expression(G) = (x^2/4 + y^2/3 = 1);PointOnCurve(P, G)",
+      "coords": {}
+    },
+    {
+      "index": 5839,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5839.png",
+      "answer": "6",
+      "fact_expr": "G: Ellipse;H: Circle;C:Circle;P: Point;Q: Point;R: Point;Expression(G) = (x^2/4 + y^2/3 = 1);Expression(H) = (y^2 + (x + 1)^2 = 1);Expression(C) = ((x-1)^2+y^2=1);PointOnCurve(P,G);PointOnCurve(Q,H);PointOnCurve(R,C)",
+      "coords": {}
+    },
+    {
+      "index": 5840,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5841,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5841.png",
+      "answer": "sqrt(5)/2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;e: Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (2*x + y + 1 = 0);IsPerpendicular(OneOf(Asymptote(G)), H);Eccentricity(G) = e",
+      "coords": {}
+    },
+    {
+      "index": 5842,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5842.png",
+      "answer": "y=pm*(sqrt(3)/3)*x",
+      "fact_expr": "G: Hyperbola;b: Number;H: Ellipse;F1: Point;F2: Point;Expression(G) = (x^2/3 - y^2/b^2 = 1);Expression(H) = (x^2/6 + y^2/2 = 1);Focus(H) = {F1, F2};Focus(G) = {F1, F2}",
+      "coords": {}
+    },
+    {
+      "index": 5843,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5843.png",
+      "answer": "3",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;F: Point;Focus(G) = F;l: Line;Directrix(G) = l;A: Point;Coordinate(A) = (3, y0);y0: Number;PointOnCurve(A, G);L: Line;PointOnCurve(A, L);IsPerpendicular(L, l);FootPoint(L, l) = B;B: Point;C: Point;Coordinate(C) = ((7/2)*p, 0);Intersection(LineSegmentOf(A, F), LineSegmentOf(B, C)) = E;E: Point;Abs(LineSegmentOf(F, E)) = 2*Abs(LineSegmentOf(A, E))",
+      "coords": {}
+    },
+    {
+      "index": 5844,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5844.png",
+      "answer": "5/3",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;F1: Point;F2: Point;P: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;IsDiameter(LineSegmentOf(F1, F2), G);OneOf(Intersection(G,Asymptote(C)))=P;Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 5845,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 5.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5845.png",
+      "answer": "8",
+      "fact_expr": "E: Hyperbola;Expression(E) = (25*x^2 - 16*y^2 = 400)",
+      "coords": {}
+    },
+    {
+      "index": 5846,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5846.png",
+      "answer": "9",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;E: Circle;M: Point;N: Point;F2: Point;F1: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(E) = (x^2 + (y + sqrt(6))^2 = 1);LeftFocus(C) =F1;RightFocus(C)=F2;Length(RealAxis(C)) = 6;Expression(Asymptote(C)) = (y = pm*(x/3));PointOnCurve(M, LeftPart(C));PointOnCurve(N, E)",
+      "coords": {}
+    },
+    {
+      "index": 5847,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.0,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5847.png",
+      "answer": "{(2*x - 3*y = 0), (2*x + 3*y = 0)}",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/4 = 1)",
+      "coords": {}
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+      "index": 5848,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 5849,
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+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5849.png",
+      "answer": "5*sqrt(3)",
+      "fact_expr": "G: Parabola;M: Point;P: Point;Expression(G) = (y^2 = 10*x);Coordinate(M) = (10, 0);PointOnCurve(P, G)",
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+          0.0
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+    {
+      "index": 5850,
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+      "error": null,
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+        "b": 2.4270472526550293,
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+        "converged": true,
+        "iterations": 125
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5850.png",
+      "answer": "y=pm*(1/2)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = sqrt(5)/2",
+      "coords": {}
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+    {
+      "index": 5851,
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+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 3.0,
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+        "x_coef": 25.0,
+        "y_coef": 9.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5851.png",
+      "answer": "8",
+      "fact_expr": "G: Ellipse;H: Line;F2: Point;A: Point;B: Point;F1: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Focus(G) ={F1,F2} ;PointOnCurve(F1, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(F2, A)) + Abs(LineSegmentOf(F2, B)) = 12",
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+    {
+      "index": 5852,
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+      "error": null,
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+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5852.png",
+      "answer": "12+8*sqrt(3)",
+      "fact_expr": "C: Hyperbola;b: Number;A: Point;F1: Point;F2: Point;Expression(C) = (x^2/4 - y^2/b = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(A, C);IsPerpendicular(LineSegmentOf(A, F1), LineSegmentOf(A, F2));AngleOf(A, F1, F2) = ApplyUnit(30, degree)",
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+    {
+      "index": 5853,
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+      "conic_type": "hyperbola",
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+        "a": 8.0,
+        "b": 6.0,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5853.png",
+      "answer": "32/5",
+      "fact_expr": "G: Hyperbola;P: Point;Expression(G) = (x^2/64 - y^2/36 = 1);PointOnCurve(P, G);Distance(P, RightFocus(G)) = 8",
+      "coords": {}
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+    {
+      "index": 5854,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "p": 2.0,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5854.png",
+      "answer": "1",
+      "fact_expr": "l: Line;C: Parabola;A: Point;B: Point;P: Point;E: Point;M: Point;F: Point;Q: Point;Expression(C) = (x^2 = 8*y);Coordinate(P) = (0, -1);Coordinate(E) = (3, 0);Focus(C) = F;PointOnCurve(P,l);Slope(l) = k;Intersection(l, C) = {A, B};MidPoint(LineSegmentOf(A,B))=Q;Distance(Q, xAxis) = 3;PointOnCurve(M, l);k:Number;k>0",
+      "coords": {
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+          -1.0
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+        "E": [
+          3.0,
+          0.0
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+    {
+      "index": 5855,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 5856,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5856.png",
+      "answer": "y^2=-8*x",
+      "fact_expr": "H: Parabola;Expression(H) = (y^2 = 2*(p*x));p: Number;G: Hyperbola;Expression(G) = (x^2/3 - y^2 = 1);Focus(H) = LeftFocus(G)",
+      "coords": {}
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+    {
+      "index": 5857,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 5858,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5858.png",
+      "answer": "3*sqrt(2)/4",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*p*x);p: Number;A: Point;F: Point;B: Point;p>0;Coordinate(A) = (0, 2);Focus(C) = F;PointOnCurve(B, C);MidPoint(LineSegmentOf(A,F))=B",
+      "coords": {
+        "A": [
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+          2.0
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+    },
+    {
+      "index": 5859,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5859.png",
+      "answer": "3",
+      "fact_expr": "C: Parabola;A: Point;B: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(A, C);PointOnCurve(B, C);Abs(LineSegmentOf(A, F)) = 2;Abs(LineSegmentOf(B, F)) = 6;IsChordOf(LineSegmentOf(A, B), C)",
+      "coords": {}
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+    {
+      "index": 5860,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5861,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5861.png",
+      "answer": "y^2/64 - x^2/36 = 1",
+      "fact_expr": "G: Hyperbola;C:Hyperbola;A: Point;Expression(G) = (x^2/9 - y^2/16 = 1);Coordinate(A) = (6, 8*sqrt(2));Asymptote(G) = Asymptote(C);PointOnCurve(A, C)",
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+    {
+      "index": 5862,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_focus_mismatch",
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5862.png",
+      "answer": "y = pm*sqrt(3)*x",
+      "fact_expr": "G: Hyperbola;b: Number;H: Point;Expression(G) = (x^2/4 - y^2/b^2 = 1);Coordinate(H) = (4, 0);RightFocus(G) = H",
+      "coords": {
+        "H": [
+          4.0,
+          0.0
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+      "validation_reasons": [
+        "hyperbola_focus_mismatch"
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+    {
+      "index": 5863,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5864,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "vertical"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5864.png",
+      "answer": "sqrt(5)/2",
+      "fact_expr": "C1: Parabola;Expression(C1) = (x^2 = 2*p*y);p: Real;p>0;C2: Hyperbola;Expression(C2) = (y^2/9 - x^2/16 = 1);Distance(Focus(C1), Asymptote(C2)) = sqrt(5)/5",
+      "coords": {}
+    },
+    {
+      "index": 5865,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5865.png",
+      "answer": "(12 - sqrt(41))/5",
+      "fact_expr": "C: Ellipse;G: Line;P: Point;Expression(C) = (x^2/2 + y^2 = 1);Expression(G) = (x/3 + y/4 = 1);PointOnCurve(P, C)",
+      "coords": {}
+    },
+    {
+      "index": 5866,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5866.png",
+      "answer": "(1/2, pm*sqrt(2))",
+      "fact_expr": "G: Parabola;O: Origin;Vertex(G) = O;F: Point;Coordinate(F) = (1, 0);Focus(G) = F;H: Line;PointOnCurve(Focus(G), H);A: Point;B: Point;Intersection(H, G) = {A, B};M: Point;MidPoint(LineSegmentOf(A, B)) = M;P: Point;Z: Line;PointOnCurve(M, Z);IsPerpendicular(Z, Directrix(G));Intersection(Z, G) = P;Abs(LineSegmentOf(A, B)) = 6",
+      "coords": {
+        "F": [
+          1.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 5867,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 2.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5867.png",
+      "answer": "1/2",
+      "fact_expr": "G: Hyperbola;P: Point;F: Point;O: Origin;M: Point;Expression(G) = (x^2 - y^2/4 = 1);RightFocus(G) = F;H:Line;PointOnCurve(F,H);IsParallel(H,OneOf(Asymptote(G)));Intersection(H,G)=P;PointOnCurve(M,LineOf(P,F));DotProduct(VectorOf(O, M), VectorOf(P, F)) = 0",
+      "coords": {}
+    },
+    {
+      "index": 5868,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5869,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 4.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5869.png",
+      "answer": "x-2*y-4=0",
+      "fact_expr": "G: Ellipse;M:Point;A: Point;B: Point;Expression(G) = (x^2/16 + y^2/4 = 1);Coordinate(M) = (2, 1);MidPoint(LineSegmentOf(A,B))=M;IsChordOf(LineSegmentOf(A,B),G)",
+      "coords": {
+        "M": [
+          2.0,
+          1.0
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+      }
+    },
+    {
+      "index": 5870,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5870.png",
+      "answer": "pm*2*sqrt(2)/3",
+      "fact_expr": "G: Parabola;H: Line;F: Point;A: Point;B: Point;k:Number;Expression(G) = (y^2 = 8*x);k>0;Expression(H) = (y = k*(x + 2));Intersection(H, G) = {A, B};Focus(G) = F;LineSegmentOf(F, A) = 2*LineSegmentOf(F, B)",
+      "coords": {}
+    },
+    {
+      "index": 5871,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.0,
+        "major_axis": "y",
+        "x_coef": 9.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5871.png",
+      "answer": "3",
+      "fact_expr": "l: Line;C: Ellipse;P: Point;M: Point;N: Point;Expression(C) = (x^2/9 + y^2/16 = 1);PointOnCurve(P, C);Expression(l)=(x-y+1=0);Intersection(l,xAxis)=M;Intersection(l,yAxis)=N",
+      "coords": {}
+    },
+    {
+      "index": 5872,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.0,
+        "major_axis": "y",
+        "x_coef": 4.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5872.png",
+      "answer": "(-3*sqrt(2),3*sqrt(2))",
+      "fact_expr": "l: Line;C: Ellipse;A: Point;B: Point;m: Real;Expression(C) = (x^2/4 + y^2/9 = 1);Expression(l)=(y = m + (3/2)*x);Intersection(C, l) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 5873,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5873.png",
+      "answer": "(\\sqrt{7}-2,1)",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;a > b;b > 0;P:Point;e:Number;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);XCoordinate(P)=a/3;Distance(P,LeftFocus(G))>Distance(P,RightDirectrix(G));Eccentricity(G)=e;PointOnCurve(P,G)",
+      "coords": {}
+    },
+    {
+      "index": 5874,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5874.png",
+      "answer": "{2, 22}",
+      "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/25 - y^2/9 = 1);Focus(G)={F1,F2};PointOnCurve(P,G);Abs(LineSegmentOf(P,F1))=12",
+      "coords": {}
+    },
+    {
+      "index": 5875,
+      "success": false,
+      "error": "Circle from diameter: missing point coordinates"
+    },
+    {
+      "index": 5876,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5877,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5878,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5879,
+      "success": false,
+      "error": "Circle from diameter: missing point coordinates"
+    },
+    {
+      "index": 5880,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "vertical"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5880.png",
+      "answer": "y=pm*(2/3)*x",
+      "fact_expr": "G: Hyperbola;a: Number;a>0;Expression(G) = (-x^2/9 + y^2/a^2 = 1);Coordinate(OneOf(Focus(G))) = (0, sqrt(13))",
+      "coords": {}
+    },
+    {
+      "index": 5881,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5882,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5882.png",
+      "answer": "4*sqrt(13)/13",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Ellipse;F1: Point;P: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2/25 + y^2/9 = 1);Focus(G)={F1,F2};Focus(H)={F1,F2};OneOf(Intersection(G,H))=P;AngleOf(F1, P, F2) = pi/3",
+      "coords": {}
+    },
+    {
+      "index": 5883,
+      "success": true,
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+      "params": {
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+        "direction": "right"
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5883.png",
+      "answer": "0",
+      "fact_expr": "G: Parabola;A: Point;B: Point;P: Point;F:Point;Expression(G) = (y^2 = 4*x);PointOnCurve(P, Directrix(G));IsChordOf(LineSegmentOf(A,B),G);PointOnCurve(F,LineSegmentOf(A,B));Focus(G) = F",
+      "coords": {}
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+      "index": 5884,
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5884.png",
+      "answer": "x-y-1=0",
+      "fact_expr": "l: Line;C: Parabola;M: Point;A: Point;B: Point;F: Point;Expression(C) = (y^2 = 4*x);Coordinate(M) = (-1, 2);Focus(C)=F;PointOnCurve(F, l);Intersection(l, C) = {A, B};AngleOf(A, M, B) = ApplyUnit(90, degree)",
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+          2.0
+        ]
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+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5885.png",
+      "answer": "[\\sqrt{2}/2,1)",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;H: Circle;Expression(H) = (x^2 + y^2 = a^2 + b^2);L:Line;P: Point;A: Point;B: Point;a > b;b > 0;PointOnCurve(P, G);Intersection(L,H)={A,B};Abs(LineSegmentOf(P, A))*Abs(LineSegmentOf(P, B)) = (a^2 - b^2);PointOnCurve(P,L)",
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+    {
+      "index": 5886,
+      "success": true,
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+      "error": null,
+      "params": {
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+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 36.0,
+        "y_coef": 9.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5886.png",
+      "answer": "24",
+      "fact_expr": "G: Ellipse;H: Line;A: Point;F1: Point;B: Point;F2: Point;Expression(G) = (x^2/36 + y^2/9 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2, H);Intersection(H, G) = {A, B}",
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+    {
+      "index": 5887,
+      "success": true,
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+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5887.png",
+      "answer": "(\\sqrt{5}/2,+\\infty)",
+      "fact_expr": "H: Line;Expression(H) = (y = x/2);G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;NumIntersection(H, G) = 2",
+      "coords": {}
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+    {
+      "index": 5888,
+      "success": true,
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+        "b": 1.5,
+        "orientation": "vertical"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5888.png",
+      "answer": "y=pm*((sqrt(3)/2)*x)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/4 + y^2/3 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 5889,
+      "success": true,
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+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5889.png",
+      "answer": "3/4",
+      "fact_expr": "C: Ellipse;A: Point;F1: Point;F2: Point;O: Origin;B: Point;Expression(C) = (x^2/4 + y^2/3 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(A, C);IsPerpendicular(LineSegmentOf(A, F2), LineSegmentOf(F1, F2));Intersection(LineSegmentOf(A, F1), yAxis) = B",
+      "coords": {}
+    },
+    {
+      "index": 5890,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5890.png",
+      "answer": "5",
+      "fact_expr": "G: Parabola;H: Line;B: Point;A: Point;M: Point;x1:Number;x2:Number;y1:Number;y2:Number;Expression(G) = (y^2 = 8*x);PointOnCurve(Focus(G),H);Coordinate(A)=(x1,y1);Coordinate(B)=(x2,y2);Intersection(H, G) = {A,B};x1+x2=6;MidPoint(LineSegmentOf(A,B)) = M",
+      "coords": {}
+    },
+    {
+      "index": 5891,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 3.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5891.png",
+      "answer": "1",
+      "fact_expr": "G: Hyperbola;H: Circle;F: Point;P: Point;M: Point;N: Point;O: Origin;Expression(G) = (x^2/16 - y^2/9 = 1);Expression(H) = (x^2 + y^2 = 16);LeftFocus(G) = F;PointOnCurve(P, RightPart(G));TangentPoint(LineSegmentOf(P,F),H)=N;MidPoint(LineSegmentOf(P,F))=M",
+      "coords": {}
+    },
+    {
+      "index": 5892,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5893,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 5894,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5894.png",
+      "answer": "(1,-1)",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = x);P: Point;PointOnCurve(P, G);A: Point;Coordinate(A) = (3, -1);WhenMin(Distance(P, A) + Distance(P, Focus(G)))",
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+        "A": [
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+          -1.0
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+    {
+      "index": 5895,
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+        "b": 2.23606797749979,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 5.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5895.png",
+      "answer": "10",
+      "fact_expr": "C: Ellipse;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/9 + y^2/5 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C)",
+      "coords": {}
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+    {
+      "index": 5896,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5896.png",
+      "answer": "pm*2*sqrt(2)",
+      "fact_expr": "C: Parabola;G: Line;M: Point;A: Point;B: Point;l: Line;k: Number;Expression(C) = (y^2 = 4*x);Coordinate(M) = (1, 0);Directrix(C) = l;PointOnCurve(M,G);Slope(G)=k;Intersection(l,G)=A;OneOf(Intersection(G, C)) = B;VectorOf(A, M) = 2*VectorOf(M, B)",
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+        "M": [
+          1.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 5897,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 4.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5897.png",
+      "answer": "12",
+      "fact_expr": "G: Ellipse;H: Line;A: Point;B: Point;F2: Point;F1: Point;Expression(G) = (x^2/9 + y^2/4 = 1);Focus(G) = {F1, F2};PointOnCurve(F1, H);Intersection(H, G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 5898,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "vertical"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5898.png",
+      "answer": "sqrt(6)/2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/3 + y^2/6 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 5899,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5899.png",
+      "answer": "2*x^2-2*y^2=1",
+      "fact_expr": "G: Hyperbola;H: Ellipse;O: Origin;Expression(H) = (x^2/2 + y^2 = 1);Center(G) = O;Focus(G) = Focus(H);InterReciprocal(Eccentricity(G), Eccentricity(H))",
+      "coords": {}
+    },
+    {
+      "index": 5900,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5901,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5902,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5902.png",
+      "answer": "6",
+      "fact_expr": "G: Ellipse;H: Line;A: Point;F2: Point;B: Point;F1: Point;Expression(G) = (x^2/25 + y^2/16 = 1);Focus(G)={F1,F2};PointOnCurve(F1, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, F2)) - Abs(LineSegmentOf(B, F1)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 5903,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 4.898979485566356,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5903.png",
+      "answer": "24",
+      "fact_expr": "F1: Point;F2: Point;Focus(G) = {F1, F2};G: Hyperbola;Expression(G) = (x^2 - y^2/24 = 1);P: Point;PointOnCurve(P, G) = True;Abs(LineSegmentOf(P, F1)) + Abs(LineSegmentOf(P, F2)) = 14",
+      "coords": {}
+    },
+    {
+      "index": 5904,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5904.png",
+      "answer": "4*sqrt(3)",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;l: Line;Directrix(G) = l;P: Point;PointOnCurve(P, G);A: Point;IsPerpendicular(LineSegmentOf(P, A), l);FootPoint(LineSegmentOf(P, A), l) = A;Slope(LineOf(A, F)) = -sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 5905,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5905.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;LeftFocus(G) = F;H: Circle;Expression(H) = (x^2 + y^2 = a^2);Z: Line;TangentOfPoint(F, H) = Z;E: Point;TangentPoint(Z, H) = E;P: Point;Intersection(OverlappingLine(LineSegmentOf(F, E)), G) = P;O: Origin;VectorOf(O, E) = (1/2)*(VectorOf(O, F) + VectorOf(O, P))",
+      "coords": {}
+    },
+    {
+      "index": 5906,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5907,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5908,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "vertical"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5908.png",
+      "answer": "17",
+      "fact_expr": "G: Hyperbola;P: Point;F1:Point;F2:Point;Expression(G) = (-x^2/16 + y^2/64 = 1);PointOnCurve(P, G);OneOf(Focus(G))=F1;OneOf(Focus(G))=F2;Negation(F1=F2);Distance(P,F1) = 1",
+      "coords": {}
+    },
+    {
+      "index": 5909,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5909.png",
+      "answer": "3/2",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2/b^2 = 1);b: Number;0 < b;b < 2;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C);Abs(LineSegmentOf(P, F1)) = 3;AngleOf(F1, P, F2) = pi/3",
+      "coords": {}
+    },
+    {
+      "index": 5910,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5910.png",
+      "answer": "sqrt(3)/3",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/b^2 + y^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;UpperFocus(C) = F1;LowerFocus(C) = F2;B1: Point;B2: Point;LeftVertex(C) = B1;RightVertex(C) = B2;P: Point;PointOnCurve(P, C) ;VectorOf(F2, P) = VectorOf(F1, B1)/2",
+      "coords": {}
+    },
+    {
+      "index": 5911,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5912,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5912.png",
+      "answer": "5",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;G: Line;PointOnCurve(F, G);A: Point;B: Point;Intersection(G, C) = {A, B};M: Point;Coordinate(M) = (-1, 1);IsPerpendicular(LineSegmentOf(M, A), LineSegmentOf(M, B));IsChordOf(LineSegmentOf(A, B), C)",
+      "coords": {
+        "M": [
+          -1.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 5913,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 6.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 36.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5913.png",
+      "answer": "Answer missing",
+      "fact_expr": "l: Line;G: Ellipse;Expression(G) = (x^2/36 + y^2/9 = 1);A: Point;B: Point;Intersection(l, G) = {A, B};I: Point;Coordinate(I) = (4, 2);MidPoint(LineSegmentOf(A, B)) = I",
+      "coords": {
+        "I": [
+          4.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 5914,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5914.png",
+      "answer": "3",
+      "fact_expr": "G: Parabola;l1:Line;l2:Line;P:Point;Expression(G) = (y^2 = 4*x);Expression(l1)=(4*x-3*y+11=0);Expression(l2)=(x=-1);PointOnCurve(P,G)",
+      "coords": {}
+    },
+    {
+      "index": 5915,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5915.png",
+      "answer": "x^2=8*y",
+      "fact_expr": "C: Parabola;Expression(C) = (x^2 = 2*(p*y));p: Number;p>0;P: Point;PointOnCurve(P, C);Distance(P, Focus(C)) = 8;Distance(P, xAxis) = 6",
+      "coords": {}
+    },
+    {
+      "index": 5916,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5916.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;H: Circle;P: Point;Q:Point;Expression(G) = (y^2 = 4*x);Expression(H) = ((x + 3)^2 + (y - 3)^2 = 1);PointOnCurve(P, G);Distance(P,Directrix(G)) = d1;PointOnCurve(Q, H);Distance(P,Q)=d2;d1:Number;d2:Number",
+      "coords": {}
+    },
+    {
+      "index": 5917,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5918,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 2.8284271247461903,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5918.png",
+      "answer": "y^2/8 - x^2/4 = 1",
+      "fact_expr": "C: Hyperbola;G: Point;C1:Hyperbola;Expression(C) = (x^2/4 - y^2/8 = 1);Coordinate(G) = (2, 4);PointOnCurve(G, C1);Asymptote(C1)=Asymptote(C)",
+      "coords": {
+        "G": [
+          2.0,
+          4.0
+        ]
+      }
+    },
+    {
+      "index": 5919,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5919.png",
+      "answer": "5+sqrt(2)",
+      "fact_expr": "N: Parabola;A: Point;B: Point;P: Point;Expression(N) = (x^2 = 8*y);Coordinate(A) = (-1, 3);Coordinate(B) = (0, 2);PointOnCurve(P, N)",
+      "coords": {
+        "A": [
+          -1.0,
+          3.0
+        ],
+        "B": [
+          0.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 5920,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5920.png",
+      "answer": "3",
+      "fact_expr": "C: Parabola;l1: Line;F: Point;l2: Line;A: Point;Expression(C) = (x^2 = 4*y);Expression(l1)=(sqrt(3)*x + y + 5 = 0);Focus(C) = F;Directrix(C) = l2;PointOnCurve(A, C);d1:Number;d2:Number;Distance(A,l1)=d1;Distance(A,l2)=d2",
+      "coords": {}
+    },
+    {
+      "index": 5921,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5921.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "G: Line;Expression(G) = (y = k*x + m);k: Number;k>0;m: Number;C: Parabola;Expression(C) = (y^2 = 4*x);M: Point;Intersection(G, C) = M;N: Point;Intersection(G, Directrix(C)) = N;F: Point;Focus(C) = F;3*VectorOf(F, M) = VectorOf(M, N)",
+      "coords": {}
+    },
+    {
+      "index": 5922,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5923,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5923.png",
+      "answer": "pm*2*sqrt(2)",
+      "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;A: Point;F1: Point;F2: Point;B: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F2, l);Intersection(l, RightPart(C)) = {A, B};Radius(InscribedCircle(TriangleOf(A,F1,F2)))=r1;Radius(InscribedCircle(TriangleOf(B,F1,F2)))=r2;r1:Number;r2:Number;r1 = 2*r2",
+      "coords": {}
+    },
+    {
+      "index": 5924,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5925,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5925.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(C) = F;A: Point;B: Point;Quadrant(A) = 1;Quadrant(B) = 1;PointOnCurve(A, C);PointOnCurve(B, C);O: Origin;AngleOf(A, F, O) = AngleOf(A, F, B);AngleOf(A, F, B) = pi/3;Abs(LineSegmentOf(A, B)) = 2*sqrt(7)",
+      "coords": {}
+    },
+    {
+      "index": 5926,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5927,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5927.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "l: Line;C: Parabola;Expression(C) = (y^2 = x);Intersection(l, C) = {A, B};A: Point;B: Point;l1: Line;Expression(l1) = (y = 1);PointOnCurve(MidPoint(LineSegmentOf(A,B)),l1) = True;DotProduct(VectorOf(O, A), VectorOf(O, B)) = 0;O: Origin",
+      "coords": {}
+    },
+    {
+      "index": 5928,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 2.23606797749979,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5928.png",
+      "answer": "8",
+      "fact_expr": "G: Hyperbola;A: Point;B: Point;Expression(G) = (x^2/4 - y^2/5 = 1);PointOnCurve(A, RightPart(G));PointOnCurve(B, RightPart(G));Distance(MidPoint(LineSegmentOf(A,B)),yAxis)=4",
+      "coords": {}
+    },
+    {
+      "index": 5929,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5929.png",
+      "answer": "(\\sqrt{2}-1,1)",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;e: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) =F1;RightFocus(G)=F2;Abs(LineSegmentOf(P,F1))=e*Abs(LineSegmentOf(P,F2));Eccentricity(G) = e;PointOnCurve(P,G)",
+      "coords": {}
+    },
+    {
+      "index": 5930,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5930.png",
+      "answer": "8*sqrt(3)/5",
+      "fact_expr": "l: Line;C: Ellipse;a: Number;F: Point;A: Point;M: Point;a>sqrt(3);Expression(C) = (y^2/3 + x^2/a^2 = 1);LowerVertex(C) = A;LeftFocus(C) = F;PointOnCurve(A, l);Inclination(l) = ApplyUnit(60, degree);Intersection(l, C) = M;Negation(M=A);Perimeter(TriangleOf(F, A, M)) = 4*a",
+      "coords": {}
+    },
+    {
+      "index": 5931,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5931.png",
+      "answer": "pm*sqrt(2)",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, C) = {A, B};A1: Point;B1: Point;Projection(A, yAxis) = A1;Projection(B, yAxis) = B1;Abs(LineSegmentOf(A, B)) = (3/2)*(Abs(LineSegmentOf(A, A1)) + Abs(LineSegmentOf(B, B1)))",
+      "coords": {}
+    },
+    {
+      "index": 5932,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5933,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5934,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5935,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5936,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5936.png",
+      "answer": "2*sqrt(8-m^2)",
+      "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2/8 + y^2/m^2 = 1);PointOnCurve(Focus(G), xAxis)",
+      "coords": {}
+    },
+    {
+      "index": 5937,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.366584062576294,
+        "b": 2.364173412322998,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.00017838670464698225,
+        "converged": true,
+        "iterations": 89
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5937.png",
+      "answer": "sqrt(3)/2",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Eccentricity(C) = 2;G: Ellipse;Expression(G) = (x^2/5 + y^2 = 1);Focus(C) = Focus(G);P: Point;PointOnCurve(P, C);Z1: Line;Z2: Line;L1: Line;L2: Line;Asymptote(C) = {L1, L2};PointOnCurve(P, Z1);PointOnCurve(P, Z2);IsPerpendicular(Z1, L1);IsPerpendicular(Z2, L2);A: Point;B: Point;FootPoint(Z1, L1) = A;FootPoint(Z2, L2) = B",
+      "coords": {}
+    },
+    {
+      "index": 5938,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5938.png",
+      "answer": "1",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);H: Line;M: Point;m: Number;Coordinate(M) = (0, m);PointOnCurve(M, H);A: Point;B: Point;Intersection(H, G) = {A,B};x1: Number;x2: Number;y1: Number;y2: Number;Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);x1*x2 = -4",
+      "coords": {}
+    },
+    {
+      "index": 5939,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5939.png",
+      "answer": "36",
+      "fact_expr": "E: Parabola;Expression(E) = (y^2 = 4*x);F: Point;Focus(E) = F;l1: Line;l2: Line;PointOnCurve(F, l1) = True;PointOnCurve(F, l2) = True;IsPerpendicular(l1, l2);Intersection(l1, E) = {A, B};Intersection(l2, E) = {C, D};A: Point;B: Point;C: Point;D: Point",
+      "coords": {}
+    },
+    {
+      "index": 5940,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5940.png",
+      "answer": "a - 2*p",
+      "fact_expr": "G: Parabola;p: Number;M: Point;p>0;a: Number;a > p/2;Expression(G) = (x^2 = 2*(p*y));PointOnCurve(M, G);Distance(M, Focus(G)) = a",
+      "coords": {}
+    },
+    {
+      "index": 5941,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5942,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5942.png",
+      "answer": "(1, \\sqrt{2}]",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/4 + x^2/a^2 = 1);a: Number;a>2;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C) = True;IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2)) = True;Gamma: Hyperbola;Expression(Gamma) = (x^2/a^2 - y^2/8 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 5943,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5943.png",
+      "answer": "sqrt(3)/3",
+      "fact_expr": "P: Point;PointOnCurve(P, G);G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;F1: Point;F2: Point;Focus(G) = {F1, F2};WhenMax(AngleOf(F1, P, F2));Cos(AngleOf(F1, P, F2)) = 1/3",
+      "coords": {}
+    },
+    {
+      "index": 5944,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "y",
+        "x_coef": 1.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5944.png",
+      "answer": "1/4",
+      "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2 + y^2/m = 1);PointOnCurve(Focus(G), xAxis);FocalLength(G)=sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 5945,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5945.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "l: Line;C: Parabola;A: Point;F: Point;B: Point;O: Origin;m:Number;Expression(C) = (y^2 = 2*x);Focus(C) = F;Expression(l)=(y=m*(2*x-1));Intersection(l, C) = {A, B};Quadrant(A)=1;Abs(LineSegmentOf(A, F)) = 2*Abs(LineSegmentOf(B, F))",
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+      "answer": "sqrt(5)",
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+      "answer": "y=(pm*1/2)*x",
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+        "note": "using explicit params"
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+      "answer": "2",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5950.png",
+      "answer": "4",
+      "fact_expr": "C: Hyperbola;a: Number;Expression(C) = (-y^2/3 + x^2/a^2 = 1);Expression(OneOf(Asymptote(C))) = (sqrt(3)*x + y = 0)",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+        "y_coef": 3.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5953.png",
+      "answer": "(0,2)",
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+    {
+      "index": 5954,
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+        "b": 2.449489742783178,
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+        "x_coef": 16.0,
+        "y_coef": 6.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5954.png",
+      "answer": "x^2/8 - y^2/2 = 1",
+      "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/16 + y^2/6 = 1);Focus(G) = Focus(H);Expression(Asymptote(G)) = (y = pm*(x/2))",
+      "coords": {}
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+      "index": 5955,
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+      "error": null,
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5955.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
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+        "y_coef": 3.0
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5956.png",
+      "answer": "2*sqrt(5)/5",
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+    {
+      "index": 5957,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5957.png",
+      "answer": "4*p",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;H: Line;PointOnCurve(Focus(G), H);A: Point;B: Point;x1: Number;x2: Number;y1: Number;y2: Number;Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);Intersection(H, G) = {A, B};x1 + x2 = 3*p",
+      "coords": {}
+    },
+    {
+      "index": 5958,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5958.png",
+      "answer": "sqrt(3)/2",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;G: Line;A: Point;B: Point;M: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(M) = (2, 1);PointOnCurve(M, G);Slope(G) = -1/2;Intersection(G, C) = {A, B};MidPoint(LineSegmentOf(A,B)) = M",
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+          1.0
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+      "index": 5959,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "p": 1.0,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5959.png",
+      "answer": "y=-1",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y)",
+      "coords": {}
+    },
+    {
+      "index": 5960,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5960.png",
+      "answer": "y^2 - x^2/3 = 1",
+      "fact_expr": "C: Hyperbola;G: Parabola;Expression(G) = (x^2 = 8*y);Eccentricity(C) = 2;OneOf(Focus(C)) = Focus(G)",
+      "coords": {}
+    },
+    {
+      "index": 5961,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5961.png",
+      "answer": "4",
+      "fact_expr": "C: Parabola;p: Number;G: Circle;H: Line;A: Point;B: Point;M:Point;p>0;Expression(C) = (y^2 = 2*(p*x));PointOnCurve(Focus(C), H);Slope(H) = 2;Intersection(H, C) = {A, B};IsDiameter(LineSegmentOf(A,B),G);Intersection(Directrix(C),G)=M;YCoordinate(M) = 2",
+      "coords": {}
+    },
+    {
+      "index": 5962,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5962.png",
+      "answer": "ApplyUnit(120,degree)",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 + y^2/2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 5963,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5963.png",
+      "answer": "5",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;P: Point;y0: Number;Coordinate(P) = (4, y0);PointOnCurve(P, G)",
+      "coords": {}
+    },
+    {
+      "index": 5964,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5964.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "C: Parabola;p: Number;M: Point;O: Origin;F: Point;p>0;y0:Number;Expression(C) = (x^2 = 2*p*y);Coordinate(M) = (2,y0);Focus(C) = F;PointOnCurve(M,C);Abs(LineSegmentOf(M,O))=Abs(LineSegmentOf(M,F))",
+      "coords": {}
+    },
+    {
+      "index": 5965,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5966,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5966.png",
+      "answer": "5/6",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x);F: Point;Focus(G) = F;Z: Line;PointOnCurve(F, Z);A: Point;B: Point;Intersection(Z, G) = {A, B};Abs(LineSegmentOf(A, B)) = 25/12;Abs(LineSegmentOf(A, F))<Abs(LineSegmentOf(B, F))",
+      "coords": {}
+    },
+    {
+      "index": 5967,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5967.png",
+      "answer": "2*sqrt(5)",
+      "fact_expr": "G: Hyperbola;b: Number;H: Parabola;Expression(G) = (x^2/4 - y^2/b^2 = 1);Expression(H) = (y^2 = 12*x);RightFocus(G) = Focus(H)",
+      "coords": {}
+    },
+    {
+      "index": 5968,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5968.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;A: Point;B: Point;M: Point;l1: Line;l2:Line;Expression(G) = (y^2 = 4*x);Coordinate(M) = (-1, 2);TangentOfPoint(M,G)={l1,l2};TangentPoint(l1,G)=A;TangentPoint(l2,G)=B",
+      "coords": {
+        "M": [
+          -1.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 5969,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5969.png",
+      "answer": "-sqrt(11)/4",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;A: Point;B: Point;F1: Point;D: Point;F2: Point;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;UpperVertex(G) = D;AngleOf(F1, D, F2) = ApplyUnit(120, degree);PointOnCurve(B, C);PointOnCurve(A, C);Quadrant(A) = 1;Quadrant(B) = 1;Abs(LineSegmentOf(A, F2)) = 2;Abs(LineSegmentOf(B, F2)) = 4;Abs(LineSegmentOf(A, B)) = 3",
+      "coords": {}
+    },
+    {
+      "index": 5970,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_5970.png",
+      "answer": "sqrt(2)-1",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;M: Point;F2: Point;F1: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1,H);Inclination(H)=ApplyUnit(45,degree);OneOf(Intersection(H,G))=M;IsPerpendicular(LineSegmentOf(M,F2),xAxis)",
+      "coords": {}
+    },
+    {
+      "index": 5971,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 5972,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5972.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;H: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y = 2*x);Coordinate(H) = (a, b);PointOnCurve(H, G)",
+      "coords": {}
+    },
+    {
+      "index": 5973,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_5973.png",
+      "answer": "4",
+      "fact_expr": "C: Parabola;a: Number;Expression(C) = (y^2 = a*x);Expression(Directrix(C)) = (x = -1)",
+      "coords": {}
+    },
+    {
+      "index": 5974,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_5974.png",
+      "answer": "{sqrt(10), sqrt(10)/3}",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;A: Point;RightVertex(G) = A;m: Line;Slope(m) = 1;PointOnCurve(A, m);Z1: Line;Z2: Line;Asymptote(G) = {Z1, Z2};B: Point;Intersection(m, Z1) = B;C: Point;Intersection(m, Z2) = C;Abs(LineSegmentOf(A, B)) = 2*Abs(LineSegmentOf(A, C))",
+      "coords": {}
+    },
+    {
+      "index": 5975,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6012.png",
+      "answer": "pm*2*sqrt(2)",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*p*x);p: Number;p>0;F: Point;Focus(C) = F;P: Point;Coordinate(P) = (2, m);m: Number;PointOnCurve(P, C);Abs(LineSegmentOf(P, F)) = 3",
+      "coords": {}
+    },
+    {
+      "index": 6013,
+      "success": true,
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+      "error": null,
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6013.png",
+      "answer": "6",
+      "fact_expr": "G: Parabola;p: Number;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(Focus(G)) = (3, 0)",
+      "coords": {}
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+    {
+      "index": 6014,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_focus_mismatch",
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+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6014.png",
+      "answer": "2",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;Coordinate(F) = (2, 0);RightFocus(C) = F;PointOnCurve(F, H);IsPerpendicular(H, Asymptote(C));H: Line;G: Circle;Expression(G) = (4*x + x^2 + y^2 = 0);IsTangent(H, G)",
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+    {
+      "index": 6015,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 4.0,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6015.png",
+      "answer": "y^2/16 - x^2/9 = 1",
+      "fact_expr": "G1: Hyperbola;Expression(G1) = (x^2/9 - y^2/16 = 1);G2: Hyperbola;Asymptote(G1) = Asymptote(G2);A: Point;Coordinate(A) = (-3, 4*sqrt(2));PointOnCurve(A, G2) = True",
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+    {
+      "index": 6016,
+      "success": true,
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+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6016.png",
+      "answer": "8",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;O: Origin;D: Point;E: Point;a>0;b>0;L1:Line;L2:Line;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x = a);Asymptote(C)={L1,L2};Intersection(G,L1)=D;Intersection(G,L2)=E;Area(TriangleOf(O, D, E)) = 8",
+      "coords": {}
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+    {
+      "index": 6017,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.0,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6017.png",
+      "answer": "1",
+      "fact_expr": "A: Point;B: Point;G: Hyperbola;H: Parabola;Expression(G) = (x^2/4 - y^2 = 1);Expression(H) = (y^2 = 4*x);l:Line;Directrix(H)=l;L1:Line;L2:Line;Asymptote(G)={L1,L2};Intersection(L1,l)=A;Intersection(L2,l)=B",
+      "coords": {}
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+    {
+      "index": 6018,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6018.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x)",
+      "coords": {}
+    },
+    {
+      "index": 6019,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.4142135623730951,
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6019.png",
+      "answer": "4*x-y-7=0",
+      "fact_expr": "l: Line;G: Hyperbola;A: Point;B: Point;M: Point;Expression(G) = (x^2 - y^2/2 = 1);Coordinate(M) = (2, 1);PointOnCurve(M, l);Intersection(l, G) = {A, B};MidPoint(LineSegmentOf(A, B)) = M",
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+    {
+      "index": 6020,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 6021,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "x_coef": 25.0,
+        "y_coef": 15.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6021.png",
+      "answer": "x^2/10 - y^2/15 = 1",
+      "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/25 + y^2/15 = 1);Vertex(H)=Focus(G);Focus(H)=Vertex(G)",
+      "coords": {}
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+    {
+      "index": 6022,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
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+        "radius": 2.23606797749979
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_6022.png",
+      "answer": "x^2/12+y^2/3=1",
+      "fact_expr": "Gamma:Ellipse;C: Circle;A: Point;B: Point;b:Number;Expression(Gamma) = (x^2+4*y^2=4*b^2);Expression(C) = ((x + 2)^2 + (y - 1)^2 = 5/2);Intersection(Gamma, C)={A,B};IsDiameter(LineSegmentOf(A, B), C)",
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+    {
+      "index": 6023,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 6024,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6024.png",
+      "answer": "3/4",
+      "fact_expr": "C1: Ellipse;Expression(C1) = (x^2/a1^2+y^2/b1^2=1);a1: Number;b1: Number;a1>b1;b1>0;F1: Point;F2: Point;C2: Hyperbola;Expression(C2) = (x^2/a2^2-y^2/b2^2=1);a2: Number;b2: Number;a2>0;b2>0;Focus(C1) = {F1, F2};Focus(C2) = {F1, F2};M: Point;Quadrant(M) = 1;Intersection(C1, C2) = M;AngleOf(F1, M, F2) = ApplyUnit(90, degree);e1: Number;e2: Number;Eccentricity(C1) = e1;Eccentricity(C2) = e2;e2 = (3*sqrt(2))/2",
+      "coords": {}
+    },
+    {
+      "index": 6025,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6025.png",
+      "answer": "sqrt(37)/5",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;l: Line;PointOnCurve(F2, l);A: Point;B: Point;Intersection(l, RightPart(C)) = {A, B};DotProduct(VectorOf(A, B), VectorOf(A, F1)) = 0;12*Abs(VectorOf(A, B)) = 5*Abs(VectorOf(A, F1))",
+      "coords": {}
+    },
+    {
+      "index": 6026,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6026.png",
+      "answer": "4/5",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;H: Line;PointOnCurve(F, H);Inclination(H) = pi/4;A: Point;B: Point;Intersection(H, G) = {A, B};IsChordOf(LineSegmentOf(A, B), G);I: Point;Coordinate(I) = (0, 2);PointOnCurve(I, PerpendicularBisector(LineSegmentOf(A, B)))",
+      "coords": {
+        "I": [
+          0.0,
+          2.0
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+      }
+    },
+    {
+      "index": 6027,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6027.png",
+      "answer": "2",
+      "fact_expr": "l: Line;C:Parabola;p: Number;A: Point;B: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Expression(l) = (y = b + 2*x);PointOnCurve(Focus(C), l);Intersection(l,C)={A,B};Abs(LineSegmentOf(A, B)) = 5;b:Number",
+      "coords": {}
+    },
+    {
+      "index": 6028,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 6029,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6029.png",
+      "answer": "1",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2 + x^2/m = 1);H: Hyperbola;Expression(H) = (-y^2 + x^2/n = 1);m: Number;n: Number;m>1;n>0;F1: Point;F2: Point;Focus(G) = {F1, F2};Focus(H) = {F1, F2};P: Point;OneOf(Intersection(G, H)) = P",
+      "coords": {}
+    },
+    {
+      "index": 6030,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 2.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6030.png",
+      "answer": "2\n4*sqrt(3)",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 + y^2/2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 6031,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.125,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6031.png",
+      "answer": "(1, 2)",
+      "fact_expr": "P: Point;PointOnCurve(P, G);G: Parabola;Expression(G) = (y = 2*x^2);l: Line;Expression(l) = (4*x - y - 6 = 0);WhenMin(Distance(P, l))",
+      "coords": {}
+    },
+    {
+      "index": 6032,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 6033,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6033.png",
+      "answer": "6",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;O: Origin;M: Point;F2: Point;Expression(G) = (x^2/25 + y^2/16 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);MidPoint(LineSegmentOf(P,F1)) = M;Abs(LineSegmentOf(O, M)) = 2",
+      "coords": {}
+    },
+    {
+      "index": 6034,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 6035,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 6036,
+      "success": false,
+      "error": "Circle from diameter: missing point coordinates"
+    },
+    {
+      "index": 6037,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
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+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6037.png",
+      "answer": "12",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;l: Line;PointOnCurve(Focus(C), l);IsPerpendicular(l, SymmetryAxis(C));A: Point;B: Point;Intersection(l, C) = {A, B};P: Point;PointOnCurve(P, Directrix(C));Area(TriangleOf(A, B, P)) = 36;H: LineSegment;PointOnCurve(Focus(C), H);IsChordOf(H, C)",
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+    },
+    {
+      "index": 6038,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6038.png",
+      "answer": "(x-2)^2+(y-1)^2=4",
+      "fact_expr": "l: Line;C: Parabola;G: Circle;A: Point;Expression(C) = (x^2 = 4*y);Expression(l)=(y = b + x) ;TangentPoint(l,C)=A;Center(G)=A;b:Number;IsTangent(Directrix(C),G)",
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+    {
+      "index": 6039,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6039.png",
+      "answer": "y^2=2*x",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(C) = F;l: Line;Directrix(C) = l;G: Line;Inclination(G) = ApplyUnit(120, degree);PointOnCurve(F, G);A: Point;Intersection(G, l) = A;B: Point;Intersection(LineSegmentOf(A, F), C) = B;Abs(LineSegmentOf(A, B)) = 4/3",
+      "coords": {}
+    },
+    {
+      "index": 6040,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6040.png",
+      "answer": "6",
+      "fact_expr": "G: Parabola;C: Circle;M: Point;A: Point;F: Point;Expression(G) = (y^2 = 4*x);Expression(C) = ((x - 6)^2 + (y + 1)^2 = 1);PointOnCurve(M, G);PointOnCurve(A, C);Focus(G)=F",
+      "coords": {}
+    },
+    {
+      "index": 6041,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6041.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;M: Point;A: Point;F: Point;C: Curve;Expression(G)=(y = x^2/4);PointOnCurve(M, G);Focus(G) = F;Expression(C)=((x - 1)^2 + (y - 4)^2 = 1);PointOnCurve(A, C)",
+      "coords": {}
+    },
+    {
+      "index": 6042,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 6043,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6043.png",
+      "answer": "16",
+      "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 - y^2/16 = 1);Focus(G) = {F1,F2};PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 6044,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.2135674953460693,
+        "b": 2.4270472526550293,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 4.453341261978494e-06,
+        "converged": true,
+        "iterations": 125
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6044.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/4 + x^2/a^2 = 1);a: Number;a>0;Eccentricity(G) = sqrt(5)/2",
+      "coords": {}
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+      "index": 6045,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6045.png",
+      "answer": "(-3/2, 3/2)",
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+      "index": 6046,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6046.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
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+      "index": 6047,
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+      "answer": "2",
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+      "index": 6048,
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+      "error": null,
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6048.png",
+      "answer": "x=-2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x)",
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+    },
+    {
+      "index": 6049,
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+      "error": null,
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+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6049.png",
+      "answer": "sqrt(5)",
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+    {
+      "index": 6050,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 6051,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6051.png",
+      "answer": "11",
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+    {
+      "index": 6052,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 2.30381441116333,
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+        "iterations": 122
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6052.png",
+      "answer": "x^2 - y^2 / 3 = 1",
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+      "index": 6053,
+      "success": true,
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+      "error": null,
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6053.png",
+      "answer": "sqrt(2)/2",
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+    {
+      "index": 6054,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 6055,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6055.png",
+      "answer": "sqrt(6)/3",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y/b + x/a = 1);Focus(G)={F1,F2};Distance(F1,H)+Distance(F2,H)=sqrt(3)*b;Eccentricity(G)=e;e:Number;F1: Point;F2: Point",
+      "coords": {}
+    },
+    {
+      "index": 6056,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 2.0,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6056.png",
+      "answer": "8*sqrt(3)/3",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2/4 = 1);L: LineSegment;IsChordOf(L,G) = True;PointOnCurve(Focus(G),L) = True;IsPerpendicular(L,xAxis) = True",
+      "coords": {}
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+    {
+      "index": 6057,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 2.23606797749979,
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+        "x_coef": 9.0,
+        "y_coef": 5.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6057.png",
+      "answer": "15",
+      "fact_expr": "A: Point;G: Ellipse;Expression(G) = (x^2/9 + y^2/5 = 1);PointOnCurve(A, G);M: Point;N: Point;H: Circle;Expression(H) = (y^2 + (x - 1)^2 = 1);IsDiameter(LineSegmentOf(M, N), H) = True",
+      "coords": {}
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+    {
+      "index": 6058,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "up"
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+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6058.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "l: Line;C: Parabola;p: Number;A: Point;O: Origin;B: Point;p>0;Expression(C) = (x^2 = 2*p*y);Inclination(l)=pi/4;PointOnCurve(F,l);Intersection(l,C)={A,B};Abs(LineSegmentOf(A, B)) = 8;F:Point;Focus(C)=F",
+      "coords": {}
+    },
+    {
+      "index": 6059,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6059.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "G: Curve;A: Point;P: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (-1, 2);PointOnCurve(P, G);l:Line;Expression(l)=(x=-1)",
+      "coords": {
+        "A": [
+          -1.0,
+          2.0
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+    {
+      "index": 6060,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 6061,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6061.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(G) = F;FocalLength(G) = 8;A: Point;LeftVertex(G) = A;B: Point;Coordinate(B) = (0, b);PointOnCurve(B, yAxis) = True;DotProduct(VectorOf(B, A), VectorOf(B, F)) = 2*a",
+      "coords": {}
+    },
+    {
+      "index": 6062,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6062.png",
+      "answer": "32",
+      "fact_expr": "G: Parabola;H: Line;P: Point;A: Point;B: Point;Expression(G) = (y^2 = 4*x);Coordinate(P) = (4, 0);PointOnCurve(P, H);x1: Number;y1: Number;x2: Number;y2: Number ;Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);Intersection(H,G)={A,B}",
+      "coords": {
+        "P": [
+          4.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 6063,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6063.png",
+      "answer": "11",
+      "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 - y^2/16 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 5",
+      "coords": {}
+    },
+    {
+      "index": 6064,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 4.898979485566356,
+        "major_axis": "x",
+        "x_coef": 49.0,
+        "y_coef": 24.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6064.png",
+      "answer": "24",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/49 + y^2/24 = 1);P: Point;PointOnCurve(P,G) = True;F1: Point;F2: Point;Focus(G) = {F1,F2}\t;IsPerpendicular(LineSegmentOf(P,F1),LineSegmentOf(P,F2)) = True",
+      "coords": {}
+    },
+    {
+      "index": 6065,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6065.png",
+      "answer": "2",
+      "fact_expr": "C: Parabola;A: Point;M: Point;F: Point;l:Line;Expression(C) = (y = x^2/4);Focus(C) = F;Directrix(C)=l;Intersection(l, yAxis) = A;PointOnCurve(M, C);Abs(LineSegmentOf(M,A))/Abs(LineSegmentOf(M,F))=sqrt(2)",
+      "coords": {}
+    },
+    {
+      "index": 6066,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 6067,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6067.png",
+      "answer": "3",
+      "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;C: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B};Intersection(l,Directrix(G))=C;VectorOf(A, B) = 2*VectorOf(B, C)",
+      "coords": {}
+    },
+    {
+      "index": 6068,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6068.png",
+      "answer": "2*sqrt(6)/3",
+      "fact_expr": "G: Ellipse;P: Point;Q: Point;Expression(G) = (x^2/16 + y^2/4 = 1);Coordinate(Q) = (2, 0);PointOnCurve(P, G)",
+      "coords": {
+        "Q": [
+          2.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 6069,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6069.png",
+      "answer": "{(x+y-2=0),(x-y-2=0)}",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);F: Point;Focus(G) = F;P: Point;PointOnCurve(P, Directrix(G)) = True;OneOf(Intersection(LineOf(P, F), G)) = Q;Q: Point;VectorOf(P, Q) = sqrt(2)*VectorOf(Q, F)",
+      "coords": {}
+    },
+    {
+      "index": 6070,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6070.png",
+      "answer": "5*sqrt(2)",
+      "fact_expr": "G: Parabola;H: Line;P: Point;A: Point;B: Point;Expression(G) = (y^2 = x);Coordinate(P) = (2, 2);PointOnCurve(P, H);Slope(H) = -1;Intersection(H, G) = {A, B}",
+      "coords": {
+        "P": [
+          2.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 6071,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 6072,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6072.png",
+      "answer": "16",
+      "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/25 + y^2/16 = 1);PointOnCurve(P, G);Focus(G)={F1,F2};AngleOf(F1, P, F2) = ApplyUnit(90, degree)",
+      "coords": {}
+    },
+    {
+      "index": 6073,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6073.png",
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+      "answer": "2*x-3*y-2*sqrt(5)=0",
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+      "answer": "[5,21]",
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+      "answer": "sqrt(6)/2",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6097.png",
+      "answer": "{x^2/75+y^2/50=1,y^2/75+x^2/50=1}",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6098.png",
+      "answer": "x^2/5+y^2/4=1",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;H: Circle;A: Point;B: Point;J: Point;L1:Line;L2:Line;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2 + y^2 = 1);Coordinate(J) = (1, 1/2);TangentOfPoint(J, H)={L1,L2};TangentPoint(L1,H)=A;TangentPoint(L2,H)=B;PointOnCurve(RightFocus(G),LineOf(A,B));PointOnCurve(UpperVertex(G),LineOf(A,B))",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6099.png",
+      "answer": "2*sqrt(5)",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6100.png",
+      "answer": "6*sqrt(5)/5-1",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6101.png",
+      "answer": "{x^2/4+y^2/3=1,x^2/4+3*y^2/16=1}",
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+      "index": 6102,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 6103,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6103.png",
+      "answer": "4*sqrt(3)",
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+      "answer": "sqrt(2)/2",
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+      "answer": "(x-2)^2+(y-5/3)^2=25/9",
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+      "answer": "(-1,0)+(0,1)",
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+      "answer": "3",
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+      "answer": "1-(\\sqrt{3}/3)",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6109.png",
+      "answer": "sqrt(6)/2",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2/4 - y^2/2 = 1)",
+      "coords": {}
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+    {
+      "index": 6110,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6110.png",
+      "answer": "2*sqrt(5)",
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+      "index": 6111,
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+      "index": 6112,
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+      "answer": "12",
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+      "index": 6113,
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+      "error": "Unsupported or unparseable expression"
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+      "error": "validation: hyperbola_focus_mismatch",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6114.png",
+      "answer": "3/2",
+      "fact_expr": "G: Hyperbola;a: Number;H: Point;Expression(G) = (-y^2/5 + x^2/a^2 = 1);Coordinate(H) = (3, 0);RightFocus(G) = H",
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+    {
+      "index": 6115,
+      "success": true,
+      "conic_type": "parabola",
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+        "direction": "up"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6115.png",
+      "answer": "(0,-1/8)",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = y/2)",
+      "coords": {}
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+      "index": 6116,
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+        "b": 3.0,
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+        "x_coef": 16.0,
+        "y_coef": 9.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6116.png",
+      "answer": "5",
+      "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/16 + y^2/9 = 1);Coordinate(P) = (x1, y1);x1:Number;y1:Number;PointOnCurve(P, G)",
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+    {
+      "index": 6117,
+      "success": true,
+      "conic_type": "ellipse",
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+        "y_coef": 1.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6117.png",
+      "answer": "y = x + pm*1",
+      "fact_expr": "l: Line;C: Ellipse;A: Point;B: Point;Expression(C) = (x^2/3 + y^2 = 1);Slope(l) = 1;Intersection(l, C) = {A, B};Abs(LineSegmentOf(A, B)) = 3*sqrt(2)/2",
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+    {
+      "index": 6118,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
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+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6118.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;A1: Point;A2: Point;LeftVertex(C) = A1;RightVertex(C) = A2;M: Point;PointOnCurve(M, C);Negation(M = A1);Negation(M = A2);Slope(LineOf(M, A1)) * Slope(LineOf(M, A2)) = -1/2",
+      "coords": {}
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+    {
+      "index": 6119,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6119.png",
+      "answer": "3/2",
+      "fact_expr": "G: Parabola;Q: Point;M: Point;P: Point;O:Origin;Expression(G) = (y^2 = x);PointOnCurve(P,G);Negation(P=O);IsPerpendicular(LineSegmentOf(P,Q),xAxis);FootPoint(LineSegmentOf(P,Q),xAxis)=Q;L:Line;PointOnCurve(MidPoint(LineSegmentOf(P,Q)),L);IsParallel(L,xAxis);Intersection(L,G)=M;Intersection(LineOf(Q,M),yAxis)=N;N:Point",
+      "coords": {}
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+    {
+      "index": 6120,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6120.png",
+      "answer": "sqrt(5)/2",
+      "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;N: Point;M: Point;a > b;b > 0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(l)=(x-3*y+c=0);l1:Line;l2:Line;Asymptote(C)={l1,l2};Intersection(l,l1)=M;Intersection(l,l2)=N;PointOnCurve(RightFocus(C),PerpendicularBisector(LineSegmentOf(M,N)));Coordinate(RightFocus(C)) = (c, 0);c:Number",
+      "coords": {}
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+    {
+      "index": 6121,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6121.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;A: Point;F1: Point;O: Origin;B: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(A, Asymptote(G));IsPerpendicular(LineSegmentOf(A, F1), LineSegmentOf(A, O));Intersection(LineSegmentOf(A, F1), G) = B;Abs(LineSegmentOf(A, B)) = Abs(LineSegmentOf(B, F1))",
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+    {
+      "index": 6122,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
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+          0.0
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_6122.png",
+      "answer": "10",
+      "fact_expr": "P: Point;C3: Curve;Expression(C3) = (x^2/16 - y^2/9 = 1);PointOnCurve(P, C3);Q: Point;C1: Curve;Expression(C1) = (y^2 + (x + 5)^2 = 1);PointOnCurve(Q, C1);R: Point;C2: Curve;Expression(C2) = (y^2 + (x - 5)^2 = 1);PointOnCurve(R, C2)",
+      "coords": {}
+    },
+    {
+      "index": 6123,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6123.png",
+      "answer": "(1,2)",
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+      "coords": {}
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+    {
+      "index": 6124,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6124.png",
+      "answer": "[sqrt(5)/5, 1/2]",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;c: Number;F1: Point;F2: Point;M: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);Focus(G) = {F1, F2};PointOnCurve(M, G);DotProduct(VectorOf(F1, M), VectorOf(F2, M)) = 3*c^2",
+      "coords": {}
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+    {
+      "index": 6125,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6125.png",
+      "answer": "6",
+      "fact_expr": "G: Parabola;p: Number;H: Ellipse;p>0;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (x^2/25 + y^2/16 = 1);PointOnCurve(LeftFocus(H), Directrix(G))",
+      "coords": {}
+    },
+    {
+      "index": 6126,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6126.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;P: Point;PointOnCurve(P, G) = True;AngleOf(F1, P, F2) = ApplyUnit(90, degree)",
+      "coords": {}
+    },
+    {
+      "index": 6127,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 6128,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6128.png",
+      "answer": "y**2=16*x",
+      "fact_expr": "G: Parabola;H: Line;O: Origin;Expression(H) = (3*x - 4*y - 12 = 0);Vertex(G) = O;SymmetryAxis(G)=xAxis;PointOnCurve(Focus(G),H)",
+      "coords": {}
+    },
+    {
+      "index": 6129,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 6130,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6130.png",
+      "answer": "[sqrt(6)/2, 2*sqrt(10)/5]",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;F: Point;B: Point;A: Point;l1: Line;l2: Line;l: Line;lambda: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;Asymptote(C) = {l1, l2};PointOnCurve(F, l);IsPerpendicular(l, l1);FootPoint(l, l1) = A;Intersection(l, l2) = B;VectorOf(F, B) = lambda*VectorOf(A, F);3 <= lambda;lambda <= 4",
+      "coords": {}
+    },
+    {
+      "index": 6131,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+      "answer": "pm*(2/5)",
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+      "answer": "sqrt(3)",
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+      "answer": "x^2 - y^2 / 3 = 1",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;c:Number;Q: Point;P: Point;F1: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(Q) = (0, sqrt(3)*c);Eccentricity(C) = 2;LeftFocus(C) = F1;HalfFocalLength(C)=c;PointOnCurve(P, RightPart(C));Min(Abs(LineSegmentOf(P, F1)) + Abs(LineSegmentOf(P, Q))) = 6",
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+      "answer": "x^2/18+y^2/9=1",
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+        "direction": "up"
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+      "answer": "y=-2",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 8*y)",
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+      "index": 6140,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6140.png",
+      "answer": "5/2",
+      "fact_expr": "C: Parabola;Expression(C) = (x^2 = 2*y);F: Point;Focus(C) = F;Slope(l) = 1/2;PointOnCurve(F, l) = True;l: Line;Intersection(l, C) = {A, B};A: Point;B: Point",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 6143,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 6144,
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+      "conic_type": "hyperbola",
+      "error": null,
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+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6145.png",
+      "answer": "2",
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+      "index": 6146,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 6147,
+      "success": true,
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6147.png",
+      "answer": "(3,pm*2*sqrt(3))",
+      "fact_expr": "C: Parabola;P: Point;N: Point;F: Point;Q: Point;Expression(C) = (y^2 = 4*x);PointOnCurve(P, C);Focus(C) = F;Intersection(Directrix(C), xAxis) = N;Intersection(LineOf(N,P),C)=Q;Abs(LineSegmentOf(P, F)) = 3*Abs(LineSegmentOf(Q, F))",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6148.png",
+      "answer": "x^2/36 - y^2/12 = 1",
+      "fact_expr": "C: Hyperbola;G: Ellipse;H: Line;Expression(G) = (x^2 + 4*y^2 = 64);Expression(H) = (x + sqrt(3)*y = 0);Focus(C) = Focus(G);OneOf(Asymptote(C))=H",
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+    {
+      "index": 6149,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6149.png",
+      "answer": "4/5",
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+        "y_coef": 3.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6150.png",
+      "answer": "(1,4)",
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+    {
+      "index": 6151,
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+      "conic_type": "parabola",
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+        "direction": "right"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6151.png",
+      "answer": "(x-2)^2 + (y - 2\\sqrt{2})^2 = 9",
+      "fact_expr": "C: Parabola;G: Circle;A: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(A,C);Quadrant(A)=1;Abs(LineSegmentOf(A, F)) = 3;Center(G)=A;Radius(G)=LineSegmentOf(A,F)",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6152.png",
+      "answer": "(a^2/c,0)",
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+    {
+      "index": 6153,
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+      "conic_type": "ellipse",
+      "error": null,
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+        "x_coef": 2.0,
+        "y_coef": 1.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6153.png",
+      "answer": "-1/3",
+      "fact_expr": "C: Ellipse;G: Line;O: Origin;A: Point;B: Point;Expression(C) = (x^2/2 + y^2 = 1);Slope(G)=1;PointOnCurve(LeftFocus(C),G);Intersection(G,C)={A,B}",
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+    {
+      "index": 6154,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 6155,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "major_axis": "x",
+        "x_coef": 5.0,
+        "y_coef": 4.0
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6155.png",
+      "answer": "5/3",
+      "fact_expr": "G: Ellipse;H: Line;O: Origin;A: Point;B: Point;Expression(G) = (x^2/5 + y^2/4 = 1);PointOnCurve(RightFocus(G), H);Slope(H)=2;Intersection(H,G) = {B, A}",
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+    {
+      "index": 6156,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 3.0,
+        "y_coef": 1.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6156.png",
+      "answer": "(0, pm*1)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/3 + y^2 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;A: Point;B: Point;PointOnCurve(A, G);PointOnCurve(B, G);VectorOf(F1, A) = 5*VectorOf(F2, B)",
+      "coords": {}
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+    {
+      "index": 6157,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6157.png",
+      "answer": "3",
+      "fact_expr": "l: Line;G: Parabola;O: Origin;P: Point;A: Point;B: Point;m:Number;n:Number;m<=n;Expression(G) = (y^2 = 4*x);Expression(l) = (sqrt(3)*x - y - sqrt(3) = 0);Intersection(l, G) = {A, B};Intersection(l,xAxis)=P;VectorOf(O, P) = m*VectorOf(O, A) + n*VectorOf(O, B)",
+      "coords": {}
+    },
+    {
+      "index": 6158,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6158.png",
+      "answer": "0",
+      "fact_expr": "G: Hyperbola;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 - y^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);Area(TriangleOf(F1, P, F2)) = 1",
+      "coords": {}
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+    {
+      "index": 6159,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 6160,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6160.png",
+      "answer": "2",
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+      "answer": "y=pm*x",
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+      "answer": "16/5",
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+      "answer": "{sqrt(5),sqrt(5)/2}",
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+      "answer": "y^2/12 - x^2/3 = 1",
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+      "answer": "25",
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+      "answer": "sqrt(5)",
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+      "success": true,
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+        "b": 1.5,
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+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
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+      "success": true,
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+      "answer": "6",
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+    {
+      "index": 6177,
+      "success": true,
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+        "b": 0.5,
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+        "y_coef": 1.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6177.png",
+      "answer": "1-sqrt(3)/2",
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+        "direction": "up"
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+      "answer": "{1, 2}",
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+      "success": true,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6179.png",
+      "answer": "sqrt(3)/3",
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+      "success": true,
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+      "error": null,
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6180.png",
+      "answer": "4",
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+    {
+      "index": 6181,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "vertical"
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+        "converged": true,
+        "note": "using explicit params"
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+      "answer": "17",
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+    {
+      "index": 6182,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "success": true,
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+      "error": null,
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+        "direction": "up"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6183.png",
+      "answer": "5/2",
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+      "index": 6184,
+      "success": true,
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+        "b": 1.1157990585381212,
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+        "x_coef": 2.25,
+        "y_coef": 1.2500000000000002
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+      "optimization": {
+        "final_loss": 1.232595164407831e-31,
+        "converged": true,
+        "iterations": 2
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6184.png",
+      "answer": "6",
+      "fact_expr": "G: Ellipse;H: Line;F1: Point;F2: Point;A: Point;B: Point;Coordinate(F1) = (-1, 0);Coordinate(F2) = (1, 0);Focus(G) = {F1, F2};Eccentricity(G) = 2/3;PointOnCurve(F1, H);Intersection(H, G) = {A, B}",
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+          0.0
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+        "F2": [
+          1.0,
+          0.0
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+    },
+    {
+      "index": 6185,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "p": 0.5,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6185.png",
+      "answer": "(3/2, sqrt(3))",
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+    {
+      "index": 6186,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6186.png",
+      "answer": "32/3",
+      "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;Expression(G) = (y^2 = 8*x);PointOnCurve(Focus(G), H);Inclination(H)=pi/3;Intersection(H, G) = {A, B}",
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+    {
+      "index": 6187,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "a": 1.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6187.png",
+      "answer": "6",
+      "fact_expr": "G: Hyperbola;H: Line;B: Point;A: Point;P: Point;Expression(G) = (x^2 - y^2/3 = 1);Coordinate(P) = (2, 1);PointOnCurve(P, H);Intersection(H, G) = {A, B};MidPoint(LineSegmentOf(A, B)) = P",
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+        "P": [
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+          1.0
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+    },
+    {
+      "index": 6188,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6188.png",
+      "answer": "y^2=4*x",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;G: Ellipse;Expression(G) = (x^2/2 + y^2 = 1);Focus(C) = RightFocus(G)",
+      "coords": {}
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+      "index": 6189,
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+      "answer": "(2,4)",
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+      "answer": "2*sqrt(3)",
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+      "answer": "y^2 = 8*x",
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+        "b": 3.0,
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+      "answer": "3",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/9 = 1)",
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+      "answer": "(-sqrt(3)/3, sqrt(3)/3)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6203.png",
+      "answer": "3*sqrt(26)/13",
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+      "answer": "7/4",
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+      "answer": "y^2=4*x",
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+      "answer": "20",
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+        "y_coef": 36.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6214.png",
+      "answer": "y^2/4 - x^2/5 = 1",
+      "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/27 + y^2/36 = 1);IsIntersect(G, H);Focus(G) = Focus(H);Coordinate(OneOf(Intersection(G, H))) = (sqrt(15), 4)",
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+    {
+      "index": 6215,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 3.4641016151377544,
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6215.png",
+      "answer": "9",
+      "fact_expr": "G: Hyperbola;A: Point;P: Point;F: Point;Expression(G) = (x^2/4 - y^2/12 = 1);Coordinate(A) = (1, 4);LeftFocus(G) = F;PointOnCurve(P, RightPart(G))",
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+      "index": 6216,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 6217,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6217.png",
+      "answer": "sqrt(10)/3",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(OneOf(Asymptote(G)))=(x+3*y=0)",
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+    {
+      "index": 6218,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 6219,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 6220,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_6220.png",
+      "answer": "x^2/4 + y^2/3 = 1",
+      "fact_expr": "G: Ellipse;PointOnCurve(Focus(G), xAxis);Eccentricity(G) = 0.5;C: Circle;Expression(C) = (-2*x + x^2 + y^2 - 15 = 0);Length(MajorAxis(G)) = Radius(C)",
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+    {
+      "index": 6221,
+      "success": true,
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+      "error": null,
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+        "direction": "up"
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+      "answer": "y=pm*x",
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+      "answer": "y=-1",
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+      "answer": "x^2/16 - y^2/9 = 1",
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+      "answer": "x^2+2*y^2-8*x-4*y+17=0",
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+      "answer": "4/5",
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+      "answer": "3*sqrt(6)",
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+      "answer": "1",
+      "fact_expr": "G: Hyperbola;H: Parabola;Expression(G) = (x^2/3 - y^2 = 1);Expression(H) = (y^2 = 8*x)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6250.png",
+      "answer": "6",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6251.png",
+      "answer": "(1, 3]",
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+      "answer": "x^2-4*y^2=1",
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+      "index": 6263,
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+      "answer": "y=-1/4",
+      "fact_expr": "C: Parabola;Expression(C) = (y = x^2)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6265.png",
+      "answer": "(0, 1/2]",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6267.png",
+      "answer": "9",
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+    {
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+      "answer": "x + 2*y - 8 = 0",
+      "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/36 + y^2/9 = 1);Coordinate(P) = (4, 2);Z: LineSegment;MidPoint(Z) = P;IsChordOf(Z, G)",
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+        "direction": "left"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6269.png",
+      "answer": "(-2, -1)",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6270.png",
+      "answer": "5/3",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1)",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6271.png",
+      "answer": "4*sqrt(3)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/10 - y^2/2 = 1)",
+      "coords": {}
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+    {
+      "index": 6272,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6272.png",
+      "answer": "x=-2",
+      "fact_expr": "G: Hyperbola;H: Parabola;p: Number;Expression(G) = (x^2 - y^2/3 = 1);Expression(H) = (y^2 = 2*p*x);Focus(H) = RightFocus(G)",
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+    {
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6273.png",
+      "answer": "1/3",
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+    {
+      "index": 6274,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6274.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;F2: Point;P: Point;F1: Point;e: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(P,LeftPart(G));Abs(LineSegmentOf(P, F1)) = Abs(LineSegmentOf(F1, F2));Distance(F1,LineOf(P,F2)) = sqrt(7)*a;Eccentricity(G) = e",
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+      "index": 6275,
+      "success": false,
+      "error": "Circle from diameter: cannot parse diameter pattern"
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+    {
+      "index": 6276,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 6277,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 6278,
+      "success": true,
+      "conic_type": "parabola",
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6278.png",
+      "answer": "sqrt(2)",
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+    {
+      "index": 6279,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_6279.png",
+      "answer": "x^2 + y^2/(3/4) = 1",
+      "fact_expr": "G: Circle;P: Point;P0: Point;M: Point;Expression(G) = (x^2 + y^2 = 1);PointOnCurve(P, G);PointOnCurve(P,LineSegmentOf(P,P0));IsPerpendicular(xAxis,LineSegmentOf(P,P0));FootPoint(xAxis,LineSegmentOf(P,P0))=P0;VectorOf(M, P0) = (sqrt(3)/2)*VectorOf(P, P0);Locus(M)=C;C:Curve",
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+    {
+      "index": 6280,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 6281,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6281.png",
+      "answer": "b",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1)",
+      "coords": {}
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+    {
+      "index": 6282,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6282.png",
+      "answer": "1/2",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;A: Point;B: Point;F: Point;P:Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F;RightVertex(G)=A;PointOnCurve(B,G);IsPerpendicular(LineSegmentOf(B,F),xAxis);Intersection(LineOf(A,B),yAxis)=P;VectorOf(A,P)=2*VectorOf(P,B)",
+      "coords": {}
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+    {
+      "index": 6283,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6283.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;B: Point;F: Point;A: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);OneOf(Focus(G))=F;OneOf(Endpoint(ImageinaryAxis(G)))=B;Intersection(LineSegmentOf(B,F),OneOf(Asymptote(G)))= A;VectorOf(F, A) = 2*VectorOf(A, B)",
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+    {
+      "index": 6284,
+      "success": true,
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+      "answer": "{(x+2*y-1=0),(x-2*y+1=0)}",
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+      "answer": "y^2 = -12x",
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+      "answer": "x^2/3 - y^2/12 = 1",
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+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/25 - y^2/11 = 1)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6314.png",
+      "answer": "sqrt(5)",
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+      "answer": "-8",
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+      "answer": "(\\sqrt{5}-1)/2",
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+      "answer": "[4*sqrt(2),+oo)",
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+      "answer": "[\\sqrt{2}/2, 1)",
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+        "note": "using explicit params"
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+      "answer": "sqrt(5)",
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+      "answer": "-3",
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+        "note": "using explicit params"
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+      "answer": "3*sqrt(2)",
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+      "answer": "3*sqrt(2)/4",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6327.png",
+      "answer": "sqrt(5)/2",
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+      "success": false,
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+      "answer": "4*x-y-1=0",
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+      "answer": "y^2/4 - x^2/5 = 1",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6333.png",
+      "answer": "y = pm*sqrt(3)*x",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2 - y^2/3 = 1)",
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+    {
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+      "answer": "7",
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+    {
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6335.png",
+      "answer": "{(x-1)^2+y^2=2,(x+1)^2+y^2=2}",
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+    {
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+        "direction": "up"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6336.png",
+      "answer": "{(x=1),(y=4*x-2)}",
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+        "y_coef": 2.0
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+        "note": "using explicit params"
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+      "answer": "3",
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+      "answer": "27*sqrt(13)/13",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6339.png",
+      "answer": "8",
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+      "answer": "sqrt(2)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6343.png",
+      "answer": "y=pm*sqrt(3)*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/12 = 1)",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6344.png",
+      "answer": "sqrt(3)+sqrt(2)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6345.png",
+      "answer": "2*sqrt(13)",
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+      "index": 6346,
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "answer": "8",
+      "fact_expr": "G: Parabola;H: Line;Expression(G) = (y^2 = 4*x);Expression(H) = (y = x - 1)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6347.png",
+      "answer": "x^2/3 - y^2/6 = 1",
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+      "answer": "sqrt(3)-1",
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+      "answer": "26",
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+      "index": 6378,
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+      "answer": "9",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/m = 1);m: Number;Eccentricity(G) = 5/4",
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+      "answer": "12",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6380.png",
+      "answer": "(pm*sqrt(5)/2, 0)",
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+      "answer": "y^2/15 - x^2/12 = 1",
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+      "index": 6401,
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+      "answer": "[-9/20,9/20]",
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+      "answer": "x^2/8 - y^2/2 = 1",
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+      "answer": "4",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/16 = 1)",
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+        "note": "using explicit params"
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+      "answer": "sqrt(3)",
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+      "answer": "8",
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+      "answer": "17/2",
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+      "answer": "{1, 16}",
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+      "index": 6410,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6410.png",
+      "answer": "2*sqrt(3)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6411.png",
+      "answer": "x^2/4+y^2/3=1",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6412.png",
+      "answer": "3\n8\n6/5",
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+      "index": 6413,
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+        "note": "using explicit params"
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+      "answer": "24/7",
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+      "answer": "20",
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+      "answer": "[-1,17/8]",
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+      "answer": "[2*sqrt(2),4)",
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+      "answer": "(1+sqrt(5))/2",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6421.png",
+      "answer": "6-sqrt(2)",
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+      "answer": "y = pm*2*x",
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+      "answer": "7*sqrt(2)/8",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6425.png",
+      "answer": "x^2/4-y^2=1",
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+      "answer": "[0, 12]",
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+        "note": "using explicit params"
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+      "answer": "sqrt(2)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6428.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "G: Hyperbola;a: Number;a>0;Expression(G) = (-y^2 + x^2/a^2 = 1);Expression(OneOf(Asymptote(G))) = (y = sqrt(2)*x)",
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+        "direction": "right"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6429.png",
+      "answer": "16 + pm*8*sqrt(3)",
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6430.png",
+      "answer": "2*sqrt(5)",
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6431.png",
+      "answer": "(-16/5,16/5)",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 6433,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6434.png",
+      "answer": "[1/3, 1)",
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+      "index": 6435,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 6436,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6436.png",
+      "answer": "(6, pm*6*sqrt(2))",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 12*x);P:Point;PointOnCurve(P, G);Distance(P, Focus(G)) = 9",
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6437.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;H: Circle;P: Point;Q: Point;F: Point;Expression(G) = (y^2 = 4*x);Expression(H) = ((x - 2)^2 + (y - 1)^2 = 1);Focus(G) = F;PointOnCurve(P, G);PointOnCurve(Q, H)",
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+        "y_coef": 7.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6438.png",
+      "answer": "[1, 7]",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/7 = 1);P: Point;PointOnCurve(P, G);F: Point;LeftFocus(G) = F",
+      "coords": {}
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+      "index": 6439,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 6440,
+      "success": true,
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+      "error": null,
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+        "direction": "up"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6440.png",
+      "answer": "pi/6",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);F: Point;Focus(G) = F;M: Point;Intersection(Directrix(G), yAxis) = M;N: Point;PointOnCurve(N, G);Abs(LineSegmentOf(N, F)) = (sqrt(3)/2)*Abs(LineSegmentOf(M, N))",
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+    {
+      "index": 6441,
+      "success": true,
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+        "y_coef": 3.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6441.png",
+      "answer": "-1",
+      "fact_expr": "F: Point;RightFocus(G) = F;G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);P: Point;PointOnCurve(P, G) = True;A: Point;PointOnCurve(A, H) = True;H: Line;Expression(H) = (3*x - 4*y - 12 = 0)",
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+    {
+      "index": 6442,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_focus_mismatch",
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+        "b": 1.7320508075688772,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6442.png",
+      "answer": "sqrt(6)/3",
+      "fact_expr": "l: Line;G: Ellipse;b: Number;a: Number;B: Point;A: Point;F: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(MidPoint(LineSegmentOf(A,B))) = (-3, 1);Coordinate(F) = (-4, 0);OneOf(Focus(G)) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B}",
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+      "index": 6443,
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+        "converged": true,
+        "iterations": 2
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6443.png",
+      "answer": "(5,0)",
+      "fact_expr": "G: Hyperbola;a: Number;a>0;Expression(G) = (-y^2/9 + x^2/a^2 = 1);Eccentricity(G) = 5/4",
+      "coords": {}
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+    {
+      "index": 6444,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6444.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2 - y^2/3 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;M: Point;N: Point;PointOnCurve(M, Asymptote(C));PointOnCurve(N, LeftPart(C));WhenMin(Abs(LineSegmentOf(M,N))+Abs(LineSegmentOf(N,F2)))",
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+      "index": 6445,
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+      "conic_type": "parabola",
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+      "answer": "6",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6470.png",
+      "answer": "sqrt(5)/2",
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+      "index": 6471,
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 6474,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6475.png",
+      "answer": "x^2/3 - y^2/5 = 1",
+      "fact_expr": "G: Hyperbola;H: Ellipse;I: Point;Expression(H) = (x^2/8 + y^2/5 = 1);Coordinate(I) = (3, sqrt(10));PointOnCurve(I, G);Endpoint(MajorAxis(H))=Focus(G)",
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+      "index": 6477,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6477.png",
+      "answer": "sqrt(2)",
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+      "index": 6478,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 6479,
+      "success": true,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6479.png",
+      "answer": "4",
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+    {
+      "index": 6480,
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+      "answer": "5/3",
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+      "answer": "x^2/2 - y^2/2 = 1",
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+      "answer": "ApplyUnit(30, degree)",
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+      "answer": "6+3*sqrt(3)",
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+        "note": "using explicit params"
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+      "answer": "2*sqrt(2)/3",
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+        "note": "using explicit params"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6499.png",
+      "answer": "(\\sqrt{5}-1)/2",
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+      "answer": "4/3",
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+      "error": "Unsupported or unparseable expression"
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+        "note": "using explicit params"
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+        "note": "using explicit params"
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+      "answer": "{11,29}",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6505.png",
+      "answer": "2*sqrt(2)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6507.png",
+      "answer": "{2, 6}",
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+      "answer": "y^2=4*x",
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+        "note": "using explicit params"
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+      "answer": "1\n2",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6511.png",
+      "answer": "sqrt(2)",
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+        "y_coef": 7.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6512.png",
+      "answer": "[1,7]",
+      "fact_expr": "G: Ellipse;P: Point;F: Point;Expression(G) = (x^2/16 + y^2/7 = 1);PointOnCurve(P, G);LeftFocus(G) = F",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6513.png",
+      "answer": "(26/3, -1)",
+      "fact_expr": "P: Point;Coordinate(P) = (1, -1);G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);F: Point;RightFocus(G) = F;M: Point;PointOnCurve(M, G);WhenMin(Abs(LineSegmentOf(M, P)) + 2*Abs(LineSegmentOf(M, F)))",
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+      "answer": "sqrt(2)",
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+      "answer": "[1/2,1)",
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+      "answer": "3/2",
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+      "answer": "y^2=(x-1)/4",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6521.png",
+      "answer": "8/5",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6523.png",
+      "answer": "sqrt(2)",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6524.png",
+      "answer": "x^2 - y^2 / 3 = 1",
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+      "answer": "6",
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+      "answer": "{4, 9/4}",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6528.png",
+      "answer": "y=pm*x",
+      "fact_expr": "G: Hyperbola;b: Number;H: Parabola;Expression(G) = (x^2/4 - y^2/b^2 = 1);Expression(H) = (y^2 = 8*sqrt(2)*x);RightFocus(G) = Focus(H)",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6529.png",
+      "answer": "sqrt(13)/3",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "success": true,
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+        "direction": "up"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6531.png",
+      "answer": "(0,1)",
+      "fact_expr": "G: Parabola;Expression(G) = (y = x^2/4)",
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+      "index": 6532,
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+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6532.png",
+      "answer": "4",
+      "fact_expr": "C: Ellipse;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/4 + y^2/3 = 1);Focus(C)={F1,F2};PointOnCurve(P,C)",
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+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6533.png",
+      "answer": "y^2/3 - x^2/6 = 1",
+      "fact_expr": "G: Hyperbola;C:Hyperbola;H: Point;Expression(G) = (x^2/2 - y^2 = 1);Coordinate(H) = (sqrt(2), -2);Asymptote(G)=Asymptote(C);PointOnCurve(H,C)",
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6534.png",
+      "answer": "sqrt(5)",
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+          1.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6535.png",
+      "answer": "x^2/9 - y^2/16 = 1",
+      "fact_expr": "C: Hyperbola;G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);Focus(C) = Endpoint(MajorAxis(G));Endpoint(RealAxis(C)) = Focus(G)",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 6537,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+        "note": "using explicit params"
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+      "answer": "sqrt(13)/4",
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+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6539.png",
+      "answer": "(1,2]",
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6540.png",
+      "answer": "8",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);l: Line;PointOnCurve(Focus(G), l);A: Point;B: Point;Intersection(l, G) = {A, B};XCoordinate(MidPoint(LineSegmentOf(A, B))) = 3",
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+    {
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+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6541.png",
+      "answer": "pm*1",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;A1: Point;B: Point;A2: Point;C: Point;F: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F;LeftVertex(G) = A1;RightVertex(G) = A2;L:Line;PointOnCurve(F,L);IsPerpendicular(L,xAxis);Intersection(L,G)={B,C};IsPerpendicular(LineSegmentOf(A1, B), LineSegmentOf(A2, C))",
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+    {
+      "index": 6542,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "iterations": 89
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6542.png",
+      "answer": "(pm*2,0)\nsqrt(3)*x+pm*y=0",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 8*x);Eccentricity(G) = 2;OneOf(Focus(G))=Focus(H)",
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+    {
+      "index": 6543,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 6544,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6544.png",
+      "answer": "2*a^2*b^2/a^2+b^2",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;O: Origin;P: Point;Q: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P,G);PointOnCurve(Q,G);IsPerpendicular(LineSegmentOf(O, P), LineSegmentOf(O, Q))",
+      "coords": {}
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+    {
+      "index": 6545,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6545.png",
+      "answer": "6",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;A: Point;B: Point;C: Point;PointOnCurve(A, G);PointOnCurve(B, G);PointOnCurve(C, G);VectorOf(F, A) + VectorOf(F, B) + VectorOf(F, C) = 0",
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+    {
+      "index": 6546,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 6547,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6558.png",
+      "answer": "-3/4",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6559.png",
+      "answer": "x^2/10 - y^2/40 = 1",
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+      "answer": "ApplyUnit(120,degree)",
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+      "answer": "y^2=2*x",
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+      "conic_type": "ellipse",
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+        "y_coef": 9.0
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+      "answer": "x^2/4+y^2/9=1",
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+      "answer": "10",
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 6566,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "orientation": "horizontal"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6567.png",
+      "answer": "y=pm*(sqrt(3)*x)",
+      "fact_expr": "H: Parabola;Expression(H) = (y^2 = 8*x);G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;PointOnCurve(LeftFocus(G), Directrix(H));Length(InterceptChord(Directrix(H), G)) = 6",
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+      "index": 6568,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 6569,
+      "success": true,
+      "conic_type": "ellipse",
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6569.png",
+      "answer": "{4,1/4}",
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+      "index": 6570,
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+        "y_coef": 2.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6570.png",
+      "answer": "2",
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+      "index": 6571,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 6572,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "x_coef": 12.0,
+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6572.png",
+      "answer": "x-2*y+4=0",
+      "fact_expr": "G: Ellipse;H: Circle;A: Point;B: Point;Expression(G) = (x^2/12 + y^2/3 = 1);Expression(H) = ((x + 2)^2 + (y - 1)^2 = 5/2);IsDiameter(LineSegmentOf(A, B), H);PointOnCurve(A,G);PointOnCurve(B,G)",
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+    {
+      "index": 6573,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6573.png",
+      "answer": "y = pm*sqrt(2)*x",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;FocalLength(C) = 2*c;c: Number;P: Point;PointOnCurve(P, C);Perimeter(TriangleOf(F1, P, F2)) = 6*a + 2*c;Area(TriangleOf(F1, P, F2)) = 2*a*c",
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+    {
+      "index": 6574,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 6575,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 6576,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6576.png",
+      "answer": "-1/4",
+      "fact_expr": "G: Parabola;A: Point;P: Point;B: Point;Expression(G) = (x^2 = y);PointOnCurve(A, G);PointOnCurve(B, G);l1: Line;l2: Line;TangentOnPoint(A, G) = l1;TangentOnPoint(B, G) = l2;Intersection(l1, l2) = P;AngleOf(A, P, B) = ApplyUnit(90, degree)",
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+    {
+      "index": 6577,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 6578,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 6579,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6579.png",
+      "answer": "1",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x);H: Line;P: Point;Coordinate(P) = (2, 1);PointOnCurve(P, H);A: Point;B: Point;Intersection(H, G) = {A, B};IsChordOf(LineSegmentOf(A, B), G);MidPoint(LineSegmentOf(A, B)) = P",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6607.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Distance(RightFocus(G),RightDirectrix(G))=(1/3)*FocalLength(G)",
+      "coords": {}
+    },
+    {
+      "index": 6608,
+      "success": true,
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+      "error": null,
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+        "direction": "right"
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+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6608.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;G: Circle;Center(G) = Vertex(C);A: Point;B: Point;Intersection(G, C) = {A, B};D: Point;E: Point;Intersection(G, Directrix(C)) = {D, E};Abs(LineSegmentOf(A, B)) = 2*sqrt(6);Abs(LineSegmentOf(D, E)) = 2*sqrt(10)",
+      "coords": {}
+    },
+    {
+      "index": 6609,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 6610,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+      "optimization": {
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+        "iterations": 122
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6610.png",
+      "answer": "x^2/4 - y^2/12 = 1",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 16*x);Expression(OneOf(Asymptote(G))) = (y = sqrt(3)*x);OneOf(Focus(G))=Focus(H)",
+      "coords": {}
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+    {
+      "index": 6611,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 6612,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "left"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6612.png",
+      "answer": "(-1/4, 1)",
+      "fact_expr": "A: Point;Coordinate(A) = (-2, 1);G: Curve;Expression(G) = (y^2 = -4*x);F: Point;Focus(G) = F;P: Point;PointOnCurve(P, G);WhenMin(Abs(LineSegmentOf(P, A))+Abs(LineSegmentOf(P, F)))",
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+          1.0
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+      }
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+    {
+      "index": 6613,
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+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 4.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6613.png",
+      "answer": "12/5",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2/9 - y^2/16 = 1);P: Point;PointOnCurve(P, C);M: Point;O: Origin;Abs(VectorOf(O, M)) = 1;DotProduct(VectorOf(O, M), VectorOf(P, M)) = 0;WhenMin(Abs(VectorOf(P, M)))",
+      "coords": {}
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+    {
+      "index": 6614,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 6615,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6615.png",
+      "answer": "12",
+      "fact_expr": "H: Parabola;Expression(H) = (y^2 = 16*x);D: Point;PointOnCurve(D, H) ;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;Distance(D, RightFocus(G)) = Distance(D, LeftDirectrix(G)) + 1",
+      "coords": {}
+    },
+    {
+      "index": 6616,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 6617,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 6618,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6618.png",
+      "answer": "2*sqrt(5)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;p: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);p>0;Expression(H) = (y^2 = 2*(p*x));Coordinate(Intersection(OneOf(Asymptote(G)),Directrix(H))) = (-2, -1);Distance(LeftVertex(G),Focus(H)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 6619,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6619.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(OneOf(Asymptote(G))) = (y = x)",
+      "coords": {}
+    },
+    {
+      "index": 6620,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6620.png",
+      "answer": "5",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);P : Point;PointOnCurve(P, G);Distance(P, Focus(G)) = 6",
+      "coords": {}
+    },
+    {
+      "index": 6621,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 6622,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.4641016151377544,
+        "b": 2.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6622.png",
+      "answer": "2",
+      "fact_expr": "H: Parabola;Expression(H) = (y^2 = 16*x);G: Hyperbola;Expression(G) = (x^2/12 - y^2/4 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 6623,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6623.png",
+      "answer": "y^2 = 8*x",
+      "fact_expr": "G: Parabola;m: Number;H: Line;m>0;Expression(G) = (y^2 = m*x);Expression(H) = (x = 1);Distance(Directrix(G) , H) = 3",
+      "coords": {}
+    },
+    {
+      "index": 6624,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 6625,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 6626,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 6627,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 6628,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 6629,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 6630,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6630.png",
+      "answer": "1",
+      "fact_expr": "l: Line;Expression(l) = (4*x - 3*y + 6 = 0);C: Parabola;Expression(C) = (y^2 = 4*x);P: Point;PointOnCurve(P, C)",
+      "coords": {}
+    },
+    {
+      "index": 6631,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6631.png",
+      "answer": "9",
+      "fact_expr": "P: Point;G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);PointOnCurve(P, G);H1: Circle;Expression(H1) = (y^2 + (x - 5)^2 = 4);H2: Circle;Expression(H2) = (y^2 + (x + 5)^2 = 1);M: Point;N: Point;PointOnCurve(M, H1)  ;PointOnCurve(N,H2)",
+      "coords": {}
+    },
+    {
+      "index": 6632,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 6633,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6633.png",
+      "answer": "(\\sqrt{2}/2, 1)",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;O: Origin;M: Point;A: Point;e: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftVertex(G) = A;PointOnCurve(M, G);IsPerpendicular(LineSegmentOf(O, M), LineSegmentOf(M, A));Eccentricity(G) = e",
+      "coords": {}
+    },
+    {
+      "index": 6634,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6634.png",
+      "answer": "2*p",
+      "fact_expr": "E: Parabola;Expression(E) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(E) = F;l: Line;Inclination(l) = ApplyUnit(30, degree);PointOnCurve(F, l);A: Point;B: Point;Intersection(l, E) = {A, B};L1: Line;L2: Line;C: Point;D: Point;PointOnCurve(A, L1);IsPerpendicular(L1, Directrix(E));FootPoint(L1, Directrix(E)) = C;PointOnCurve(B, L2);IsPerpendicular(L2, Directrix(E));FootPoint(L2, Directrix(E)) = D;M: Point;MidPoint(LineSegmentOf(C, D)) = M",
+      "coords": {}
+    },
+    {
+      "index": 6635,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6635.png",
+      "answer": "sqrt(3)/3",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*p*x);p: Number;p>0;F: Point;Focus(C) = F;l: Line;Directrix(C) = l;K: Point;Intersection(l, xAxis) = K;G: Line;PointOnCurve(F, G);A: Point;B: Point;Intersection(G, C) = {A, B};alpha: Number;Slope(G) = alpha;AngleOf(A, K, B) = ApplyUnit(60, degree)",
+      "coords": {}
+    },
+    {
+      "index": 6636,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6636.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (3, 1);Focus(G) = F;PointOnCurve(P, G)",
+      "coords": {
+        "A": [
+          3.0,
+          1.0
+        ]
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+    {
+      "index": 6637,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 6638,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6638.png",
+      "answer": "5/4",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;E: Circle;Expression(E) = ((x-5)^2 + y^2 = 9);IsTangent(Asymptote(C), E) = True",
+      "coords": {}
+    },
+    {
+      "index": 6639,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6639.png",
+      "answer": "(\\sqrt{6}+\\sqrt{2})/2",
+      "fact_expr": "G: Parabola;p: Number;P: Point;Q: Point;F: Point;M: Point;M1: Point;p>0;Expression(G) = (x^2 = 2*(p*y));Focus(G) = F;PointOnCurve(P, G);PointOnCurve(Q, G);AngleOf(P, F, Q) = 5*pi/6;MidPoint(LineSegmentOf(P, Q)) = M;L:Line;PointOnCurve(M,L);Directrix(G)=l;IsPerpendicular(L,l);FootPoint(L,l)=M1;IsChordOf(LineSegmentOf(P,Q),G);l:Line",
+      "coords": {}
+    },
+    {
+      "index": 6640,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.125,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6640.png",
+      "answer": "11/8",
+      "fact_expr": "G: Parabola;Expression(G) = (y = 2*x^2);A: Point;B: Point;IsChordOf(LineSegmentOf(A,B),G) = True;MidPoint(LineSegmentOf(A,B)) = M;M: Point;Length(LineSegmentOf(A,B)) = 3",
+      "coords": {}
+    },
+    {
+      "index": 6641,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 6642,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6642.png",
+      "answer": "sqrt(3)+2",
+      "fact_expr": "E: Hyperbola;b: Number;a: Number;F2: Point;P: Point;F1: Point;a>0;b>0;Expression(E) = (-y^2/b + x^2/a^2 = 1);LeftFocus(E) = F1;RightFocus(E) = F2;PointOnCurve(P, E);PointOnCurve(MidPoint(LineSegmentOf(P,F2)), yAxis);AngleOf(P, F2, F1) = ApplyUnit(60, degree)",
+      "coords": {}
+    },
+    {
+      "index": 6643,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6643.png",
+      "answer": "7",
+      "fact_expr": "G: Ellipse;H: Circle;C:Circle;P: Point;M: Point;N: Point;Expression(G) = (x^2/25 + y^2/16 = 1);Expression(H) = (y^2 + (x + 3)^2 = 1);Expression(C) = (y^2 + (x - 3)^2 = 4);PointOnCurve(P, G);PointOnCurve(M, H);PointOnCurve(N, C)",
+      "coords": {}
+    },
+    {
+      "index": 6644,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 2.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6644.png",
+      "answer": "8*sqrt(2)/3",
+      "fact_expr": "G: Hyperbola;H: Line;Expression(G) = (x^2 - y^2/4 = 1);Expression(H) = (y = x + 1)",
+      "coords": {}
+    },
+    {
+      "index": 6645,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6645.png",
+      "answer": "(1, 5/4]",
+      "fact_expr": "E: Hyperbola;b: Number;a: Number;C: Parabola;P: Point;A: Point;F: Point;a>0;b>0;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);Expression(C) = (y^2 = 16*(a*x));RightVertex(E) = A;Focus(C) = F;PointOnCurve(P, Asymptote(E));DotProduct(VectorOf(P, A), VectorOf(P, F)) = 0",
+      "coords": {}
+    },
+    {
+      "index": 6646,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6646.png",
+      "answer": "sqrt(10)/2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;F1: Point;A: Point;F2: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(A, G);AngleOf(F1, A, F2) = ApplyUnit(90, degree);Abs(LineSegmentOf(A, F1)) = 3*Abs(LineSegmentOf(A, F2))",
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+    {
+      "index": 6647,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 2.364173412322998,
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+      "optimization": {
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+        "iterations": 89
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6647.png",
+      "answer": "2",
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+    {
+      "index": 6648,
+      "success": false,
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+      "error": "validation: ellipse_point_off_curve",
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+        "y_coef": 5.000000000000001
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+        "iterations": 62
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6648.png",
+      "answer": "x^2/25 + y^2/5 = 1",
+      "fact_expr": "G: Ellipse;P: Point;Q: Point;Coordinate(P) = (sqrt(3), -2);Coordinate(Q) = (-2*sqrt(3), 1);PointOnCurve(Q, G);PointOnCurve(P, G)",
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+    {
+      "index": 6649,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 4.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6649.png",
+      "answer": "8",
+      "fact_expr": "G: Hyperbola;P: Point;A: Point;O: Origin;B: Point;t: Real;Expression(G) = (x^2/9 - y^2/16 = 1);PointOnCurve(P, G);VectorOf(P, A) = VectorOf(O, P)*(t - 1);DotProduct(VectorOf(O, A), VectorOf(O, P)) = 60;Coordinate(VectorOf(O, B)) = (0, 1)",
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+    {
+      "index": 6650,
+      "success": true,
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+        "b": 1.4142135623730951,
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+        "x_coef": 9.0,
+        "y_coef": 2.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6650.png",
+      "answer": "-4",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/2 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G) = True;Abs(LineSegmentOf(P, F1)) = 4",
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+    {
+      "index": 6651,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6651.png",
+      "answer": "3*sqrt(2)/16",
+      "fact_expr": "G: Parabola;H: Line;P: Point;Q: Point;O: Origin;M: Point;Expression(G) = (y = x^2);Expression(H) = (y = x + 1);PointOnCurve(P, H);PointOnCurve(Q, G);MidPoint(LineSegmentOf(P, Q)) = M",
+      "coords": {}
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+    {
+      "index": 6652,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.7320508075688772,
+        "b": 1.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6652.png",
+      "answer": "sqrt(3)*2/3\npm*4",
+      "fact_expr": "C: Hyperbola;G: Parabola;m: Number;Expression(C) = (x^2/3 - y^2 = 1);Expression(G) = (y^2 = 2*(m*x));Focus(G) = Focus(C)",
+      "coords": {}
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+    {
+      "index": 6653,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 4.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6653.png",
+      "answer": "4*sqrt(38)/5",
+      "fact_expr": "G: Ellipse;H: Line;Expression(G) = (x^2 + 4*y^2 = 16);Expression(H) = (y = x + 1)",
+      "coords": {}
+    },
+    {
+      "index": 6654,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6654.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
+      "fact_expr": "G: Parabola;H: Line;P: Point;Expression(G) = (y^2 = 2*x);Coordinate(P) = (0, 1);PointOnCurve(P,H);NumIntersection(G, H) = 1",
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+        "P": [
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+          1.0
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+    {
+      "index": 6655,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_focus_mismatch",
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6655.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "C: Hyperbola;a: Number;b: Number;F: Point;O: Origin;E: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (2, 0);RightFocus(C) = F;l:Line;PointOnCurve(F,l);IsPerpendicular(l,OneOf(Asymptote(C)));FootPoint(l,OneOf(Asymptote(C)))=E;WhenMax(Area(TriangleOf(O,E,F)))",
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+          2.0,
+          0.0
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+      "validation_reasons": [
+        "hyperbola_focus_mismatch"
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+    {
+      "index": 6656,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6656.png",
+      "answer": "(6, 4*sqrt(3))",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 8*x);F: Point;Focus(C) = F;P: Point;PointOnCurve(P, C);Quadrant(P) = 1;Abs(LineSegmentOf(P, F)) = 8",
+      "coords": {}
+    },
+    {
+      "index": 6657,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6657.png",
+      "answer": "4",
+      "fact_expr": "M: Point;G: Parabola;Expression(G) = (y^2 = 4*x);PointOnCurve(M, G) = True;F: Point;Focus(G) = F;A: Point;C: Circle;Expression(C) = ((x - 4)^2 + (y - 1)^2 = 1);PointOnCurve(A, C) = True",
+      "coords": {}
+    },
+    {
+      "index": 6658,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 8.0,
+        "y_coef": 4.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6658.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
+      "fact_expr": "G: Ellipse;H: Line;P0: Point;Expression(G) = (x^2/8 + y^2/4 = 1);Expression(H) = (x - y + 6 = 0);PointOnCurve(P0, G)",
+      "coords": {}
+    },
+    {
+      "index": 6659,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6659.png",
+      "answer": "pm*y+x=0",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;F1: Point;P: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);AngleOf(F1, P, F2) = pi/2;Area(TriangleOf(F1, P, F2)) = a^2",
+      "coords": {}
+    },
+    {
+      "index": 6660,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 6661,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6661.png",
+      "answer": "sqrt(37)",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);L: Line;PointOnCurve(Focus(G), L);Inclination(L) = ApplyUnit(60, degree);H: Circle;Expression(H) = (4*(sqrt(3)*y) - 4*x + x^2 + y^2 = 0)",
+      "coords": {}
+    },
+    {
+      "index": 6662,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6662.png",
+      "answer": "-3/8",
+      "fact_expr": "l: Line;C: Ellipse;P: Point;A2: Point;A1: Point;Expression(C) = (x^2/4 + y^2/3 = 1);LeftVertex(C)=A1;RightVertex(C)=A2;PointOnCurve(A1,l);Intersection(l,C)={A1, P};Slope(l)=2",
+      "coords": {}
+    },
+    {
+      "index": 6663,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 6664,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.23606797749979,
+        "b": 2.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6664.png",
+      "answer": "2*sqrt(5)\n4\n(pm*3,0)\n3*sqrt(5)/5\ny=pm*(2*sqrt(5)/5)*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/5 - y^2/4 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 6665,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "vertical"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6665.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-x^2/b^2 + y^2/a^2 = 1);Expression(OneOf(Asymptote(G))) = (x - 2*y = 0);Eccentricity(G)=e;e:Number",
+      "coords": {}
+    },
+    {
+      "index": 6666,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6666.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
+      "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(Focus(G), H);Inclination(H) = pi/3;Intersection(H, G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 6667,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6667.png",
+      "answer": "7/4",
+      "fact_expr": "l: Line;G: Parabola;Expression(G) = (y^2 = x);PointOnCurve(Focus(G), l);A: Point;B: Point;Intersection(l, G) = {A, B};Abs(LineSegmentOf(A, B)) = 4;IsChordOf(LineSegmentOf(A, B), G)",
+      "coords": {}
+    },
+    {
+      "index": 6668,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6668.png",
+      "answer": "(2, 0)",
+      "fact_expr": "G: Ellipse;M: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 + y^2/3 = 1);PointOnCurve(M, G);LeftFocus(G) = F1;RightFocus(G) = F2;Abs(LineSegmentOf(M, F1)) = 3*Abs(LineSegmentOf(M, F2))",
+      "coords": {}
+    },
+    {
+      "index": 6669,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6669.png",
+      "answer": "(\\sqrt{5}, +\\infty)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;H: Line;Expression(H) = (y = 2*x);IsIntersect(G, H)",
+      "coords": {}
+    },
+    {
+      "index": 6670,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6670.png",
+      "answer": "x^2 - y^2/4 = 1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Expression(OneOf(Asymptote(G))) = (2*x + y = 0);H: Point;Coordinate(H) = (sqrt(5), 0);OneOf(Focus(G)) = H",
+      "coords": {
+        "H": [
+          2.23606797749979,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 6671,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6671.png",
+      "answer": "1",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x)",
+      "coords": {}
+    },
+    {
+      "index": 6672,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6672.png",
+      "answer": "(4, pm*4*sqrt(2))",
+      "fact_expr": "G: Parabola;A: Point;F: Point;Expression(G) = (y^2 = 8*x);Focus(G) = F;PointOnCurve(A, G);Abs(LineSegmentOf(A, F))=6",
+      "coords": {}
+    },
+    {
+      "index": 6673,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6673.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;H: Line;Q: Point;A: Point;B: Point;F: Point;D: Point;Expression(G) = (y^2 = 4*x);Coordinate(Q) = (0, 3);Focus(G) = F;PointOnCurve(Q, H);Intersection(H, G) = {A, B};Intersection(PerpendicularBisector(LineSegmentOf(A, B)), xAxis) = D;Abs(LineSegmentOf(A, F)) + Abs(LineSegmentOf(B, F)) = 6",
+      "coords": {
+        "Q": [
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+          3.0
+        ]
+      }
+    },
+    {
+      "index": 6674,
+      "success": true,
+      "conic_type": "ellipse",
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+        "b": 1.7320508075688772,
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+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6674.png",
+      "answer": "1/2",
+      "fact_expr": "l: Line;C: Ellipse;b: Number;a: Number;O: Circle;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Expression(O) = (x^2 + y^2 = b^2);IsTangent(l, O);e:Number;Eccentricity(C) = e;Inclination(l) = pi/3;PointOnCurve(RightVertex(C),l)",
+      "coords": {}
+    },
+    {
+      "index": 6675,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6675.png",
+      "answer": "x-sqrt(3)*y+1=0",
+      "fact_expr": "l: Line;C: Parabola;G: Circle;t: Number;A: Point;B: Point;M: Point;F: Point;N: Point;Expression(C) = (y^2 = 4*x);t>0;Expression(l) = (-t*y + x + 1 = 0);Intersection(l, C) = {A, B};Negation(A=B);MidPoint(LineSegmentOf(A, B)) = M;Focus(C) = F;IsDiameter(LineSegmentOf(M,F),G);Intersection(l,G)={M,N};Abs(LineSegmentOf(M,N))=3*sqrt(3)*Abs(LineSegmentOf(N,F))",
+      "coords": {}
+    },
+    {
+      "index": 6676,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 6677,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.8284271247461903,
+        "b": 2.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6677.png",
+      "answer": "sqrt(6)/2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/8 - y^2/4 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 6678,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 10.0,
+        "b": 6.0,
+        "major_axis": "x",
+        "x_coef": 100.0,
+        "y_coef": 36.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6678.png",
+      "answer": "36",
+      "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/100 + y^2/36 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(90, degree)",
+      "coords": {}
+    },
+    {
+      "index": 6679,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 3.4641016151377544,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6679.png",
+      "answer": "(pm*4, 0)\nsqrt(3)*x + pm*y = 0",
+      "fact_expr": "G: Hyperbola;a: Number;b: Number;H: Ellipse;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2/25 + y^2/9 = 1);Eccentricity(G) = 2;Focus(G) = Focus(H)",
+      "coords": {}
+    },
+    {
+      "index": 6680,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6680.png",
+      "answer": "1",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;H: Point;Intersection(Directrix(G), xAxis) = H;P: Point;PointOnCurve(P, G);Abs(LineSegmentOf(P, H)) = sqrt(2)*Abs(LineSegmentOf(P, F))",
+      "coords": {}
+    },
+    {
+      "index": 6681,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6681.png",
+      "answer": "sqrt(3)+1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, RightPart(G));O: Origin;DotProduct((VectorOf(O, F2) + VectorOf(O, P)), VectorOf(F2, P)) = 0;Abs(LineSegmentOf(P, F1)) = sqrt(3)*Abs(LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 6682,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 6.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 36.0,
+        "y_coef": 9.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6682.png",
+      "answer": "23/3",
+      "fact_expr": "G: Ellipse;K: Point;M: Point;N: Point;Expression(G) = (x^2/36 + y^2/9 = 1);Coordinate(K) = (2, 0);DotProduct(VectorOf(K, M), VectorOf(K, N)) = 0;PointOnCurve(M, G);PointOnCurve(N, G)",
+      "coords": {
+        "K": [
+          2.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 6683,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.23606797749979,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 5.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6683.png",
+      "answer": "y^2=4*x",
+      "fact_expr": "G: Parabola;H: Ellipse;Expression(H) = (x^2/5 + y^2/4 = 1);Focus(G) = RightFocus(H);Vertex(G)=Center(H)",
+      "coords": {}
+    },
+    {
+      "index": 6684,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6684.png",
+      "answer": "x^2/3-y^2=1",
+      "fact_expr": "E: Hyperbola;a: Number;b: Number;a>0;b>0;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(OneOf(Focus(E))) = (2, 0);Distance(OneOf(Focus(E)),Asymptote(E))=1",
+      "coords": {}
+    },
+    {
+      "index": 6685,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6685.png",
+      "answer": "2*p",
+      "fact_expr": "C: Parabola;p: Number;G: Line;Q: Point;B: Point;F: Point;A: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;PointOnCurve(F, G);Intersection(Directrix(C), xAxis) = Q;Intersection(G, C) = {A, B};AngleOf(Q, B, F) = ApplyUnit(90, degree)",
+      "coords": {}
+    },
+    {
+      "index": 6686,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6686.png",
+      "answer": "pm*(3/2)",
+      "fact_expr": "G: Hyperbola;H: Line;k: Real;A: Point;B: Point;l1:Line;l2:Line;Expression(G) = (x^2 - y^2/3 = 1);Expression(H) = (y = k*x);Intersection(H, G) = {A, B};IsPerpendicular(l1,xAxis);IsPerpendicular(l2,xAxis);PointOnCurve(A,l1);PointOnCurve(B,l2);FootPoint(l1,xAxis)=F1;FootPoint(l2,xAxis)=F2;Focus(G)={F1,F2};F1:Point;F2:Point",
+      "coords": {}
+    },
+    {
+      "index": 6687,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6687.png",
+      "answer": "7/4",
+      "fact_expr": "P: Point;G: Parabola;Expression(G) = (y^2 = x);PointOnCurve(P, G);M: Circle;Expression(M) = (y^2 + (x - 3)^2 = 1);l1: Line;A: Point;TangentOfPoint(P, M) = l1;TangentPoint(l1, M) = A",
+      "coords": {}
+    },
+    {
+      "index": 6688,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.125,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6688.png",
+      "answer": "(5/4,+oo)",
+      "fact_expr": "l: Line;G: Parabola;B: Point;A: Point;Expression(G) = (y = 2*x^2);PointOnCurve(A, G);PointOnCurve(B, G);PerpendicularBisector(LineSegmentOf(A,B))=l;Slope(l) = 1/2",
+      "coords": {}
+    },
+    {
+      "index": 6689,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6689.png",
+      "answer": "8",
+      "fact_expr": "G: Ellipse;H: Line;F2: Point;A: Point;B: Point;F1: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Focus(G) = {F1, F2};PointOnCurve(F1, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(F2, A)) + Abs(LineSegmentOf(F2, B)) = 12",
+      "coords": {}
+    },
+    {
+      "index": 6690,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 6691,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6691.png",
+      "answer": "9",
+      "fact_expr": "G: Hyperbola;A: Point;B: Point;F2: Point;F1: Point;Expression(G) = (x^2/9 - y^2/16 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;IsChordOf(LineSegmentOf(A,B),G);PointOnCurve(F1,LineSegmentOf(A,B));PointOnCurve(A,LeftPart(G));PointOnCurve(B,LeftPart(G));Perimeter(TriangleOf(A, B, F2)) = 30",
+      "coords": {}
+    },
+    {
+      "index": 6692,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6692.png",
+      "answer": "[(sqrt(5)-1)/2, 1)",
+      "fact_expr": "E: Ellipse;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(E) = F1;RightFocus(E) = F2;A: Point;B: Point;RightVertex(E) = A;UpperVertex(E) = B;P: Point;PointOnCurve(P, LineOf(A, B));IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2));e: Number;Eccentricity(E) = e",
+      "coords": {}
+    },
+    {
+      "index": 6693,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6693.png",
+      "answer": "16/3",
+      "fact_expr": "P: Point;Coordinate(P) = (1, 0);l: Line;PointOnCurve(P, l);Inclination(l) = ApplyUnit(60, degree);G: Parabola;Expression(G) = (y^2 = 4*x);A: Point;B: Point;Intersection(l, G) = {A, B};IsChordOf(LineSegmentOf(A, B), G)",
+      "coords": {
+        "P": [
+          1.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 6694,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6694.png",
+      "answer": "-sqrt(13)",
+      "fact_expr": "P: Point;G:Curve;x1:Number;y1:Number;Coordinate(P) = (x1, y1);Expression(G)=(x^2/25 + y^2/16 = 1);PointOnCurve(P,G)",
+      "coords": {}
+    },
+    {
+      "index": 6695,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6695.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;b > a;a > 0;c: Number;FocalLength(G) = 2*c;l: Line;Q: Point;W: Point;Coordinate(Q) = (a, 0);Coordinate(W) = (0, b);PointOnCurve(Q, l);PointOnCurve(W, l);E: Point;Coordinate(E) = (-a, 0);Distance(E, l) = c*(2*sqrt(2)/3)",
+      "coords": {}
+    },
+    {
+      "index": 6696,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6696.png",
+      "answer": "12/5",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);P: Point;PointOnCurve(P, G);d1: Number;Distance(P, Directrix(G)) = d1;H: Line;Expression(H) = (3*x - 4*y + 9 = 0);d2: Number;Distance(P, H) = d2",
+      "coords": {}
+    },
+    {
+      "index": 6697,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 6698,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6698.png",
+      "answer": "4",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 + y^2 = 1);PointOnCurve(P, G);Focus(G) = {F1, F2}",
+      "coords": {}
+    },
+    {
+      "index": 6699,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6699.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "G: Parabola;P: Point;F: Point;A: Point;Expression(G) = (y = x^2/4);Focus(G) = F;Intersection(SymmetryAxis(G),Directrix(G))=A;PointOnCurve(P,G);Abs(LineSegmentOf(P,F))=m*Abs(LineSegmentOf(P,A));m:Number",
+      "coords": {}
+    },
+    {
+      "index": 6700,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 6701,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6701.png",
+      "answer": "1/2",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;LeftFocus(C) = F;G: Line;Expression(G) = (x = m);m: Number;A: Point;B: Point;Intersection(G, C) = {A, B};WhenMax(Perimeter(TriangleOf(A, B, F)));Area(TriangleOf(A, B, F)) = b^2;e: Number;Eccentricity(C) = e",
+      "coords": {}
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+    {
+      "index": 6702,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 6703,
+      "success": true,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6703.png",
+      "answer": "(0,2]",
+      "fact_expr": "C: Parabola;p: Number;M: Point;P: Point;Q: Point;p>0;Expression(C) = (x^2 = 2*p*y);Coordinate(M) = (0, -1);PointOnCurve(P,C);PointOnCurve(Q, C);DotProduct(VectorOf(M, P), VectorOf(M, Q)) >= 0",
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+        "M": [
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+          -1.0
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+    {
+      "index": 6704,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.0,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6704.png",
+      "answer": "(3, 6)",
+      "fact_expr": "H: Line;Expression(H) = (y = 2*x);P: Point;Quadrant(P) = 1;PointOnCurve(P, H);G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1);L1: Line;L2: Line;Asymptote(G) = {L1, L2};Distance(P, L1)*Distance(P, L2) = 27",
+      "coords": {}
+    },
+    {
+      "index": 6705,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6705.png",
+      "answer": "sqrt(26)-2*sqrt(3)",
+      "fact_expr": "G: Hyperbola;A: Point;M: Point;F: Point;Expression(G) = (x^2/3 - y^2 = 1);Coordinate(A) = (3, 1);RightFocus(G) = F;PointOnCurve(M,RightPart(G))",
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+          1.0
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+      "index": 6706,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 6707,
+      "success": true,
+      "conic_type": "ellipse",
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6707.png",
+      "answer": "1-sqrt(5)/5",
+      "fact_expr": "C: Ellipse;Expression(C)=(x^2/a^2 + y^2/b^2 = 1);b:Number;a>b;F: Point;LeftFocus(C)=F;A:Point;RightVertex(C)=A;PointOnCurve(P,C);IsPerpendicular(LineSegmentOf(P,F),xAxis);Slope(LineOf(P,A))=sqrt(5)/5;b>0;a:Number;P:Point",
+      "coords": {}
+    },
+    {
+      "index": 6708,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6708.png",
+      "answer": "1+3*sqrt(2)/4",
+      "fact_expr": "l: Line;E: Ellipse;A: Point;F: Point;B: Point;Expression(E) = (x^2/2 + y^2 = 1);LeftFocus(E) = F;PointOnCurve(F, l);Intersection(l, E) = {A, B}",
+      "coords": {}
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+    {
+      "index": 6709,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6709.png",
+      "answer": "(1, 1)",
+      "fact_expr": "G: Parabola;H: Line;Expression(G) = (y = x^2);Expression(H) = (2*x - y - 4 = 0);P: Point;PointOnCurve(P, G);WhenMin(Distance(P, H))",
+      "coords": {}
+    },
+    {
+      "index": 6710,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 6711,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6711.png",
+      "answer": "(3, 2*sqrt(3))",
+      "fact_expr": "C: Parabola;p: Number;M: Point;F: Point;p>0;y0: Number;y0 > 0;Expression(C) = (y^2 = 2*(p*x));Coordinate(M) = (3, y0);Focus(C) = F;PointOnCurve(M, C);Distance(M, Focus(C)) = Abs(LineSegmentOf(M, F));Abs(LineSegmentOf(M, F)) = 4",
+      "coords": {}
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+    {
+      "index": 6712,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6712.png",
+      "answer": "(-1, 1/4)",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);P: Point;PointOnCurve(P, G);F: Point;Focus(G) = F;A: Point;Coordinate(A) = (-1, 3);WhenMin(Abs(LineSegmentOf(P, F))+Abs(LineSegmentOf(P, A)))",
+      "coords": {
+        "A": [
+          -1.0,
+          3.0
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+    },
+    {
+      "index": 6713,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.9960020780563354,
+        "b": 1.4970015287399292,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "iterations": 2
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6713.png",
+      "answer": "7*sqrt(2)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;P: Point;PointOnCurve(P, G);Eccentricity(G) = 5/4;F1: Point;F2: Point;Focus(G) = {F1, F2};DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0;Area(TriangleOf(F1, P, F2)) = 18",
+      "coords": {}
+    },
+    {
+      "index": 6714,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 6715,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6715.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
+      "fact_expr": "G: Ellipse;a: Number;b: Number;H: Line;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(Focus(G), H);IsPerpendicular(H, xAxis);Length(InterceptChord(H, G)) = a",
+      "coords": {}
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+    {
+      "index": 6716,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6716.png",
+      "answer": "5",
+      "fact_expr": "l: Line;C: Parabola;G: Line;B: Point;A: Point;F: Point;M: Point;Expression(C) = (y^2 = 4*x);Expression(G) = (x + 1 = 0);Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};Abs(LineSegmentOf(A, B)) = 10;MidPoint(LineSegmentOf(A,B)) = M",
+      "coords": {}
+    },
+    {
+      "index": 6717,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "down"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6717.png",
+      "answer": "2*sqrt(5)",
+      "fact_expr": "C: Parabola;G: Circle;A: Point;F: Point;Expression(C) = (x^2 = -4*y);Focus(C) = F;PointOnCurve(A,C);Abs(LineSegmentOf(A,F))=3;Center(G)=A;Radius(G)=LineSegmentOf(A,F)",
+      "coords": {}
+    },
+    {
+      "index": 6718,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6718.png",
+      "answer": "(1,2)",
+      "fact_expr": "G: Parabola;H: Line;P: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (y = x + 4);PointOnCurve(P, G);WhenMin(Distance(P,H))",
+      "coords": {}
+    },
+    {
+      "index": 6719,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6719.png",
+      "answer": "1",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;M: Point;PointOnCurve(M, G) = True;AngleOf(F1, M, F2) = 2*pi/3;DotProduct(VectorOf(M, F1), VectorOf(M, F2)) = -2",
+      "coords": {}
+    },
+    {
+      "index": 6720,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6720.png",
+      "answer": "1",
+      "fact_expr": "F1: Point;F2: Point;G: Hyperbola;Expression(G) = (x^2 - 4*y^2 = 4);Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G) = True;DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0",
+      "coords": {}
+    },
+    {
+      "index": 6721,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 6722,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.8284271247461903,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 8.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6722.png",
+      "answer": "[0,sqrt(2)]",
+      "fact_expr": "C: Ellipse;P: Point;F1: Point;F2: Point;O: Origin;Expression(C) = (x^2/8 + y^2/4 = 1);PointOnCurve(P, C);LeftFocus(C)=F1;RightFocus(C)=F2",
+      "coords": {}
+    },
+    {
+      "index": 6723,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6723.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;p: Number;Q: Point;p>0;Expression(G) = (y^2 = 2*(p*x));PointOnCurve(Q, G);Min(Distance(Q, Focus(G))) = 1",
+      "coords": {}
+    },
+    {
+      "index": 6724,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6724.png",
+      "answer": "x=pm*3",
+      "fact_expr": "G: Parabola;H: Ellipse;a: Number;Expression(G) = (y^2 = 4*x);Expression(H) = (y^2/2 + x^2/a^2 = 1);PointOnCurve(Focus(H), Directrix(G))",
+      "coords": {}
+    },
+    {
+      "index": 6725,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6725.png",
+      "answer": "{2*sqrt(5), 2*sqrt(13)}",
+      "fact_expr": "G: Ellipse;k: Real;k > 0;Expression(G) = (x^2/36 + y^2/k^2 = 1);FocalLength(G) = 8",
+      "coords": {}
+    },
+    {
+      "index": 6726,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 6727,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          -3.0,
+          -0.0
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+        "radius": 2.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_6727.png",
+      "answer": "x^2/36+y^2/27=1",
+      "fact_expr": "G: Circle;C:Circle;H:Circle;Expression(G) = (6*x + x^2 + y^2 + 5 = 0);Expression(C) = (-6*x + x^2 + y^2 - 91 = 0);IsOutTangent(H,G);IsInTangent(H,C)",
+      "coords": {}
+    },
+    {
+      "index": 6728,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6728.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/m^2 = 1);m: Number;m>0;Z: Line;Expression(Z) = (y = (sqrt(2)/2)*x)\t;Projection(OneOf(Intersection(Z, G)), xAxis) = RightFocus(G)",
+      "coords": {}
+    },
+    {
+      "index": 6729,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6729.png",
+      "answer": "9",
+      "fact_expr": "G: Hyperbola;H: Circle;Z: Circle;P: Point;M: Point;N: Point;Expression(G) = (x^2/9 - y^2/16 = 1);Expression(H) = (y^2 + (x + 5)^2 = 4);PointOnCurve(P, RightPart(G));Expression(Z) = (y^2 + (x - 5)^2 = 1);PointOnCurve(M, H);PointOnCurve(N, Z)",
+      "coords": {}
+    },
+    {
+      "index": 6730,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6730.png",
+      "answer": "2/3",
+      "fact_expr": "G: Ellipse;H: Line;A: Point;O: Origin;B: Point;Expression(G) = (x^2/2 + y^2 = 1);PointOnCurve(RightFocus(G), H);Slope(H) = 1;Intersection(H, G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 6731,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6731.png",
+      "answer": "sqrt(65)/15",
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+      "answer": "9/2",
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+        "note": "using explicit params"
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+      "answer": "(-2, 1/2)",
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+      "answer": "20",
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+      "answer": "y = pm*sqrt(3)*x",
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+      "answer": "9/32",
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+      "answer": "(1, 0)",
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+      "answer": "7",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6781.png",
+      "answer": "8*sqrt(3)/3",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6783.png",
+      "answer": "x^2/9 - y^2/7 = 1",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;l: Line;Expression(l) = (7*x - 24*y + 28 = 0);PointOnCurve(F1, l);A: Point;B: Point;Intersection(l, C) = {A, B};IsPerpendicular(LineSegmentOf(A, F2), xAxis)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6784.png",
+      "answer": "6",
+      "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;M:Point;Expression(G) = (y^2 = 4*x);PointOnCurve(Focus(G), H);Intersection(H, G) = {A, B};MidPoint(LineSegmentOf(A,B))=M;XCoordinate(M)=2",
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+      "index": 6785,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "success": true,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6786.png",
+      "answer": "pm*sqrt(3)",
+      "fact_expr": "G: Ellipse;Z: Ellipse;a: Real;Expression(G) = (x^2/4 + y^2/a^2 = 1);Expression(Z) = (y^2/2 + x^2/a^2 = 1);Focus(G) = Focus(Z)",
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+      "index": 6787,
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+        "y_coef": 5.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6787.png",
+      "answer": "y^2=8*x",
+      "fact_expr": "G: Parabola;H: Ellipse;Expression(H) = (x^2/9 + y^2/5 = 1);Center(H)=Vertex(G);RightFocus(H)=Focus(G)",
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+        "direction": "up"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6788.png",
+      "answer": "x^2=12*y",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 2*p*y);p: Number;p>0;D: Point;PointOnCurve(D, G) = True;YCoordinate(D) = 2;Distance(D, Focus(G)) = 5",
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+      "index": 6789,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 6790,
+      "success": true,
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+        "x_coef": 16.0,
+        "y_coef": 9.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6790.png",
+      "answer": "5/2",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;M: Point;Expression(G) = (x^2/16 + y^2/9 = 1);PointOnCurve(P,G);LeftFocus(G)=F1;Distance(P,F1)=3;MidPoint(LineSegmentOf(P, F1)) = M;O:Origin",
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+    {
+      "index": 6791,
+      "success": true,
+      "conic_type": "parabola",
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6791.png",
+      "answer": "(1, \\pm 2)",
+      "fact_expr": "G: Parabola;O: Origin;A: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(A, G);DotProduct(VectorOf(O, A), VectorOf(A, F)) = -4",
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+    {
+      "index": 6792,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
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+        "x_coef": 9.0,
+        "y_coef": 3.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6792.png",
+      "answer": "sqrt(10)/4",
+      "fact_expr": "P: Point;PointOnCurve(P, G) = True;G: Ellipse;Expression(G) = (x^2/9 + y^2/3 = 1);Q: Point;PointOnCurve(Q, H) = True;H: Circle;Expression(H) = (y^2 + (x - 1)^2 = 5/8)",
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+    {
+      "index": 6793,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "vertical"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6793.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/5 + y^2/a^2 = 1);a: Number;a>0;H: Point;Coordinate(H) = (0, 3);OneOf(Focus(G))=H",
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+    {
+      "index": 6794,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6794.png",
+      "answer": "(\\sqrt{5}+1)/2",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2/a^2-y^2/b^2=1);a: Number;b: Number;a>0;b>0;A: Point;LeftVertex(C) = A;F: Point;RightFocus(C) = F;B: Point;Coordinate(B) = (0, b);DotProduct(VectorOf(B, A), VectorOf(B, F)) = 0",
+      "coords": {}
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+    {
+      "index": 6795,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 6796,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6796.png",
+      "answer": "8",
+      "fact_expr": "C: Parabola;G: Line;A: Point;B: Point;M: Point;E: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;Intersection(Directrix(C), xAxis) = E;PointOnCurve(F, G);Intersection(G, C) = {A, B};MidPoint(LineSegmentOf(A, B)) = M;Abs(LineSegmentOf(M, E)) = sqrt(20)",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6809.png",
+      "answer": "2*sqrt(7)-2",
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+        "note": "using explicit params"
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+      "answer": "{0,1}",
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+        "note": "using explicit params"
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+      "answer": "sqrt(3)",
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+      "success": false,
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6813.png",
+      "answer": "5",
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+      "success": false,
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6815.png",
+      "answer": "sqrt(2)",
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+      "error": null,
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+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6816.png",
+      "answer": "sqrt(3)",
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+    {
+      "index": 6817,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 3.872983346207417,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6817.png",
+      "answer": "5",
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+      "index": 6818,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6819.png",
+      "answer": "1/5",
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+      "index": 6820,
+      "success": true,
+      "conic_type": "parabola",
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6820.png",
+      "answer": "x = -4",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 16*x)",
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+      "error": null,
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+        "direction": "up"
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+        "note": "using explicit params"
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+      "answer": "1",
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+      "success": true,
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+        "b": 1.7320508075688772,
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+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6822.png",
+      "answer": "(0, pm*sqrt(m - 4))",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/m = 1);PointOnCurve(Focus(G), yAxis);m: Number",
+      "coords": {}
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+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6823.png",
+      "answer": "4/5",
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+      "success": true,
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6824.png",
+      "answer": "[0,2)",
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+          0.0
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+        "B": [
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+          0.0
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+      "success": true,
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6825.png",
+      "answer": "3/2",
+      "fact_expr": "G: Parabola;P: Point;A: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(P,G);Coordinate(A)=(1/2,1)",
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+          1.0
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+      "success": true,
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6826.png",
+      "answer": "2",
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+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+      "answer": "{y^2 = 4*x, y^2 = -36*x}",
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+      "answer": "8*sqrt(5)/5",
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+        "note": "using explicit params"
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+      "answer": "sqrt(6)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2/6 = 1)",
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+    {
+      "index": 6830,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6830.png",
+      "answer": "5*x^2-5*y^2/4=1",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 4*x);Expression(OneOf(Asymptote(G)))=(y=2*x);OneOf(Focus(G))=Focus(H)",
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+      "index": 6831,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6832.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6833.png",
+      "answer": "x^2/16 - y^2/9 = 1",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Distance(RightFocus(C),Asymptote(C))=3;P: Point;F1: Point;F2: Point;Focus(C) = {F1, F2};PointOnCurve(P, RightPart(C));Distance(P, F1) - Distance(P, F2) = (4/3)*Length(ImageinaryAxis(C))",
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+      "index": 6834,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 6835,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6836.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;H: Line;N: Point;F: Point;M: Point;Expression(G) = (y^2 = 12*x);Focus(G)=F;PointOnCurve(M,G);PointOnCurve(M,H);IsPerpendicular(H,Directrix(G));FootPoint(H,Directrix(G))=N;Slope(LineOf(N,F)) = -sqrt(3)/3",
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+    {
+      "index": 6837,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6837.png",
+      "answer": "x^2/2+y^2=1",
+      "fact_expr": "G: Hyperbola;H: Ellipse;O: Origin;Expression(G) = (2*x^2 - 2*y^2 = 1);Center(H) = O;Focus(H) = Focus(G);InterReciprocal(Eccentricity(G), Eccentricity(H))",
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+    {
+      "index": 6838,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 4.898979485566356,
+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6838.png",
+      "answer": "16",
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+      "index": 6839,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 6840,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6840.png",
+      "answer": "sqrt(6)",
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+      "coords": {}
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+    {
+      "index": 6841,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 6842,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "major_axis": "y",
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+        "y_coef": 4.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6842.png",
+      "answer": "-3",
+      "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (x^2 + y^2/m = 1);Inclination(OneOf(Asymptote(G)))=ApplyUnit(60,degree)",
+      "coords": {}
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+    {
+      "index": 6843,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
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+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6843.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;c: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);HalfFocalLength(G)=c",
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+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6844.png",
+      "answer": "3/5",
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+    {
+      "index": 6845,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 8.0,
+        "y_coef": 2.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6845.png",
+      "answer": "10",
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+    },
+    {
+      "index": 6846,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 3.1622776601683795,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6846.png",
+      "answer": "8",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/6 - y^2/10 = 1);H: Parabola;Expression(H) = (y^2 = 2*(p*x));p: Real;p>0;Focus(H) = RightFocus(G)",
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+      "index": 6847,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6847.png",
+      "answer": "-2",
+      "fact_expr": "G: Ellipse;H: Line;P: Point;F: Point;Q: Point;Expression(G) = (x^2/4 + y^2/3 = 1);RightFocus(G) = F;PointOnCurve(Center(G), H);Intersection(H, G) = {P, Q};WhenMax(Area(TriangleOf(P,F,Q)))",
+      "coords": {}
+    },
+    {
+      "index": 6848,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6848.png",
+      "answer": "2*sqrt(5)",
+      "fact_expr": "G: Ellipse;H: Line;Expression(G) = (x^2 + 4*y^2 = 16);Expression(H) = (x - 2*y + 4 = 0)",
+      "coords": {}
+    },
+    {
+      "index": 6849,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6849.png",
+      "answer": "8",
+      "fact_expr": "G: Hyperbola;H: Parabola;p: Number;Expression(G) = (-x^2 + y^2/3 = 1);Expression(H) = (x^2 = p*y);UpperFocus(G)=Focus(H)",
+      "coords": {}
+    },
+    {
+      "index": 6850,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6850.png",
+      "answer": "x^2/16+y^2/7=1",
+      "fact_expr": "G: Ellipse;C:Ellipse;H: Point;Expression(G) = (x^2/25 + y^2/16 = 1);Coordinate(H) = (-3, 7/4);Focus(G)=Focus(C);PointOnCurve(H,C)",
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+          1.75
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+      }
+    },
+    {
+      "index": 6851,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6851.png",
+      "answer": "2",
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+      "coords": {}
+    },
+    {
+      "index": 6852,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 6853,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6853.png",
+      "answer": "sqrt(2)+1",
+      "fact_expr": "M: Hyperbola;b: Number;a: Number;G: Line;F2: Point;A: Point;B: Point;F1: Point;e: Number;a>0;b>0;Expression(M) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(M) = F1;RightFocus(M) = F2;PointOnCurve(F1,G);IsPerpendicular(G,xAxis);Intersection(G, M) = {A, B};DotProduct(VectorOf(F2, A), VectorOf(F2, B)) = 0;Eccentricity(M) = e",
+      "coords": {}
+    },
+    {
+      "index": 6854,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.4142135623730951,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6854.png",
+      "answer": "2",
+      "fact_expr": "A: Point;B: Point;PointOnCurve(A, G);PointOnCurve(B, G);G: Hyperbola;Expression(G) = (x^2 - y^2/2 = 1);O: Origin;DotProduct(VectorOf(O, A), VectorOf(O, B)) = 0",
+      "coords": {}
+    },
+    {
+      "index": 6855,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6855.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;M: Point;F: Point;O: Origin;Expression(G) = (y^2 = 4*x);PointOnCurve(M, G);Focus(G) = F;AngleOf(M, F, O) = ApplyUnit(120, degree)",
+      "coords": {}
+    },
+    {
+      "index": 6856,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.7320508075688772,
+        "b": 1.4142135623730951,
+        "major_axis": "y",
+        "x_coef": 2.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6856.png",
+      "answer": "sqrt(3)/3",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/2 + y^2/3 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 6857,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6857.png",
+      "answer": "3",
+      "fact_expr": "G: Parabola;P: Point;F: Point;x0:Number;y0:Number;Expression(G) = (y^2 = 4*x);Coordinate(P) = (x0,y0);PointOnCurve(P, G);Focus(G) = F;Distance(P, F) = 4",
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+      "answer": "sqrt(2)",
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+        "note": "using explicit params"
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+        "note": "using explicit params"
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+      "answer": "3/2",
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+      "answer": "1",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 6907,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 6908,
+      "success": false,
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+      "error": "validation: hyperbola_point_off_curve",
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+      "answer": "x^2/16 - y^2/9 = 1",
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+        "iterations": 2
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+      "answer": "-4/3",
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+        "ellipse_focus_mismatch"
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+      "index": 6910,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 6911,
+      "success": true,
+      "conic_type": "ellipse",
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+        "y_coef": 2.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6911.png",
+      "answer": "sqrt(6)",
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+      "error": "Unsupported or unparseable expression"
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+      "index": 6913,
+      "success": true,
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+        "y_coef": 4.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6913.png",
+      "answer": "y^2 = -4 \\sqrt{5} x",
+      "fact_expr": "G: Parabola;H: Ellipse;Expression(H) = (x^2/9 + y^2/4 = 1);Focus(G) = LeftFocus(H);Vertex(G) = Center(H)",
+      "coords": {}
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+    {
+      "index": 6914,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 6915,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6915.png",
+      "answer": "(-3, 0)",
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+    {
+      "index": 6916,
+      "success": true,
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+      "params": {
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6916.png",
+      "answer": "(1, 1)",
+      "fact_expr": "G: Parabola;H: Line;P:Point;Expression(G) = (y^2 = x);Expression(H) = (x - 2*y + 4 = 0);PointOnCurve(P, G);WhenMin(Distance(P,H))",
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+    {
+      "index": 6917,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6917.png",
+      "answer": "2",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;c: Number;H: Line;F: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (2*(c*x) + x^2 + y^2 = 0);Coordinate(F) = (c, 0);RightFocus(C) = F;PointOnCurve(F,H);IsPerpendicular(H,Asymptote(C));IsTangent(H,G)",
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+    {
+      "index": 6918,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6918.png",
+      "answer": "1",
+      "fact_expr": "C: Ellipse;a: Number;Q: Circle;P: Point;F1: Point;F2: Point;a>1;O: Origin;Expression(C) = (y^2 + x^2/a^2 = 1);Focus(C) = {F1, F2};Center(Q) = O ;Diameter(Q) = FocalLength(C);Intersection(Q, C) = P",
+      "coords": {}
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+    {
+      "index": 6919,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 6920,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 6921,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.75,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6921.png",
+      "answer": "-5/2",
+      "fact_expr": "G: Parabola;A: Point;B: Point;x1: Number;x2: Number;y1: Number;y2: Number;Expression(G) = (y^2 = 3*x);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);PointOnCurve(A, G);PointOnCurve(B, G);Slope(LineOf(A, B)) = 2;y1 = 4",
+      "coords": {}
+    },
+    {
+      "index": 6922,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.0,
+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "answer": "(-1, 1)",
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+        "direction": "right"
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+        "note": "using explicit params"
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6941.png",
+      "answer": "y=pm*2*x/3",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/a = 1);a: Number;Coordinate(RightFocus(G)) = (sqrt(13), 0)",
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+      "success": true,
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+          0.0
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+      "answer": "(x^2/4+y^2/3=1)&Negation(x=-2)",
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+      "error": null,
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6943.png",
+      "answer": "(2,2*sqrt(2))",
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+      "success": true,
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+      "error": null,
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6944.png",
+      "answer": "(1/4,1)",
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+          1.0
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+      "success": true,
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+      "error": null,
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6945.png",
+      "answer": "8",
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+          0.0
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+      "index": 6946,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 1.4142135623730951,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6946.png",
+      "answer": "3+2*sqrt(2)",
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+      "index": 6947,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 6948,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6949.png",
+      "answer": "sqrt(5)",
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+    {
+      "index": 6950,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 6951,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6951.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
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+    {
+      "index": 6952,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 6953,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6966.png",
+      "answer": "sqrt(3)/2",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6968.png",
+      "answer": "3/2",
+      "fact_expr": "G: Parabola;M: Point;O: Origin;Expression(G) = (y^2 = 2*x);PointOnCurve(M, G);Distance(M, O) = sqrt(3)",
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+        "note": "using explicit params"
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+      "answer": "2*sqrt(10)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6970.png",
+      "answer": "sqrt(3)",
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+      "index": 6971,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6971.png",
+      "answer": "1-(sqrt(3)/3)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6972.png",
+      "answer": "2",
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+        "y_coef": 3.0
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6973.png",
+      "answer": "x^2/4+y^2=1",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6974.png",
+      "answer": "3*x+4*y-7=0",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 6976,
+      "success": true,
+      "conic_type": "parabola",
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6976.png",
+      "answer": "x+y-1=0",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;Slope(l) = -1;l: Line;Intersection(l, C) = {M, N};M: Point;N: Point;Abs(LineSegmentOf(M, F)) + Abs(LineSegmentOf(N, F)) = 8",
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+    {
+      "index": 6977,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "up"
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+      "optimization": {
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6977.png",
+      "answer": "y=-2",
+      "fact_expr": "l: Line;C: Parabola;F: Point;A: Point;B: Point;P:Point;Expression(C) = (x^2 = 8*y);Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};TangentOnPoint(A,C)=l1;TangentOnPoint(B,C)=l2;Intersection(l1,l2)=P",
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+    {
+      "index": 6978,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 6979,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_6979.png",
+      "answer": "sqrt(5)",
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+      "coords": {}
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+    {
+      "index": 6980,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6980.png",
+      "answer": "[sqrt(3)/3, sqrt(2)/2]",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;F1: Point;F2: Point;c: Number;Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = c^2",
+      "coords": {}
+    },
+    {
+      "index": 6981,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_6981.png",
+      "answer": "2",
+      "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;P: Point;Expression(G) = (y^2 = 4*x);Coordinate(P) = (1, 1);PointOnCurve(P, l);Intersection(l, G) = {A, B};MidPoint(LineSegmentOf(A,B)) = P",
+      "coords": {
+        "P": [
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+          1.0
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+    {
+      "index": 6982,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 3.0,
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+      "answer": "3/4",
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+      "answer": "{0,-1/2}",
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+      "answer": "9/2",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_6997.png",
+      "answer": "(4, +oo)",
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+      "answer": "sqrt(3)",
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+      "error": "Unsupported or unparseable expression"
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+      "answer": "5 + sqrt(2)",
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+      "index": 7002,
+      "success": true,
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+      "answer": "9",
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+      "index": 7003,
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+      "answer": "2*b^2/a",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_7004.png",
+      "answer": "x^2/9 + y^2/4 = 1",
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+      "index": 7006,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 7007,
+      "success": true,
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+      "answer": "40",
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+    {
+      "index": 7008,
+      "success": true,
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+      "error": null,
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+        "direction": "up"
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+        "note": "using explicit params"
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+      "answer": "2*sqrt(3)/3",
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+      "index": 7009,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 7010,
+      "success": true,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7010.png",
+      "answer": "5/7",
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+    {
+      "index": 7011,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7011.png",
+      "answer": "6",
+      "fact_expr": "G: Parabola;l: Line;A: Point;B: Point;M: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(M) = (-1, 0);PointOnCurve(M,l);Intersection(l, G) = {A,B};Focus(G) = F;Abs(LineSegmentOf(B, F)) = (sqrt(3)/2)*Abs(LineSegmentOf(B, M))",
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+          0.0
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+    {
+      "index": 7012,
+      "success": true,
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+      "error": null,
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7012.png",
+      "answer": "(\\sqrt{33}-3)/4",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;G: Line;A: Point;P: Point;Q: Point;F: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x = a);LeftVertex(C)=A;RightFocus(C)=F;PointOnCurve(P,G);Intersection(LineOf(P,A),C) = Q;VectorOf(A, Q) = 2*VectorOf(Q, P);DotProduct(VectorOf(A, Q), VectorOf(Q, F)) = 0",
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+    {
+      "index": 7013,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7013.png",
+      "answer": "10",
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+      "answer": "8",
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+      "answer": "sqrt(22)/2",
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+      "error": "validation: ellipse_focus_mismatch",
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+      "answer": "3",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7017.png",
+      "answer": "4/5",
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7018.png",
+      "answer": "x = -2",
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+      "index": 7019,
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+        "y_coef": 9.0
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+      "answer": "1",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);P: Point;PointOnCurve(P, G)",
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+        "note": "using explicit params"
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+      "answer": "-9",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7021.png",
+      "answer": "y=pm*(sqrt(2)/2)*x",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);FocalLength(C) = 4*sqrt(3);Length(RealAxis(C)) = 4*sqrt(2)",
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+      "error": null,
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+        "x_coef": 4.0,
+        "y_coef": 1.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7022.png",
+      "answer": "2*sqrt(3)/3 - 1",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(60, degree)",
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+    {
+      "index": 7023,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7023.png",
+      "answer": "pm*4/3",
+      "fact_expr": "l: Line;C: Parabola;A: Point;M: Point;B: Point;k: Real;Expression(C) = (y^2 = 16*x);Expression(l) = (y = k*x - 4*k);Intersection(SymmetryAxis(C),Directrix(C))=M;Intersection(l, C) = {A, B};Abs(LineSegmentOf(A, M)) = 4*Abs(LineSegmentOf(B, M))",
+      "coords": {}
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+    {
+      "index": 7024,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.0625,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7024.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;A: Point;Expression(G) = (y = 4*x^2);PointOnCurve(A, G);Distance(A, Focus(G)) = 33/16",
+      "coords": {}
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+    {
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+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
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+          0.0
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+        "radius": 1.0
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+      "optimization": {
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_7025.png",
+      "answer": "(x-2)^2+y^2=1/4",
+      "fact_expr": "G: Curve;B: Point;A: Point;P: Point;Expression(G) = (x^2 + y^2 = 1);Coordinate(A) = (4, 0);PointOnCurve(B, G);MidPoint(LineSegmentOf(A,B)) = P",
+      "coords": {}
+    },
+    {
+      "index": 7026,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7026.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;O: Origin;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);DotProduct(VectorOf(P, F1),(VectorOf(O,F1)+VectorOf(O,P)))=0;Abs(VectorOf(P, F1)) = 2*Abs(VectorOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 7027,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7027.png",
+      "answer": "4",
+      "fact_expr": "G: Parabola;P: Point;F: Point;m:Number;Expression(G) = (y^2 = 4*x);Coordinate(P) = (m, 2*sqrt(3));PointOnCurve(P, G);Focus(G) = F",
+      "coords": {}
+    },
+    {
+      "index": 7028,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7028.png",
+      "answer": "(1+sqrt(5))/2",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;B: Point;F: Point;A: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(B) = (0, b);LeftFocus(C) = F;RightVertex(C) = A;DotProduct(VectorOf(F, B), VectorOf(A, B)) = 0",
+      "coords": {}
+    },
+    {
+      "index": 7029,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7029.png",
+      "answer": "28",
+      "fact_expr": "G: Hyperbola;A: Point;B: Point;F1:Point;F2: Point;Expression(G) = (x^2/16 - y^2/9 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;IsChordOf(LineSegmentOf(A,B),G);PointOnCurve(F1,LineSegmentOf(A,B));Length(LineSegmentOf(A,B))=6",
+      "coords": {}
+    },
+    {
+      "index": 7030,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7030.png",
+      "answer": "12",
+      "fact_expr": "C: Parabola;p: Number;M: Point;F: Point;N: Point;p>0;l:Line;Expression(C) = (y^2 = 2*p*x);Focus(C) = F;PointOnCurve(M, C);Directrix(C)=l;PointOnCurve(N,l);VectorOf(F, N) + 3*VectorOf(M, F) = 0;-p + 2*Abs(LineSegmentOf(M, F)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 7031,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7032,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
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+      "optimization": {
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7032.png",
+      "answer": "y = -2",
+      "fact_expr": "C: Parabola;Expression(C) = (x^2 = 2*p*y);p: Number;p>0;Distance(Focus(C), Directrix(C)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 7033,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7034,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 2.0,
+        "major_axis": "y",
+        "x_coef": 4.0,
+        "y_coef": 9.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7034.png",
+      "answer": "x^2 = -4\\sqrt{5}y",
+      "fact_expr": "G: Parabola;H: Ellipse;Expression(H) = (x^2/4 + y^2/9 = 1);Focus(G) = LowerFocus(H);Vertex(G) = Center(H)",
+      "coords": {}
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+    {
+      "index": 7035,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "down"
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+      "optimization": {
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7035.png",
+      "answer": "-1/4",
+      "fact_expr": "G: Parabola;Expression(G) = (y = a*x^2);a: Real;F: Point;Coordinate(F) = (0, -1);Focus(G) = F",
+      "coords": {
+        "F": [
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+          -1.0
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+    {
+      "index": 7036,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 5.196152422706632,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7036.png",
+      "answer": "14",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/27 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};A: Point;PointOnCurve(A, G);Abs(LineSegmentOf(A, F1)) = 8",
+      "coords": {}
+    },
+    {
+      "index": 7037,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.0416666679084301,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7037.png",
+      "answer": "(0,1/24)",
+      "fact_expr": "G: Parabola;Expression(G) = (y = 6*x^2)",
+      "coords": {}
+    },
+    {
+      "index": 7038,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7038.png",
+      "answer": "{1, 3}",
+      "fact_expr": "l: Line;C: Parabola;B: Point;D: Point;F: Point;A: Point;m: Line;p: Number;p > 0;Expression(C) = (y^2 = 2*p*x);Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};Directrix(C) = m;Intersection(l, m) = D;Abs(LineSegmentOf(B, D)) = 2*Abs(LineSegmentOf(F, B));Abs(LineSegmentOf(F, A)) = 2",
+      "coords": {}
+    },
+    {
+      "index": 7039,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7040,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7040.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "C: Parabola;p: Number;G: Line;A: Point;B: Point;F: Point;l: Line;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;Directrix(C) = l;PointOnCurve(F, G);Intersection(G, C) = {A, B};A1: Point;B1: Point;IsPerpendicular(LineSegmentOf(A, A1), l);IsPerpendicular(LineSegmentOf(B, B1), l);FootPoint(LineSegmentOf(A, A1), l) = A1;FootPoint(LineSegmentOf(B, B1), l) = B1;Abs(LineSegmentOf(A1, F)) = 2*sqrt(3);Abs(LineSegmentOf(B1, F)) = 2",
+      "coords": {}
+    },
+    {
+      "index": 7041,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "vertical"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7041.png",
+      "answer": "y = pm*(3/4)*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/16 + y^2/9 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 7042,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "answer": "x^2/(2/3)-y^2/(1/3)=1",
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+      "answer": "x=3",
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+      "answer": "x^2 - y^2/3 = 1",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Coordinate(OneOf(Focus(C))) = (2, 0);l: Line;Expression(l) = (x + sqrt(3)*y = 0);IsPerpendicular(OneOf(Asymptote(C)),l)",
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+      "answer": "2",
+      "fact_expr": "C: Ellipse;m: Number;Expression(C) = (x^2/3 + y^2/m = 1);Length(MajorAxis(C)) = 4",
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+      "index": 7048,
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+      "error": "Unsupported or unparseable expression"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7049.png",
+      "answer": "{1, 3}",
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+      "answer": "1/10",
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+      "answer": "4*a",
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+    {
+      "index": 7053,
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+        "b": 5.656854249492381,
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+        "note": "using explicit params"
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+      "answer": "2",
+      "fact_expr": "A: Point;Coordinate(A) = (x0, y0);x0: Number;y0: Number;G: Hyperbola;Expression(G) = (x^2/4 - y^2/32 = 1);PointOnCurve(A, RightPart(G)) = True;Distance(A, RightFocus(G)) = 2*x0",
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+        "y_coef": 4.0
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+        "note": "using explicit params"
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+      "answer": "3",
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+        "b": 4.0,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7055.png",
+      "answer": "16*sqrt(3)",
+      "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (16*x^2 - 9*y^2 = 144);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1))*Abs(LineSegmentOf(P, F2)) = 64",
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+    {
+      "index": 7056,
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+      "error": null,
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7056.png",
+      "answer": "10/3",
+      "fact_expr": "C: Parabola;P: Point;F: Point;Q: Point;l: Line;Expression(C) = (y^2 = 4*x);Focus(C) = F;Directrix(C) = l;PointOnCurve(P, l);Intersection(OverlappingLine(LineSegmentOf(P, F)), C) = Q;2*VectorOf(F, P) + 3*VectorOf(F, Q) =0",
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+      "index": 7057,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "success": false,
+      "conic_type": "parabola",
+      "error": "validation: parabola_point_off_curve",
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7058.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
+      "fact_expr": "H: Point;Coordinate(H) = (1, 0);G: Line;Inclination(G) = (2*pi)/3;PointOnCurve(H, G);A: Point;B: Point;Z: Curve;Expression(Z) = (y^2 = 4*x);Intersection(G, Z) = {A, B};IsChordOf(LineSegmentOf(A, B), Z)",
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+          0.0
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+        "parabola_point_off_curve"
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+      "index": 7059,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "up"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7059.png",
+      "answer": "{x^2 = -20*y, x^2 = 4*y}",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = m*y);m: Number;Z: Line;Expression(Z) = (y = 2);Distance(Directrix(G), Z) = 3",
+      "coords": {}
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+    {
+      "index": 7060,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7060.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "C: Parabola;l1:Line;l2:Line;P: Point;Expression(C) = (y^2 = 4*x);Expression(l2) = (y=x+1);Expression(l1) = (x = -1);PointOnCurve(P, C)",
+      "coords": {}
+    },
+    {
+      "index": 7061,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7062,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7062.png",
+      "answer": "2",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;G: Circle;Expression(G) = (x^2 + y^2 = a^2);L1: Line;L2: Line;TangentOfPoint(OneOf(Focus(C)), G) = {L1, L2};TangentPoint(L1, G) = A;TangentPoint(L2, G) = B;A: Point;B: Point;AngleOf(A, O, B) = ApplyUnit(120, degree);O: Origin",
+      "coords": {}
+    },
+    {
+      "index": 7063,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7063.png",
+      "answer": "4/3",
+      "fact_expr": "l: Line;G: Parabola;A: Point;F: Point;B: Point;Expression(G) = (y^2 = 4*x);Focus(G)= F;PointOnCurve(F,l);Intersection(l,G) = {A, B};Abs(LineSegmentOf(A, F)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 7064,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7064.png",
+      "answer": "(10, 0)",
+      "fact_expr": "O: Origin;C: Parabola;Expression(C) = (y^2 = 8*x);F: Point;Focus(C) = F;P: Point;PointOnCurve(P, C) ;IsPerpendicular(LineSegmentOf(P, F), xAxis) ;Q: Point;PointOnCurve(Q, xAxis) ;PointOnCurve(P, G) ;IsDiameter(LineSegmentOf(O, Q), G);G: Circle",
+      "coords": {}
+    },
+    {
+      "index": 7065,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7065.png",
+      "answer": "(0, 1]",
+      "fact_expr": "G: Parabola;p: Number;A: Point;B: Point;H: Point;N: Point;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;PointOnCurve(A,G);PointOnCurve(B,G);AngleOf(A, F, B) = ApplyUnit(60, degree);IsChordOf(LineSegmentOf(A,B),G);MidPoint(LineSegmentOf(A,B))=H;PointOnCurve(H,LineOf(H,N));IsPerpendicular(Directrix(G),LineOf(H,N));FootPoint(Directrix(G),LineOf(H,N))=N",
+      "coords": {}
+    },
+    {
+      "index": 7066,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7067,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7067.png",
+      "answer": "y=pm*(1/2)*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 7068,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7068.png",
+      "answer": "2*sqrt(3)/3",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;E: Curve;l: Line;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(E) = (y^2 + (x - 2)^2 = 4);OneOf(Asymptote(C)) = l;Slope(l) > 0;F1: Point;F2: Point;F3: Point;PointOnCurve(F1, E);PointOnCurve(F2, E);PointOnCurve(F3, E);Distance(F1, l) = 1;Distance(F2, l) = 1;Distance(F3, l) = 1;Negation(F1 = F2);Negation(F2 = F3);Negation(F1 = F3)",
+      "coords": {}
+    },
+    {
+      "index": 7069,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7070,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "y",
+        "x_coef": 1.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7070.png",
+      "answer": "[41,49]",
+      "fact_expr": "l: Line;G: Ellipse;a:Number;b:Number;C: Point;A: Point;B: Point;Expression(G) = (x^2 + y^2/9 = 1);Expression(l) = (a*x + b*y = 0);Coordinate(C) = (5, 5);Intersection(l, G) = {A, B}",
+      "coords": {
+        "C": [
+          5.0,
+          5.0
+        ]
+      }
+    },
+    {
+      "index": 7071,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7072,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7073,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7073.png",
+      "answer": "[4*sqrt(3)/3,+oo)",
+      "fact_expr": "C: Curve;Expression(C) = (x^2/4 + y^2/b^2 = 1);b: Number;b>0;l: Line;Expression(l) = (k + k*x - y + 2 = 0);k: Number;IsIntersect(C, l) = True",
+      "coords": {}
+    },
+    {
+      "index": 7074,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
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+      "answer": "(1, 2*sqrt(3)/3)",
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+      "answer": "5/3",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1)",
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+      "answer": "4",
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+      "answer": "[\\sqrt{2}/2,1)",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7085.png",
+      "answer": "sqrt(10)/2",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Focus(C)={F1,F2};PointOnCurve(P,C);IsPerpendicular(LineSegmentOf(P,F1),LineSegmentOf(P,F2));Abs(LineSegmentOf(P,F1))=3*Abs(LineSegmentOf(P,F2))",
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+    {
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+        "y_coef": 1.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7086.png",
+      "answer": "sqrt(3)/2",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F2), xAxis)",
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+    {
+      "index": 7087,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7087.png",
+      "answer": "(2*sqrt(21))/3",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);P: Point;PointOnCurve(P, G);A: Point;Coordinate(A) = (0, 2)",
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+          2.0
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+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7088.png",
+      "answer": "(2, 0)\nsqrt(3)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1)",
+      "coords": {}
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+    {
+      "index": 7089,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.0,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7089.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;C: Circle;r: Number;Expression(G) = (x^2 - y^2 = 1);r>0;Expression(C) = (y^2 + (x - 2)^2 = r^2);Length(InterceptChord(OneOf(Asymptote(G)),C))=2*sqrt(2)",
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7090.png",
+      "answer": "2*sqrt(5) - 1",
+      "fact_expr": "G: Parabola;P: Point;F: Point;Q: Point;Coordinate(Q) = (3, 4);Focus(G) = F;Expression(G) = (y^2 = 4*x);PointOnCurve(P, G)",
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+          4.0
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+      "index": 7091,
+      "success": true,
+      "conic_type": "ellipse",
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7091.png",
+      "answer": "[\\sqrt{2}/2,1)",
+      "fact_expr": "H: Circle;O: Origin;Center(H) = O;Intersection(H, xAxis) = {A, B};A: Point;B: Point;Focus(G) = {A, B};G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;IsIntersect(H, G) = True",
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+    {
+      "index": 7092,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 4.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7092.png",
+      "answer": "[-2,2]",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2 + 4*y^2 = 16);P:Point;PointOnCurve(P,G)",
+      "coords": {}
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+    {
+      "index": 7093,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 7094,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7094.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*p*x);p: Number;p>0;F: Point;Focus(C) = F;l: Line;Expression(l) = (y = m + x);m: Number;Intersection(l, C) = {M, N};M: Point;N: Point;Abs(LineSegmentOf(M, F)) = 2;Abs(LineSegmentOf(N, F)) = 3",
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+      "index": 7095,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7095.png",
+      "answer": "a/(sqrt(a^2+b^2))",
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+      "index": 7096,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "up"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7096.png",
+      "answer": "y = -1",
+      "fact_expr": "C: Parabola;p: Number;G: Line;A: Point;B: Point;p>0;Expression(C) = (x^2 = 2*(p*y));Expression(G) = (y = p/2 + x);Intersection(G, C) = {A, B};Abs(LineSegmentOf(A, B)) = 8",
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+    {
+      "index": 7097,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "up"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7097.png",
+      "answer": "x^2=pm*16*y",
+      "fact_expr": "G: Parabola;O: Origin;Vertex(G) = O;SymmetryAxis(G) = yAxis;L: LineSegment;PointOnCurve(Focus(G), L) = True;IsPerpendicular(L, yAxis) = True;IsChordOf(L, G) = True;Length(L) = 16",
+      "coords": {}
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+    {
+      "index": 7098,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7098.png",
+      "answer": "(1, 1+sqrt(3)]",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, RightPart(G));O: Origin;DotProduct((VectorOf(O, F2) + VectorOf(O, P)), VectorOf(F2, P)) = 0;Abs(LineSegmentOf(P, F1)) >= sqrt(3)*Abs(LineSegmentOf(P, F2))",
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+    {
+      "index": 7099,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7099.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "E: Ellipse;b: Number;a: Number;G: Line;F: Point;c:Number;a > b;b > 0;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);Expression(G) = (-a*y + b*x = 0);Coordinate(F) = (c, 0);RightFocus(E) = F;Distance(F, G) = (sqrt(3)/3)*c",
+      "coords": {}
+    },
+    {
+      "index": 7100,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "vertical"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7100.png",
+      "answer": "y=pm*(sqrt(2)/2)*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/8 + y^2/4 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 7101,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 4.898979485566356,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7101.png",
+      "answer": "48*pi/7",
+      "fact_expr": "F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;C: Hyperbola;Expression(C) = (x^2 - y^2/24 = 1);P: Point;PointOnCurve(P, C) = True;Quadrant(P) = 1;Abs(LineSegmentOf(P, F1))/Abs(LineSegmentOf(P, F2)) = 5/4",
+      "coords": {}
+    },
+    {
+      "index": 7102,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7102.png",
+      "answer": "18",
+      "fact_expr": "G: Ellipse;H: Line;P: Point;F: Point;Q: Point;Expression(G) = (x^2/25 + y^2/16 = 1);PointOnCurve(Center(G), H);Intersection(H, G) = {P, Q};LeftFocus(G) = F",
+      "coords": {}
+    },
+    {
+      "index": 7103,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7103.png",
+      "answer": "1",
+      "fact_expr": "G: Parabola;P: Point;F: Point;Expression(G) = (y^2 = 8*x);Focus(G) = F;PointOnCurve(P, G);Abs(LineSegmentOf(P, F)) = 3",
+      "coords": {}
+    },
+    {
+      "index": 7104,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7104.png",
+      "answer": "[sqrt(5)/5, 1/2]",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;c: Number;F1: Point;F2: Point;Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 3*c^2;e: Number;Eccentricity(C) = e",
+      "coords": {}
+    },
+    {
+      "index": 7105,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7105.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;Eccentricity(G) = sqrt(2)/2;l: Line;A: Point;B: Point;Intersection(l, G) = {A, B};M: Point;Coordinate(M) = (1, 1);MidPoint(LineSegmentOf(A, B)) = M",
+      "coords": {
+        "M": [
+          1.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 7106,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.7320508075688772,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7106.png",
+      "answer": "y=(pm*sqrt(3)/3)*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - 3*y^2 = 3)",
+      "coords": {}
+    },
+    {
+      "index": 7107,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7107.png",
+      "answer": "2*a+2*b",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;PointOnCurve(Center(G), H) = True;H: Line;Intersection(H, G) = {P, Q};P: Point;Q: Point;F: Point;OneOf(Focus(G)) = F",
+      "coords": {}
+    },
+    {
+      "index": 7108,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 2.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7108.png",
+      "answer": "x^2/12 - y^2/8 = 1",
+      "fact_expr": "G: Hyperbola;Z: Hyperbola;H: Point;Expression(G) = (x^2/16 - y^2/4 = 1);Coordinate(H) = (3*sqrt(2), 2);PointOnCurve(H, Z);Focus(G) = Focus(Z)",
+      "coords": {
+        "H": [
+          4.242640687119286,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 7109,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7109.png",
+      "answer": "pm*2*sqrt(2)",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;F: Point;Focus(G) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, G) = {A, B};C: Point;Intersection(l, Directrix(G)) = C;VectorOf(C, B) = 3*VectorOf(B, F)",
+      "coords": {}
+    },
+    {
+      "index": 7110,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7110.png",
+      "answer": "4",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Focus(C) = {F1, F2};PointOnCurve(P, C);IsPerpendicular(VectorOf(P, F2),VectorOf(P, F1));Area(TriangleOf(P, F1, F2)) = 16",
+      "coords": {}
+    },
+    {
+      "index": 7111,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7111.png",
+      "answer": "y^2=4*x",
+      "fact_expr": "F: Point;Focus(G) = F;G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;A: Point;B: Point;C: Point;D: Point;IsChordOf(LineSegmentOf(A, B), G);IsChordOf(LineSegmentOf(C, D), G);PointOnCurve(F, LineSegmentOf(A, B));PointOnCurve(F, LineSegmentOf(C, D));IsPerpendicular(LineSegmentOf(A, B), LineSegmentOf(C, D));Min(Abs(LineSegmentOf(A, B)) + Abs(LineSegmentOf(C, D))) = 16",
+      "coords": {}
+    },
+    {
+      "index": 7112,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7112.png",
+      "answer": "y=x",
+      "fact_expr": "l: Line;C: Parabola;A: Point;B: Point;F: Point;O: Origin;Coordinate(F) = (1, 0);Focus(C)=F;Vertex(C) = O;Intersection(l,C)={A,B};Coordinate(MidPoint(LineSegmentOf(A,B)))=(2,2)",
+      "coords": {
+        "F": [
+          1.0,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 7113,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.7195594310760498,
+        "b": 1.7224669456481934,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 2.391175712546101e-06,
+        "converged": true,
+        "iterations": 83
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7113.png",
+      "answer": "y=pm*x",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = sqrt(2)",
+      "coords": {}
+    },
+    {
+      "index": 7114,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7115,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7115.png",
+      "answer": "8",
+      "fact_expr": "C: Parabola;p: Number;G: Line;B: Point;A: Point;M: Point;N: Point;p>0;Expression(C) = (y^2 = 2*p*x);Expression(G) = (x = 1);Intersection(G, C) = M;Intersection(G,xAxis)=N;PointOnCurve(A, C);PointOnCurve(B, C);Slope(LineOf(A,B)) = 2;Distance(N,LineOf(M,A))=Distance(N,LineOf(M,B))",
+      "coords": {}
+    },
+    {
+      "index": 7116,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7117,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7118,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7118.png",
+      "answer": "1/2",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;O:Origin;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(OneOf(Focus(G)),H);PointOnCurve(Endpoint(MinorAxis(G)),H);Distance(OneOf(Focus(G)),H)=b/2",
+      "coords": {}
+    },
+    {
+      "index": 7119,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7119.png",
+      "answer": "4",
+      "fact_expr": "C: Hyperbola;F1: Point;P: Point;F2: Point;Expression(C) = (x^2 - y^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);AngleOf(F1, P, F2) = ApplyUnit(60, degree)",
+      "coords": {}
+    },
+    {
+      "index": 7120,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7120.png",
+      "answer": "(1, 2+sqrt(5))",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;c: Number;HalfFocalLength(G) = c;-4*a*c + b^2 < 0",
+      "coords": {}
+    },
+    {
+      "index": 7121,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7121.png",
+      "answer": "1/4",
+      "fact_expr": "l: Line;G: Curve;A: Point;Expression(G) = (y = x^2);Coordinate(A) = (t, 0);Inclination(l) = ApplyUnit(45, degree);PointOnCurve(A,l);IsTangent(l, G);t:Number",
+      "coords": {}
+    },
+    {
+      "index": 7122,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7122.png",
+      "answer": "2*sqrt(7)-2",
+      "fact_expr": "G: Hyperbola;P: Point;F: Point;A: Point;Expression(G) = (x^2 - y^2/3 = 1);Coordinate(A) = (3, sqrt(3));PointOnCurve(P, RightPart(G));RightFocus(G) = F",
+      "coords": {
+        "A": [
+          3.0,
+          1.7320508075688772
+        ]
+      }
+    },
+    {
+      "index": 7123,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.125,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7123.png",
+      "answer": "(0,1/8)",
+      "fact_expr": "G: Parabola;Expression(G) = (y = 2*x^2)",
+      "coords": {}
+    },
+    {
+      "index": 7124,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7124.png",
+      "answer": "5",
+      "fact_expr": "Slope(G) = 2;G: Line;C: Parabola;Expression(C) = (y^2 = 4*x);PointOnCurve(Focus(C), G) =  True;Intersection(G, C) = {A, B};A: Point;B: Point",
+      "coords": {}
+    },
+    {
+      "index": 7125,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7125.png",
+      "answer": "81",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/9 + x^2/m = 1);m: Number;m>0;Expression(OneOf(Asymptote(C))) = (y = x/3)",
+      "coords": {}
+    },
+    {
+      "index": 7126,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7127,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.449489742783178,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 6.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7127.png",
+      "answer": "1/3",
+      "fact_expr": "l: Line;C: Ellipse;P: Point;A: Point;B: Point;b:Number;Expression(C) = (x^2/6 + y^2/3 = 1);PointOnCurve(P,C);Coordinate(P) = (2, 1);Expression(l)=(y=-x+b);Intersection(l, C) = {A,B};IsPerpendicular(LineSegmentOf(P, A), LineSegmentOf(P, B));Negation(A=B)",
+      "coords": {
+        "P": [
+          2.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 7128,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.4142135623730951,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7128.png",
+      "answer": "4*sqrt(6)/3",
+      "fact_expr": "G: Hyperbola;P: Point;Expression(G) = (x^2/4 - y^2/2 = 1);PointOnCurve(P, G);Distance(P, RightFocus(G)) = 2",
+      "coords": {}
+    },
+    {
+      "index": 7129,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.1622776601683795,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 10.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7129.png",
+      "answer": "48",
+      "fact_expr": "l: Line;G: Ellipse;H:Circle;P: Point;A: Point;B: Point;Expression(G) = (x^2/10 + y^2 = 1);Expression(H) = (x^2 + (y - 6)^2 = 2);PointOnCurve(P, G);PointOnCurve(Center(H),l);Intersection(l, H) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 7130,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7130.png",
+      "answer": "x^2/25+y^2/16=1",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;B: Point;A: Point;F: Point;C: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftVertex(G) = A;RightFocus(G) = F;UpperVertex(G) = B;LowerVertex(G) = C;T:Point;Coordinate(T)=(3*a,16);Intersection(LineOf(A,B),LineOf(C,F))=T",
+      "coords": {}
+    },
+    {
+      "index": 7131,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7132,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "y_coef": 3.0
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+        "note": "using explicit params"
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+      "answer": "sqrt(3)/2",
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+    {
+      "index": 7133,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7133.png",
+      "answer": "3",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7134.png",
+      "answer": "pm*2*sqrt(2)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7136.png",
+      "answer": "4*x^2/9 - y^2/4 = 1",
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+      "answer": "(1, 2)",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7140.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "C: Hyperbola;a: Number;F: Point;a>0;Expression(C) = (-y^2/3 + x^2/a^2 = 1);RightFocus(C) = F",
+      "coords": {}
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+    {
+      "index": 7141,
+      "success": true,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7141.png",
+      "answer": "{sqrt(21)/3, sqrt(13)}",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;L1: Line;L2: Line;Asymptote(G) = {L1, L2};H: Parabola;Expression(H) = (y^2 = 4*x);O: Origin;A: Point;Intersection(L1, H) = A;B: Point;Intersection(L2, H) = B;Negation(O=A);Negation(O=B);F: Point;Focus(H) = F;AngleOf(A, F, B) = (2*pi)/3",
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+        "iterations": 89
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7142.png",
+      "answer": "2*sqrt(3)/3",
+      "fact_expr": "C1: Hyperbola;Expression(C1) = (x^2/a1^2 - y^2/b1^2 = 1);a1: Number;b1: Number;a1>0;b1>0;C2: Hyperbola;Expression(C2) = (y^2/a2^2 - x^2/b2^2 = 1);a2: Number;b2: Number;a2>0;b2>0;Asymptote(C1) = Asymptote(C2);Eccentricity(C1) = 2",
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+      "index": 7143,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7144.png",
+      "answer": "6",
+      "fact_expr": "G: Hyperbola;H: Parabola;p: Number;Expression(G) = (x^2/6 - y^2/3 = 1);Expression(H) = (y^2 = 2*(p*x));Focus(H) = RightFocus(G)",
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+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7145.png",
+      "answer": "3/2",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 + y^2/3 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) - Abs(LineSegmentOf(P, F2)) = 1",
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+    {
+      "index": 7146,
+      "success": true,
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+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7146.png",
+      "answer": "pm*4/3",
+      "fact_expr": "l: Line;G: Parabola;A: Point;F: Point;B: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B};VectorOf(A, F) = 4*VectorOf(F, B)",
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+    {
+      "index": 7147,
+      "success": true,
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+        "x_coef": 3.0,
+        "y_coef": 4.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7147.png",
+      "answer": "8",
+      "fact_expr": "C: Ellipse;G: Line;F2: Point;M: Point;N: Point;F1: Point;Expression(C) = (x^2/3 + y^2/4 = 1);Focus(C) = {F1, F2};PointOnCurve(F1, G);Intersection(G, C) = {M,N}",
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+    {
+      "index": 7148,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7148.png",
+      "answer": "y=pm*(sqrt(3)/3)*x",
+      "fact_expr": "G: Hyperbola;a: Number;H: Parabola;a>0;Expression(G) = (-y^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 8*x);RightFocus(G) = Focus(H)",
+      "coords": {}
+    },
+    {
+      "index": 7149,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7149.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "F1: Point;F2: Point;LeftFocus(E) = F1;RightFocus(E) = F2;E: Hyperbola;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;l: Line;PointOnCurve(F1, l);NumIntersection(l, E) = 1;O1: Circle;Expression(O1) = (x^2 + y^2 = 2*a^2);IsTangent(l,O1) = True",
+      "coords": {}
+    },
+    {
+      "index": 7150,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7150.png",
+      "answer": "a",
+      "fact_expr": "P: Point;PointOnCurve(P, RightPart(G));G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;FocalLength(G) = 2*c;c: Number",
+      "coords": {}
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+    {
+      "index": 7151,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7152,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7152.png",
+      "answer": "-1/2",
+      "fact_expr": "G: Hyperbola;H: Point;a: Number;b: Number;Expression(G) = (x^2 - y^2 = 1);Coordinate(H) = (a, b);PointOnCurve(H, LeftPart(G));Expression(L) = (y = x);Distance(H, L) = sqrt(2);OneOf(Asymptote(G)) = L;L:Line",
+      "coords": {}
+    },
+    {
+      "index": 7153,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7153.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;H: Parabola;Expression(H) = (y^2 = 4*x);OneOf(Focus(G)) = Focus(H);Distance(Focus(G), Asymptote(G)) = sqrt(3)/2",
+      "coords": {}
+    },
+    {
+      "index": 7154,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 7155,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7155.png",
+      "answer": "21/10",
+      "fact_expr": "l: Line;G: Parabola;p: Number;H:Line;A: Point;B: Point;F: Point;p>0;alpha:Number;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (x = -1/2);Coordinate(F) = (-Sin(alpha)*Cos(alpha), 0);Tan(alpha) = -2;Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B};Abs(LineSegmentOf(A, B)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 7156,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7156.png",
+      "answer": "[4, +oo)",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2/2 - y^2/lambda = 1);lambda: Number;Negation(Eccentricity(C) > sqrt(2*lambda - 5))",
+      "coords": {}
+    },
+    {
+      "index": 7157,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7157.png",
+      "answer": "7/9",
+      "fact_expr": "K: Point;Coordinate(K) = (-1, 0);PointOnCurve(K, l) = True;l: Line;C: Parabola;Expression(C) = (y^2 = 4*x);Intersection(l, C) = {A, B};A: Point;B: Point;F: Point;Focus(C) = F;Abs(LineSegmentOf(B, F)) = 2*Abs(LineSegmentOf(A, F))",
+      "coords": {
+        "K": [
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+          0.0
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+    },
+    {
+      "index": 7158,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7159,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7159.png",
+      "answer": "sqrt(3)-1",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;F2: Point;P: Point;e: Number;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);AngleOf(P, F1, F2) = ApplyUnit(30, degree);AngleOf(P, F2, F1) = ApplyUnit(60, degree);Eccentricity(G) = e",
+      "coords": {}
+    },
+    {
+      "index": 7160,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7160.png",
+      "answer": "7",
+      "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (6, 3);PointOnCurve(P, G);Focus(G) = F",
+      "coords": {
+        "A": [
+          6.0,
+          3.0
+        ]
+      }
+    },
+    {
+      "index": 7161,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7161.png",
+      "answer": "3",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;Focus(C) = {F1, F2};P: Point;PointOnCurve(P, C);IsPerpendicular(VectorOf(P, F2), VectorOf(P, F1));Area(TriangleOf(P, F1, F2)) = 9",
+      "coords": {}
+    },
+    {
+      "index": 7162,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7162.png",
+      "answer": "2*sqrt(6)/3",
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+      "answer": "y^2/64 + x^2/48 = 1",
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+      "answer": "sqrt(13)",
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+      "answer": "2",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y)",
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+      "index": 7169,
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+      "answer": "-9",
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+      "index": 7171,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7171.png",
+      "answer": "sqrt(2)/2",
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+      "index": 7172,
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+      "answer": "18",
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+      "error": "Unsupported or unparseable expression"
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+      "answer": "b^2",
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+        "y_coef": 16.0
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+        "direction": "down"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7176.png",
+      "answer": "(0,-1/2)",
+      "fact_expr": "G: Parabola;Expression(G) = (y = -x^2/2)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7177.png",
+      "answer": "x^2/1 - y^2/4 = 1",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7178.png",
+      "answer": "sqrt(3)",
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+      "index": 7179,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7180.png",
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+      "answer": "5/3",
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+      "answer": "6",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7184.png",
+      "answer": "sqrt(6)",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+        "note": "using explicit params"
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+      "answer": "8",
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7188.png",
+      "answer": "7",
+      "fact_expr": "A: Point;C: Parabola;Expression(C) = (y^2 = 12*x);PointOnCurve(A, C);Distance(A, Focus(C)) = 10",
+      "coords": {}
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+      "index": 7189,
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+      "error": "validation: ellipse_point_off_curve",
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+        "iterations": 84
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7189.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "C: Ellipse;G: Point;Coordinate(G) = (2, sqrt(2));PointOnCurve(G, C);H: Point;Coordinate(H) = (sqrt(2), sqrt(3));PointOnCurve(H, C)",
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+      "index": 7190,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 7191,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 7192,
+      "success": true,
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+        "b": 1.5,
+        "orientation": "vertical"
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+        "final_loss": 0.0,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7192.png",
+      "answer": "(0,pm*sqrt(5))",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/2 + y^2/3 = 1)",
+      "coords": {}
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+    {
+      "index": 7193,
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_7193.png",
+      "answer": "27*sqrt(7)/56",
+      "fact_expr": "Gamma:Parabola;C: Circle;O:Origin;Vertex(Gamma)=O;SymmetryAxis(Gamma)=xAxis;P:Point;Q:Point;Expression(C) = (-4*y + x^2 + y^2 = 0);Intersection(Gamma,C)={P,Q};Distance(P,Q)=3",
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+      "answer": "2",
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+    {
+      "index": 7195,
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+      "answer": "5",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7196.png",
+      "answer": "sqrt(6)/3",
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+      "answer": "2",
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+      "answer": "x + pm*sqrt(3)*y = 0",
+      "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;F:Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(l)=(x+3*y-2*b=0);RightFocus(C)=F;PointOnCurve(F,l)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7199.png",
+      "answer": "16",
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+      "answer": "[80, 120]",
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+      "answer": "3*x + 6*y + 4 = 0",
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+      "answer": "12",
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+      "answer": "-6",
+      "fact_expr": "G: Hyperbola;m: Real;Expression(G) = (x^2/m + y^2/2 = 1);Eccentricity(G) = 2",
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+      "answer": "Ellipse",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7207.png",
+      "answer": "sqrt(2)/2",
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+      "success": false,
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+      "answer": "8",
+      "fact_expr": "C: Hyperbola;G: Point;A: Point;P: Point;F: Point;PointOnCurve(G,C);Coordinate(G) = (2*sqrt(2), sqrt(3));Coordinate(A) = (0, 3);PointOnCurve(P,RightPart(C));LeftFocus(C) = F;Expression(Asymptote(C))=(y=pm*(sqrt(3)/2)*x)",
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+      "success": true,
+      "conic_type": "parabola",
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+      "params": {
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7209.png",
+      "answer": "2*sqrt(15)",
+      "fact_expr": "G: Parabola;H: Line;k: Number;A: Point;B: Point;Expression(G) = (y^2 = 8*x);Expression(H) = (y = k*x - 2);Intersection(H, G) = {A, B};XCoordinate(MidPoint(LineSegmentOf(A, B))) = 2",
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+    {
+      "index": 7210,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "x_coef": 0.5,
+        "y_coef": 2.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7210.png",
+      "answer": "8",
+      "fact_expr": "G: Ellipse;k: Real;A:Point;B:Point;Expression(G) = (4*x^2 + y^2 = 2);PointOnCurve(A,G);PointOnCurve(B,G);Max(Distance(A,B))=8",
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+      "index": 7211,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 7212,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 7213,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 7214,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.0,
+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7214.png",
+      "answer": "(0,1/8)",
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+      "coords": {}
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+    {
+      "index": 7215,
+      "success": true,
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+        "y_coef": 1.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7215.png",
+      "answer": "(0,-1)",
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+          3.0
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+      "index": 7216,
+      "success": true,
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+        "b": 2.364173412322998,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "iterations": 89
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7216.png",
+      "answer": "4",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(C) = 2;Distance(Focus(C),Asymptote(C))=sqrt(3)",
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+    {
+      "index": 7217,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 7218,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7218.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "G: Parabola;H: Line;I: Curve;P: Point;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 4*x);Expression(I) = And((y^2=-4*x), (y>=0));Focus(G) = F;PointOnCurve(F, H);Slope(H) = 1;Intersection(H, G) = {A, B};PointOnCurve(P, I)",
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+    {
+      "index": 7219,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7219.png",
+      "answer": "[2*pi, pi*10/3)",
+      "fact_expr": "G: Hyperbola;I: Line;A: Point;F1: Point;B: Point;F2: Point;Expression(G) = (x^2 - y^2/3 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2, I);Intersection(I, RightPart(G)) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 7220,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 8.0,
+        "b": 6.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7220.png",
+      "answer": "16",
+      "fact_expr": "G: Hyperbola;P: Point;Expression(G) = (x^2/64 - y^2/36 = 1);PointOnCurve(P, G);Distance(P, RightFocus(G)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 7221,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7221.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Distance(RightVertex(G), OneOf(Asymptote(G))) = b/2",
+      "coords": {}
+    },
+    {
+      "index": 7222,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7223,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7223.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Endpoint(MinorAxis(C)) = M;M: Point;PointOnCurve(N,C) = True;N: Point;VectorOf(M, F2) = 3*VectorOf(F2, N);Area(TriangleOf(M, N, F1)) = 2",
+      "coords": {}
+    },
+    {
+      "index": 7224,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 6.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 36.0,
+        "y_coef": 9.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7224.png",
+      "answer": "x+2*y-8=0",
+      "fact_expr": "l: Line;G: Ellipse;A: Point;B: Point;M: Point;Expression(G) = (x^2 + 4*y^2 = 36);InterceptChord(l,G)=LineSegmentOf(A,B);MidPoint(LineSegmentOf(A,B))=M;Coordinate(M) = (4, 2)",
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+      "answer": "3*sqrt(3)",
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+      "index": 7227,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7228.png",
+      "answer": "x^2/10 - y^2/(5/2) = 1",
+      "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (3*x^2 + 13*y^2 = 39);Focus(H)=Vertex(G);Expression(Asymptote(G)) = (y=pm*(1/2)*x)",
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+      "index": 7229,
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7229.png",
+      "answer": "y^2 = 4*x",
+      "fact_expr": "C: Parabola;p: Number;Z: Circle;H: Line;M: Point;E: Point;G: Point;F: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Expression(H) = (x = p/2);Coordinate(M) = (x0, 2*sqrt(2));x0:Number;x0>p/2;Focus(C) = F;PointOnCurve(M, C);Center(Z)=M;Radius(Z)=sqrt(3);Intersection(Z,H)={E,G};Abs(LineSegmentOf(E, G)) = 2*sqrt(2)",
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+    {
+      "index": 7230,
+      "success": true,
+      "conic_type": "ellipse",
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+        "y_coef": 16.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7230.png",
+      "answer": "(-3,0);(3,0)",
+      "fact_expr": "G: Ellipse;F1: Point;F2: Point;Expression(G) = (x^2/25 + y^2/16 = 1);LeftFocus(G) = F1;RightFocus(G) = F2",
+      "coords": {}
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+    {
+      "index": 7231,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 1.4142135623730951,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7231.png",
+      "answer": "2*sqrt(3)/3",
+      "fact_expr": "G: Hyperbola;M: Point;F1: Point;F2: Point;Expression(G) = (x^2 - y^2/2 = 1);Focus(G) = {F1, F2};PointOnCurve(M, G);DotProduct(VectorOf(M, F1), VectorOf(M, F2)) = 0",
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+    {
+      "index": 7232,
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+        "y_coef": 4.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7232.png",
+      "answer": "4/3",
+      "fact_expr": "l: Line;E: Ellipse;b: Number;A: Point;B: Point;F2: Point;F1: Point;0 < b;b < 1;Expression(E) = (x^2 + y^2/b^2 = 1);LeftFocus(E) = F1;RightFocus(E) = F2;PointOnCurve(F1, l);Intersection(l, E) = {A, B};2*Abs(LineSegmentOf(A, B)) = Abs(LineSegmentOf(A, F2)) + Abs(LineSegmentOf(B, F2))",
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+    {
+      "index": 7233,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7233.png",
+      "answer": "44",
+      "fact_expr": "P: Point;Q: Point;C: Hyperbola;A: Point;F: Point;Expression(C) = (x^2/9 - y^2/16 = 1);Coordinate(A) = (5, 0);LeftFocus(C) = F;PointOnCurve(P, C);PointOnCurve(Q, C);2*Length(ImageinaryAxis(C)) = Length(LineSegmentOf(P, Q));PointOnCurve(A, LineSegmentOf(P, Q))",
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+      "index": 7234,
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7234.png",
+      "answer": "y=pm*(sqrt(5)/2)*x",
+      "fact_expr": "C: Hyperbola;a: Number;a>0;Expression(C) = (-y^2/5 + x^2/a^2 = 1);Coordinate(OneOf(Focus(C))) = (3, 0)",
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+    {
+      "index": 7235,
+      "success": true,
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+        "y_coef": 1.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7235.png",
+      "answer": "2*x+4*y-3=0",
+      "fact_expr": "G: Ellipse;H: LineSegment;P: Point;Coordinate(P) = (1/2, 1/2);Expression(G) = (x^2/2 + y^2 = 1);IsChordOf(H,G);MidPoint(H)=P;PointOnCurve(P,H)",
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+          0.5
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+    {
+      "index": 7236,
+      "success": true,
+      "conic_type": "ellipse",
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+        "b": 3.0,
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+        "x_coef": 25.0,
+        "y_coef": 9.0
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7236.png",
+      "answer": "8",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/25 + y^2/9 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 7237,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7237.png",
+      "answer": "6",
+      "fact_expr": "G: Parabola;H: Ellipse;a: Number;P: Point;F1: Point;F2: Point;F3: Point;a>0;Expression(H) = (y^2/3 + x^2/a^2 = 1);Focus(H)={F1,F2};PointOnCurve(P,H);PointOnCurve(P,G);Focus(G) = F3;Abs(LineSegmentOf(P, F3)) = 25/16;Expression(G)=(y^2=(9/4)*x)",
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+    },
+    {
+      "index": 7238,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7239,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.0,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7239.png",
+      "answer": "4*sqrt(5)/5",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2/4 - y^2 = 1);E: Circle;Expression(E) = (-2*x + x^2 + y^2 = 0);A: Point;B: Point;Intersection(OneOf(Asymptote(C)), E) = {A, B}",
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+    },
+    {
+      "index": 7240,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.0,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7240.png",
+      "answer": "6",
+      "fact_expr": "C: Hyperbola;A: Point;P: Point;F: Point;Expression(C) = (x^2 - y^2 = 1);Coordinate(A) = (0, sqrt(2));RightFocus(C) = F;PointOnCurve(P, LeftPart(C))",
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+          1.4142135623730951
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+      "index": 7241,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 7242,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7242.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;m: Number;n: Number;H: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;m>0;n>0;Expression(G) = (-y^2/n^2 + x^2/m^2 = 1);a > b;b > 0;Expression(H) = (y^2/b^2 + x^2/a^2 = 1);Focus(H) = {F1, F2};Focus(G) = {F1, F2};OneOf(Intersection(H,G))=P;Abs(LineSegmentOf(P, F1))*Abs(LineSegmentOf(P, F2)) = 2",
+      "coords": {}
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+    {
+      "index": 7243,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7243.png",
+      "answer": "sqrt(5)/3",
+      "fact_expr": "l: Line;C: Ellipse;b: Number;a: Number;A: Point;F: Point;B: Point;a > b;b > 0;Expression(C) = (x^2/b^2 + y^2/a^2 = 1);Focus(C) = F;PointOnCurve(F, l);Slope(l) = 1/2;Intersection(l, C) = {A, B};VectorOf(A, F) = (2/3)*VectorOf(A, B)",
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+    {
+      "index": 7244,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "up"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7244.png",
+      "answer": "x+y+3=0",
+      "fact_expr": "G: Parabola;H: Line;C:Line;Expression(G) = (x^2 = 12*y);Expression(H) = (x - y = 0);IsTangent(C,G);IsPerpendicular(C,H)",
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+    {
+      "index": 7245,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 4.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7245.png",
+      "answer": "sqrt(3)/2\nx^2/9-y^2/3=1",
+      "fact_expr": "H: Ellipse;Expression(H) = (x^2 + 4*y^2 = 16);G: Hyperbola;Focus(G) = Focus(H);Expression(OneOf(Asymptote(G))) = (x + sqrt(3)*y = 0)",
+      "coords": {}
+    },
+    {
+      "index": 7246,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.117531657218933,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "iterations": 59
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7246.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;F2: Point;F1: Point;M: Point;Q: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-2, 0);Coordinate(F2) = (2, 0);LeftFocus(C) = F1;RightFocus(C)=F2;PointOnCurve(M, RightPart(C));Intersection(LineSegmentOf(M, F1), yAxis) = P;TangentPoint(InscribedCircle(TriangleOf(M,P,F2)),LineSegmentOf(P,F2))=Q;Abs(LineSegmentOf(P, Q)) = sqrt(2)",
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+        "F1": [
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+          0.0
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+        "F2": [
+          2.0,
+          0.0
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+    {
+      "index": 7247,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7248,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7249,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7249.png",
+      "answer": "x \\pm y + 1 = 0",
+      "fact_expr": "C: Parabola;P: Point;L: Line;Expression(C) = (y^2 = 4*x);Intersection(Directrix(C),SymmetryAxis(C))=P;TangentOfPoint(P, C) = L",
+      "coords": {}
+    },
+    {
+      "index": 7250,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7251,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7251.png",
+      "answer": "-1",
+      "fact_expr": "G: Parabola;a: Real;l:Line;Expression(G) = (y^2 = 4*x);Expression(l) = (a*x - y + 1 = 0);PointOnCurve(Focus(G), l)",
+      "coords": {}
+    },
+    {
+      "index": 7252,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7252.png",
+      "answer": "sqrt(6)/3",
+      "fact_expr": "H: Ellipse;Expression(H) = (y^2/2 + x^2/a^2 = 1);a: Number;G: Parabola;Expression(G) = (y^2 = 8*x);OneOf(Focus(H)) = Focus(G)",
+      "coords": {}
+    },
+    {
+      "index": 7253,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7253.png",
+      "answer": "y=pm*(sqrt(2)/2)*x",
+      "fact_expr": "G: Hyperbola;H: Ellipse;a1: Number;b1: Number;b1 > 0;a1 > b1;a2: Number;b2: Number;a2 > 0;b2 > 0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2/b^2 + x^2/a^2 = 3);Focus(G) = Focus(H)",
+      "coords": {}
+    },
+    {
+      "index": 7254,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7254.png",
+      "answer": "sqrt(34)",
+      "fact_expr": "G: Parabola;A: Point;F: Point;P: Point;Coordinate(A) = (5, 4);Expression(G) = (x^2 = 4*y);Focus(G) = F;PointOnCurve(P, G)",
+      "coords": {
+        "A": [
+          5.0,
+          4.0
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+      }
+    },
+    {
+      "index": 7255,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
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+      "answer": "{y^2 = 4*x, y^2 = 16*x}",
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+      "answer": "1/2",
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+      "index": 7260,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7260.png",
+      "answer": "7/4",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7261.png",
+      "answer": "[1/2, 2]",
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+      "index": 7265,
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+      "index": 7266,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7266.png",
+      "answer": "pm*2*sqrt(2)",
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+      "index": 7267,
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+      "index": 7268,
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+        "b": 2.23606797749979,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7268.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/5 = 1);F: Point;OneOf(Focus(G)) = F",
+      "coords": {}
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+    {
+      "index": 7269,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7269.png",
+      "answer": "1/4",
+      "fact_expr": "C: Hyperbola;P: Point;F1: Point;F2: Point;Expression(C) = (x^2 - y^2/3 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))",
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+    {
+      "index": 7270,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7270.png",
+      "answer": "x^2/2-y^2=1",
+      "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/4 + y^2 = 1);Focus(G) = Focus(H);P: Point;Coordinate(P) = (2, 1);PointOnCurve(P, G) = True",
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+      "index": 7272,
+      "success": true,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7272.png",
+      "answer": "5+sqrt(13)",
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+          2.0
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+    {
+      "index": 7273,
+      "success": false,
+      "error": "Circle from diameter: missing point coordinates"
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+      "index": 7274,
+      "success": true,
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+        "y_coef": 24.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7274.png",
+      "answer": "x^2/9 - y^2/16 = 1",
+      "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/49 + y^2/24 = 1);Focus(G)=Focus(H);Expression(Asymptote(G))=(y=(pm*4/3)*x)",
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+    {
+      "index": 7275,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 7276,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "a": 8.0,
+        "b": 6.0,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7276.png",
+      "answer": "16",
+      "fact_expr": "G: Hyperbola;P: Point;Expression(G) = (x^2/64 - y^2/36 = 1);PointOnCurve(P, G);Distance(P, RightFocus(G)) = 4",
+      "coords": {}
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+    {
+      "index": 7277,
+      "success": true,
+      "conic_type": "ellipse",
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+        "b": 1.0,
+        "major_axis": "y",
+        "x_coef": 1.0,
+        "y_coef": 4.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7277.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "G: Ellipse;H: Line;P: Point;Expression(G) = (x^2 + y^2/3 = 1);Expression(H) = (x + y - 4 = 0);PointOnCurve(P, G)",
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+    {
+      "index": 7278,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 7279,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7279.png",
+      "answer": "sqrt(3)/3",
+      "fact_expr": "F1: Point;F2: Point;C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C);PointOnCurve(MidPoint(LineSegmentOf(P, F1)), yAxis);AngleOf(P, F1, F2) = ApplyUnit(30, degree)",
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+    {
+      "index": 7280,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 7281,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7281.png",
+      "answer": "2*sqrt(7)-2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1);P: Point;PointOnCurve(P, RightPart(G));F: Point;RightFocus(G) = F;A: Point;Coordinate(A) = (3, sqrt(3))",
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+      "index": 7282,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 7283,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7283.png",
+      "answer": "sqrt(6)/3",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;H: Circle;F1: Point;F2: Point;P: Point;c:Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);Focus(G)={F1,F2};IsDiameter(Abs(LineSegmentOf(F1,F2)),H);OneOf(Intersection(H,G))=P;AngleOf(P, F1, F2) = 5*AngleOf(P, F2, F1)",
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+    {
+      "index": 7284,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7284.png",
+      "answer": "[\\sqrt{2},2]",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;a>0;b>0;L1:Line;L2:Line;d1:Number;d2:Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, G);Asymptote(G)={L1,L2};Distance(P,L1)=d1;Distance(P,L2)=d2;3*a^2+b^2<=8*d1*d2",
+      "coords": {}
+    },
+    {
+      "index": 7285,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7285.png",
+      "answer": "2*sqrt(17) - 2",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2/3 = 1);A: Point;RightVertex(C) = A;M: Point;Coordinate(M) = (2, 4);P: Point;PointOnCurve(P, C) ;L: Line;PointOnCurve(P, L) ;IsPerpendicular(L, LineOf(M, A)) ;FootPoint(L, LineOf(M, A)) = H;H: Point",
+      "coords": {
+        "M": [
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+          4.0
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+    {
+      "index": 7286,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "vertical"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7286.png",
+      "answer": "y=pm*(5/4)*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/16 + y^2/25 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 7287,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7287.png",
+      "answer": "(0, 1/2]",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C) = True;Coordinate(P) = (x0, y0);x0: Number;y0: Number;x0 >= 0;AngleOf(P, F1, F2) = ApplyUnit(60, degree)",
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+    {
+      "index": 7288,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7289,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7289.png",
+      "answer": "sqrt(5)/10",
+      "fact_expr": "m: Number;m > 0;M: Hyperbola;Expression(M) = (x^2/4 - y^2 = 1);N: Circle;Expression(N) = (x^2 + (-m + y)^2 = 5);IsTangent(M, N);A: Point;Coordinate(A) = (-sqrt(5), 0);B: Point;Coordinate(B) = (sqrt(5), 0);P: Point;PointOnCurve(P, N);Abs(LineSegmentOf(P, A)) - Abs(LineSegmentOf(P, B)) = 4",
+      "coords": {
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+          0.0
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+        "B": [
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+          0.0
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+    },
+    {
+      "index": 7290,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7290.png",
+      "answer": "44",
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+      "answer": "(1,3]",
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+      "answer": "[3,15]",
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+      "answer": "4*x+3*y-7=0",
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+      "answer": "sqrt(3)",
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+      "answer": "y=pm*sqrt(2)*x",
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+      "answer": "7",
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+      "answer": "14+sqrt(41)",
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+        "direction": "right"
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+      "answer": "16",
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "answer": "3",
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+        "y_coef": 1.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7308.png",
+      "answer": "sqrt(3)/2",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2 + 4*y^2 = 4)",
+      "coords": {}
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7309.png",
+      "answer": "sqrt(3)+1",
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+      "index": 7310,
+      "success": true,
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7310.png",
+      "answer": "(3,pm*2*sqrt(3))",
+      "fact_expr": "G: Parabola;P: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(P, G);Distance(P, Focus(G)) = 4",
+      "coords": {}
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+      "success": true,
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+        "iterations": 81
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7311.png",
+      "answer": "sqrt(3)/2",
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+        "M": [
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+          0.0
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+      "index": 7312,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 7313,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7313.png",
+      "answer": "pi/6",
+      "fact_expr": "G: Parabola;N: Point;F: Point;M: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;Intersection(Directrix(G),xAxis)=M;PointOnCurve(N,G);Abs(LineSegmentOf(N, F)) = (sqrt(3)/2)*Abs(LineSegmentOf(M, N))",
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+      "index": 7314,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 7315,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "success": true,
+      "conic_type": "ellipse",
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+        "x_coef": 3.0,
+        "y_coef": 2.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7316.png",
+      "answer": "2",
+      "fact_expr": "C: Curve;Expression(C) = (x^2/3 + y^2/2 = 1)",
+      "coords": {}
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+      "index": 7317,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "orientation": "horizontal"
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7317.png",
+      "answer": "x^2 - y^2 / 3 = 1",
+      "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(l) = (x+sqrt(3)*y=0);IsPerpendicular(OneOf(Asymptote(C)),l);Distance(OneOf(Focus(C)), l) = 1",
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+      "index": 7318,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7318.png",
+      "answer": "4/3",
+      "fact_expr": "C: Parabola;p: Number;G: Line;A: Point;F: Point;B: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;PointOnCurve(F, G);Intersection(G, C) = {A, B};Abs(LineSegmentOf(A, F)) = 2*Abs(LineSegmentOf(B, F));2*Abs(LineSegmentOf(B, F)) = 2",
+      "coords": {}
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+    {
+      "index": 7319,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7319.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, G);Focus(G) ={F1,F2};IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2));Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))",
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+    {
+      "index": 7320,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7320.png",
+      "answer": "5*sqrt(2)/4",
+      "fact_expr": "G: Parabola;H: Line;P: Point;Expression(G) = (y^2 = 10*x);Expression(H) = (x + y + 5 = 0);PointOnCurve(P, G)",
+      "coords": {}
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+    {
+      "index": 7321,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7321.png",
+      "answer": "(1, 2)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2 + (y + 2)^2 = 1);NumIntersection(Asymptote(G),H)=0",
+      "coords": {}
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+    {
+      "index": 7322,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7322.png",
+      "answer": "7/2",
+      "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;M: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (x1,y1);Coordinate(B) = (x2,y2);PointOnCurve(Focus(G), H);Intersection(H, G) = {A,B};Abs(LineSegmentOf(A, B)) = 7;MidPoint(LineSegmentOf(A, B)) = M;x1:Number;x2:Number;y1:Number;y2:Number",
+      "coords": {}
+    },
+    {
+      "index": 7323,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7323.png",
+      "answer": "(6,+oo)",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 8*y);A: Point;Intersection(Directrix(G), yAxis) = A;B: Point;PointOnCurve(B, SymmetryAxis(G));H: Line;PointOnCurve(A, H);M: Point;N: Point;Intersection(H, G) = {M, N};DotProduct(VectorOf(B, M), VectorOf(M, N)) = (-1/2)*VectorOf(M, N)^2;O: Origin",
+      "coords": {}
+    },
+    {
+      "index": 7324,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7325,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7326,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7326.png",
+      "answer": "x = -4",
+      "fact_expr": "G: Parabola;p: Number;M: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(M) = (1, m);m:Number;PointOnCurve(M, G);Distance(M, Focus(G)) = 5",
+      "coords": {}
+    },
+    {
+      "index": 7327,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "down"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7327.png",
+      "answer": "(0,-2)",
+      "fact_expr": "C: Parabola;Expression(C) = (y = -x^2/8)",
+      "coords": {}
+    },
+    {
+      "index": 7328,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7328.png",
+      "answer": "sqrt(6)/6",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 1);P: Point;a: Number;b: Number;Coordinate(P) = (a, b);PointOnCurve(P, RightPart(G));H: Line;Expression(H) = (y = x);Distance(P, H) = sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 7329,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7329.png",
+      "answer": "3",
+      "fact_expr": "l: Line;C: Parabola;P: Point;Q: Point;F: Point;Expression(C) = (y^2 = 8*x);Focus(C) = F;PointOnCurve(F,l);Intersection(l, C) = {P,Q};Abs(LineSegmentOf(P, Q)) = 32/3;Abs(LineSegmentOf(P, F)) > Abs(LineSegmentOf(Q, F))",
+      "coords": {}
+    },
+    {
+      "index": 7330,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7331,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7331.png",
+      "answer": "y^2=12*x",
+      "fact_expr": "G: Parabola;H: Ellipse;Expression(H) = (x^2/25 + y^2/16 = 1);Vertex(G) = Center(H);Focus(G) = RightFocus(H)",
+      "coords": {}
+    },
+    {
+      "index": 7332,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7332.png",
+      "answer": "(1,2]",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;e: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) =F1;RightFocus(G)=F2;PointOnCurve(P, RightPart(G));Abs(LineSegmentOf(P, F1)) = 3*Abs(LineSegmentOf(P, F2));Eccentricity(G) = e",
+      "coords": {}
+    },
+    {
+      "index": 7333,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7333.png",
+      "answer": "x^2/4+y^2=1",
+      "fact_expr": "l: Line;C: Ellipse;b: Number;a: Number;P: Point;M: Point;N: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Length(MajorAxis(C)) = 4;PointOnCurve(P, C);O:Origin;PointOnCurve(O,l);Intersection(l, C) = {M, N};Slope(LineSegmentOf(P, M)) = K1;Slope(LineSegmentOf(P, N)) = K2;K1*K2 = -1/4;K1:Number;K2:Number",
+      "coords": {}
+    },
+    {
+      "index": 7334,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7334.png",
+      "answer": "sqrt(21)/3",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;M: Point;F1: Point;F2: Point;N: Point;O: Origin;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Focus(C) = {F1, F2};PointOnCurve(M, Asymptote(C));Abs(LineSegmentOf(M, F1)) = 2*Abs(LineSegmentOf(M, F2));VectorOf(M, N) = 2*VectorOf(O, N);AngleOf(M, F2, N) = ApplyUnit(120, degree)",
+      "coords": {}
+    },
+    {
+      "index": 7335,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7335.png",
+      "answer": "sqrt(10)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;H: Line;Expression(H) = (x + 3*y + 1 = 0);IsPerpendicular(OneOf(Asymptote(G)), H)",
+      "coords": {}
+    },
+    {
+      "index": 7336,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7336.png",
+      "answer": "y=pm*2*x",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Ellipse;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2/20 + y^2/25 = 1);Distance(Focus(C),Asymptote(C))=2;FocalLength(C)=FocalLength(G)",
+      "coords": {}
+    },
+    {
+      "index": 7337,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7337.png",
+      "answer": "[1, 5)",
+      "fact_expr": "G: Ellipse;m: Number;H: Line;a: Real;Expression(G) = (x^2/5 + y^2/m = 1);Expression(H) = (y = -1 + a*x);IsIntersect(G, H) ;PointOnCurve(Focus(G),xAxis)",
+      "coords": {}
+    },
+    {
+      "index": 7338,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7338.png",
+      "answer": "sqrt(7)",
+      "fact_expr": "Gamma: Hyperbola;a: Number;b: Number;lambda:Number;A: Point;B: Point;F1: Point;C: Point;F2: Point;a>0;b>0;Expression(Gamma) = (x^2/a^2 - y^2/b^2 = 1);LeftFocus(Gamma) = F1;RightFocus(Gamma) = F2;PointOnCurve(A,LeftPart(Gamma));PointOnCurve(B,RightPart(Gamma));VectorOf(A,B)=lambda*VectorOf(F1,A);PointOnCurve(C, xAxis);VectorOf(C, B) = 3*VectorOf(F2, A);DotProduct(VectorOf(B, F2), VectorOf(B, F1))/Abs(VectorOf(B, F1)) = DotProduct(VectorOf(B, F2), VectorOf(B, C))/Abs(VectorOf(B, C))",
+      "coords": {}
+    },
+    {
+      "index": 7339,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7340,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7340.png",
+      "answer": "y=pm*2*x",
+      "fact_expr": "E: Hyperbola;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;A: Point;RightVertex(E) = A;G: Line;PointOnCurve(A, G) = True;Slope(G) = -1;Intersection(G, Asymptote(E)) = {B, C};B: Point;C: Point;2*VectorOf(B, A) + VectorOf(B, C) = 0",
+      "coords": {}
+    },
+    {
+      "index": 7341,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7342,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7342.png",
+      "answer": "3*sqrt(5)/4",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);P: Point;PointOnCurve(P, G);Q: Point;Coordinate(Q) = (1/4, 0)",
+      "coords": {
+        "Q": [
+          0.25,
+          0.0
+        ]
+      }
+    },
+    {
+      "index": 7343,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7343.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;O: Origin;F1: Point;P: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P,RightPart(C));Abs(LineSegmentOf(O,P))= Abs(LineSegmentOf(O, F1));Abs(LineSegmentOf(P, F1)) = sqrt(3)*Abs(LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 7344,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7344.png",
+      "answer": "y = pm*sqrt(3)*x",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;RightFocus(C) = F;F: Point;Coordinate(F) = (c, 0);c: Number;Distance(F, Asymptote(C)) = c*sqrt(3)/2",
+      "coords": {}
+    },
+    {
+      "index": 7345,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7345.png",
+      "answer": "2*pi/3",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 8*y);F: Point;Focus(G) = F;l: Line;Directrix(G) = l;M: Point;Intersection(l, yAxis) = M;P: Point;PointOnCurve(P, G) = True;IsPerpendicular(LineSegmentOf(P, Q), l) = True;Intersection(LineSegmentOf(P, Q), l) = Q;Q: Point;Abs(LineSegmentOf(P, F)) = 8/3;PointOnCurve(P,LineSegmentOf(P,Q))",
+      "coords": {}
+    },
+    {
+      "index": 7346,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7346.png",
+      "answer": "sqrt(3) + 1",
+      "fact_expr": "F1: Point;F2: Point;C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, RightPart(C)) = True;Abs(LineSegmentOf(O, P)) = Abs(LineSegmentOf(O, F1));O: Origin;Abs(LineSegmentOf(P, F1)) = sqrt(3)*Abs(LineSegmentOf(P, F2))",
+      "coords": {}
+    },
+    {
+      "index": 7347,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7347.png",
+      "answer": "{x=0,3*x+4*y+8=0}",
+      "fact_expr": "l: Line;C: Parabola;A: Point;Expression(C) = (y^2 = 6*x);Coordinate(A) = (0, 2);IsTangent(l,C);PointOnCurve(A,l)=True",
+      "coords": {
+        "A": [
+          0.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 7348,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7349,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7350,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7351,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7351.png",
+      "answer": "2",
+      "fact_expr": "l: Line;G: Parabola;p: Number;O: Origin;A: Point;B: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G)=F;PointOnCurve(F,l);Slope(l)=1/2;Intersection(l,G)={A,B};Area(TriangleOf(O,A,B))=2*sqrt(5);F:Point",
+      "coords": {}
+    },
+    {
+      "index": 7352,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7352.png",
+      "answer": "32",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 2*(p*y));p: Number;p>0;F: Point;Focus(G) = F;O: Origin;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, G) = {A, B};C: Point;Coordinate(C) = (2*p, 0);Intersection(l, xAxis) = C;Abs(LineSegmentOf(A, B)) = 17",
+      "coords": {}
+    },
+    {
+      "index": 7353,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7353.png",
+      "answer": "sqrt(10)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x - 3*y + 1 = 0);IsPerpendicular(OneOf(Asymptote(G)), H)",
+      "coords": {}
+    },
+    {
+      "index": 7354,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7355,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_point_off_curve",
+      "params": {
+        "a": 1.820299506187439,
+        "b": 1.6536897822836507,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 1.2588495006847936e-06,
+        "converged": true,
+        "iterations": 97
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7355.png",
+      "answer": "2",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;P: Point;Coordinate(P) = (0, sqrt(3));a > b;b > 0;PointOnCurve(P,G);Length(MajorAxis(G))=2*FocalLength(G)",
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+          1.7320508075688772
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+      },
+      "validation_reasons": [
+        "ellipse_point_off_curve"
+      ]
+    },
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+      "index": 7356,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7357,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 16.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7357.png",
+      "answer": "x^2/9 - y^2/3 = 1",
+      "fact_expr": "H: Ellipse;Expression(H) = (x^2 + 4*y^2 = 16);G: Hyperbola;Focus(G) = Focus(H);Expression(OneOf(Asymptote(G))) = (x + sqrt(3)*y = 0)",
+      "coords": {}
+    },
+    {
+      "index": 7358,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7359,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7359.png",
+      "answer": "sqrt(10)",
+      "fact_expr": "G: Parabola;Q: Point;P: Point;Expression(G) = (y^2 = 4*x);Coordinate(Q) = (2, 3);PointOnCurve(P, G)",
+      "coords": {
+        "Q": [
+          2.0,
+          3.0
+        ]
+      }
+    },
+    {
+      "index": 7360,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7360.png",
+      "answer": "sqrt(6)/2",
+      "fact_expr": "F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;C: Hyperbola;Expression(C) = (x^2 - y^2 = 1);P: Point;PointOnCurve(P, C);AngleOf(F1, P, F2) = ApplyUnit(60, degree)",
+      "coords": {}
+    },
+    {
+      "index": 7361,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7362,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "left"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7362.png",
+      "answer": "8",
+      "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;Expression(G) = (y^2 = -4*x);Slope(H)=-1;PointOnCurve(Focus(G),H);Intersection(H,G)={A,B}",
+      "coords": {}
+    },
+    {
+      "index": 7363,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7363.png",
+      "answer": "1",
+      "fact_expr": "G: Parabola;M: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(M, G);Distance(M, Focus(G)) = 2",
+      "coords": {}
+    },
+    {
+      "index": 7364,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 3.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7364.png",
+      "answer": "3/2",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 12*x);M: Point;Coordinate(M) = (-3, 4);PointOnCurve(Focus(C), G) = True;Slope(G) = k;k: Number;G: Line;Intersection(G, C) = {A, B};A: Point;B: Point;DotProduct(VectorOf(M, A), VectorOf(M, B)) = 0",
+      "coords": {
+        "M": [
+          -3.0,
+          4.0
+        ]
+      }
+    },
+    {
+      "index": 7365,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7366,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7366.png",
+      "answer": "(2*sqrt(3),7/2)",
+      "fact_expr": "C: Ellipse;Q: Point;R: Point;O: Origin;P: Point;Expression(C) = (x^2/4 + y^2/3 = 1);PointOnCurve(P,C);Quadrant(P)=2;L:Line;TangentOnPoint(P,C)=L;Intersection(L,yAxis)=Q;PointOnCurve(Q,LineSegmentOf(Q,R));IsPerpendicular(LineSegmentOf(Q,R),LineSegmentOf(O,P));FootPoint(LineSegmentOf(Q,R),LineSegmentOf(O,P))=R",
+      "coords": {}
+    },
+    {
+      "index": 7367,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7367.png",
+      "answer": "[4, +oo)",
+      "fact_expr": "H: Line;Expression(H) = (x = t);t: Number;G: Parabola;Expression(G) = (y^2 = 4*x);A: Point;B: Point;Intersection(H, G) = {A, B};C: Point;PointOnCurve(C, G);IsPerpendicular(LineSegmentOf(A, C), LineSegmentOf(B, C))",
+      "coords": {}
+    },
+    {
+      "index": 7368,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7368.png",
+      "answer": "15",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(C) = F;Distance(F, Directrix(C)) = 4;l: Line;PointOnCurve(F, l);P: Point;Q: Point;Intersection(l, C) = {P, Q};Abs(LineSegmentOf(P, F)) = 10",
+      "coords": {}
+    },
+    {
+      "index": 7369,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7369.png",
+      "answer": "3",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 7370,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7371,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 4.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7371.png",
+      "answer": "48",
+      "fact_expr": "G: Hyperbola;P: Point;F2: Point;F1: Point;Expression(G) = (x^2/9 - y^2/16 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(P,RightPart(G));Abs(LineSegmentOf(P, F2)) = Abs(LineSegmentOf(F1, F2))",
+      "coords": {}
+    },
+    {
+      "index": 7372,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7372.png",
+      "answer": "(7+pm*sqrt(17))/16",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;Focus(G) = F2;P: Point;OneOf(Intersection(C, G)) = P;Cos(AngleOf(P, F1, F2)) = 7/9",
+      "coords": {}
+    },
+    {
+      "index": 7373,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7373.png",
+      "answer": "(0,1/2)",
+      "fact_expr": "G: Parabola;Expression(G) = (y = x^2/2)",
+      "coords": {}
+    },
+    {
+      "index": 7374,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7374.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
+      "fact_expr": "G: Hyperbola;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 - y^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(90, degree)",
+      "coords": {}
+    },
+    {
+      "index": 7375,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7376,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7377,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7377.png",
+      "answer": "x^2/4 - y^2/16 = 1\nsqrt(5)",
+      "fact_expr": "C: Hyperbola;a:Number;b:Number;a>0;b>0;G: Line;P: Point;F1: Point;F2: Point;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y = 2*x);LeftFocus(C) = F1;RightFocus(C) = F2;OneOf(Asymptote(C))=G;PointOnCurve(P, C);Abs(LineSegmentOf(P,F1))-Abs(LineSegmentOf(P,F2))=4",
+      "coords": {}
+    },
+    {
+      "index": 7378,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7378.png",
+      "answer": "3/4",
+      "fact_expr": "C: Ellipse;H: Line;A: Point;B: Point;F1: Point;F2: Point;Expression(C) = (x^2/4 + y^2/3 = 1);LeftFocus(C) =F1;RightFocus(C)=F2;IsPerpendicular(xAxis,H);PointOnCurve(F1, H);Intersection(H, C) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 7379,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7379.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;RightFocus(G) = F;H: Line;O: Origin;PointOnCurve(O, H);A: Point;P: Point;Intersection(H, G) = {A, P};IsPerpendicular(LineSegmentOf(P, F), xAxis);B: Point;Intersection(LineOf(A, F), G) = B;IsPerpendicular(LineSegmentOf(P, B), LineSegmentOf(P, A));e: Number;Eccentricity(G) = e",
+      "coords": {}
+    },
+    {
+      "index": 7380,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7381,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 4.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7381.png",
+      "answer": "20",
+      "fact_expr": "C: Hyperbola;m: Number;G: Parabola;A: Point;B: Point;m>0;Expression(C) = (2*x^2 - y^2 = m);Expression(G) = (y^2 = 16*x);Intersection(C, Directrix(G)) = {A,B};Abs(LineSegmentOf(A,B))=4*sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 7382,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7382.png",
+      "answer": "(2/3)*sqrt(21)",
+      "fact_expr": "G: Ellipse;P: Point;A: Point;Expression(G) = (x^2/4 + y^2 = 1);Coordinate(A) = (0, 2);PointOnCurve(P, G)",
+      "coords": {
+        "A": [
+          0.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 7383,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7383.png",
+      "answer": "-4*sqrt(5)",
+      "fact_expr": "G: Hyperbola;H: Parabola;a: Number;Expression(G) = (x^2/4 - y^2 = 1);Expression(H) = (y^2 = a*x);Focus(H) = LeftFocus(G)",
+      "coords": {}
+    },
+    {
+      "index": 7384,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.7320508075688772,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 3.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7384.png",
+      "answer": "y^2=4*sqrt(2)*x",
+      "fact_expr": "G: Parabola;H: Ellipse;Expression(H) = (x^2/3 + y^2 = 1);Focus(G)=RightFocus(H)",
+      "coords": {}
+    },
+    {
+      "index": 7385,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7385.png",
+      "answer": "(1, 2]",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = sqrt(3)*x);NumIntersection(G, H)=0;e: Number;Eccentricity(G) = e",
+      "coords": {}
+    },
+    {
+      "index": 7386,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7386.png",
+      "answer": "sqrt(6)/3",
+      "fact_expr": "T: Ellipse;Expression(T) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;A: Point;B: Point;In(A,Vertex(T));Coordinate(A) = (a, 0);Coordinate(B) = (0, b);In(B,Vertex(T));C: Point;D: Point;L1: Line;L2: Line;PointOnCurve(A,L1);IsPerpendicular(L1,LineSegmentOf(A,B));PointOnCurve(B,L2);IsPerpendicular(L2,LineSegmentOf(A,B));Intersection(L1,T) = D;Intersection(L2,T) = C;LineSegmentOf(B, C) = 3*LineSegmentOf(A, D);Negation(C = Vertex(T))",
+      "coords": {}
+    },
+    {
+      "index": 7387,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7388,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7388.png",
+      "answer": "sqrt(5)",
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+      "coords": {}
+    },
+    {
+      "index": 7389,
+      "success": true,
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+      "error": null,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7389.png",
+      "answer": "0",
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+      "index": 7390,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 7391,
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+      "error": null,
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7391.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
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+          0.0
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+      "error": "Unsupported or unparseable expression"
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+      "index": 7393,
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+      "error": null,
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+        "y_coef": 3.0
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7393.png",
+      "answer": "{11, 29}",
+      "fact_expr": "G: Ellipse;k: Number;Expression(G) = (x^2/20 + y^2/k = 1);FocalLength(G) = 6",
+      "coords": {}
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+    {
+      "index": 7394,
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+      "error": null,
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+        "b": 2.0,
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7394.png",
+      "answer": "5*y^2/4-x^2=1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/5 - y^2/4 = 1);C: Hyperbola;Asymptote(C) = Asymptote(G);H: Point;Coordinate(H) = (2, 2);PointOnCurve(H, C)",
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+          2.0,
+          2.0
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+      "index": 7395,
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+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7395.png",
+      "answer": "pm*9/4",
+      "fact_expr": "P: Point;G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);PointOnCurve(P, G);F1: Point;F2: Point;Focus(G) = {F1, F2};AngleOf(F1, P, F2) = pi/2",
+      "coords": {}
+    },
+    {
+      "index": 7396,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 2.364173412322998,
+        "orientation": "horizontal"
+      },
+      "optimization": {
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+        "iterations": 89
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7396.png",
+      "answer": "3/16",
+      "fact_expr": "G: Hyperbola;n: Number;m: Number;H: Parabola;Expression(G) = (-y^2/n + x^2/m = 1);Expression(H) = (y^2 = 4*x);Eccentricity(G) = 2;OneOf(Focus(G)) = Focus(H)",
+      "coords": {}
+    },
+    {
+      "index": 7397,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
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+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7397.png",
+      "answer": "1",
+      "fact_expr": "G: Parabola;p: Number;H: Circle;M: Point;C: Point;N: Point;p>0;Expression(G) = (x^2 = 2*p*y);Center(H)=C;PointOnCurve(C,G);Coordinate(A) = (0, p);PointOnCurve(A,H);Intersection(xAxis,H) ={M,N};A:Point",
+      "coords": {}
+    },
+    {
+      "index": 7398,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7398.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "l: Line;E: Parabola;A: Point;B: Point;C:Point;Expression(E) = (y^2 = 4*x);PointOnCurve(Focus(E), l);Intersection(l, E) = {A, B};Intersection(l, Directrix(E)) = C;MidPoint(LineSegmentOf(A, C)) = B;Quadrant(A)=1",
+      "coords": {}
+    },
+    {
+      "index": 7399,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7400,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7400.png",
+      "answer": "2-sqrt(3)",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;H: Line;PointOnCurve(F2, H);Inclination(H) = ApplyUnit(120, degree);M: Point;OneOf(Intersection(H, G)) = M;IsPerpendicular(LineSegmentOf(M, F1), xAxis)",
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+    {
+      "index": 7401,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7402,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7402.png",
+      "answer": "x^2/4 + y^2/2 = 1",
+      "fact_expr": "G: Hyperbola;H: Parabola;I: Ellipse;b: Number;a: Number;Expression(G) = (x^2 - y^2 = 1);Expression(H) = (y^2 = 8*x);Expression(I) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(Focus(H), I);Focus(I) = Focus(G)",
+      "coords": {}
+    },
+    {
+      "index": 7403,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7403.png",
+      "answer": "x^2 - y^2 / 3 = 1",
+      "fact_expr": "H: Parabola;Expression(H) = (y^2 = 4*x);G: Hyperbola;Focus(H) = Vertex(G);Vertex(H) = Center(G);Eccentricity(G) = 2",
+      "coords": {}
+    },
+    {
+      "index": 7404,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7405,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7405.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;p: Number;H: Line;A: Point;B: Point;F: Point;p>0;Expression(G) = (y^2 = 2*p*x);Focus(G) = F;PointOnCurve(F, H);Inclination(H) = ApplyUnit(45, degree);Intersection(H, G) = {A, B};Length(LineSegmentOf(A,B))=8",
+      "coords": {}
+    },
+    {
+      "index": 7406,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.23606797749979,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7406.png",
+      "answer": "y^2/20 - x^2/36 = 1",
+      "fact_expr": "C1:Hyperbola;C2:Hyperbola;M: Point;Expression(C1)=(x^2/9-y^2/5=1);Asymptote(C1)=Asymptote(C2);Coordinate(M) = (3, 5);PointOnCurve(M, C2)",
+      "coords": {
+        "M": [
+          3.0,
+          5.0
+        ]
+      }
+    },
+    {
+      "index": 7407,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.125,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7407.png",
+      "answer": "17/8",
+      "fact_expr": "C: Parabola;Expression(C) = (y = 2*x^2);F: Point;Focus(C) = F;P: Point;x0: Number;y0: Number;Coordinate(P) = (x0, y0);PointOnCurve(P, C);x0 = 1",
+      "coords": {}
+    },
+    {
+      "index": 7408,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7409,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7409.png",
+      "answer": "2*sqrt(3)/3",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;e: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Inclination(OneOf(Asymptote(G))) = pi/3;Eccentricity(G) = e",
+      "coords": {}
+    },
+    {
+      "index": 7410,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.25,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7410.png",
+      "answer": "pm*sqrt(3)",
+      "fact_expr": "Gamma: Parabola;Expression(Gamma) = (y = x^2);M: Point;PointOnCurve(M, Gamma) = True;Distance(M, Focus(Gamma)) = 1",
+      "coords": {}
+    },
+    {
+      "index": 7411,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7412,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7413,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7414,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          3.0,
+          0.0
+        ],
+        "radius": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_7414.png",
+      "answer": "x^2/36+y^2/27=1",
+      "fact_expr": "C1: Circle;C2: Circle;Expression(C2) = (y^2 + (x + 3)^2 = 4);IsOutTangent(C1,C2) = True;C3: Circle;Expression(C3) = (y^2 + (x - 3)^2 = 100);IsInTangent(C1,C3) = True",
+      "coords": {}
+    },
+    {
+      "index": 7415,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7415.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;m: Number;k: Number;A: Point;B: Point;N: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = k*x + m);C:Curve;Expression(C)=(y=x/k);Intersection(H,Asymptote(G))={A,B};Intersection(H,C)=N;MidPoint(LineSegmentOf(A, B)) = N",
+      "coords": {}
+    },
+    {
+      "index": 7416,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7417,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7418,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7419,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7420,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7420.png",
+      "answer": "(pm*4*sqrt(2),4)",
+      "fact_expr": "G: Parabola;P: Point;Expression(G) = (x^2 = 8*y);PointOnCurve(P, G);Distance(P, Focus(G)) = 6",
+      "coords": {}
+    },
+    {
+      "index": 7421,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7421.png",
+      "answer": "(-4/3,2/3)",
+      "fact_expr": "G: Ellipse;H: Line;Expression(G) = (x^2 + 2*y^2 = 4);Expression(H) = (y = x + 2)",
+      "coords": {}
+    },
+    {
+      "index": 7422,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 6.0,
+        "b": 5.196152422706632,
+        "major_axis": "y",
+        "x_coef": 27.0,
+        "y_coef": 36.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7422.png",
+      "answer": "-x^2/5 + y^2/4 = 4",
+      "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/27 + y^2/36 = 1);Focus(G) = Focus(H);YCoordinate(OneOf(Intersection(G, H))) = 4",
+      "coords": {}
+    },
+    {
+      "index": 7423,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.5,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7423.png",
+      "answer": "6",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 6*y);H: LineSegment;IsChordOf(H, G);PointOnCurve(Focus(G), H)",
+      "coords": {}
+    },
+    {
+      "index": 7424,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7424.png",
+      "answer": "2",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;F: Point;LeftFocus(C) = F;Coordinate(F) = (-c, 0);P: Point;F1: Circle;Expression(F1) = (-2*c*x + x^2 + y^2 = 0);PointOnCurve(P, F1) = True;PerpendicularBisector(LineSegmentOf(F,P)) =  OneOf(Asymptote(C));c:Number",
+      "coords": {}
+    },
+    {
+      "index": 7425,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7425.png",
+      "answer": "[1/3,1)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7426.png",
+      "answer": "sqrt(10)/5",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;LeftFocus(C) = F;G: Line;PointOnCurve(F, G);Inclination(G) = ApplyUnit(60, degree);A: Point;B: Point;Intersection(G, C) = {A, B};M: Point;Coordinate(M) = (-5, sqrt(3));MidPoint(LineSegmentOf(A, B)) = M",
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+        "direction": "up"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7427.png",
+      "answer": "12",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);F: Point;Focus(G) = F;P: Point;Coordinate(P) = (1, 5);l: Line;PointOnCurve(P, l);A: Point;B: Point;Intersection(l, G) = {A, B};MidPoint(LineSegmentOf(A, B)) = P",
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+      "success": true,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7428.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;L1: Line;L2: Line;Asymptote(G) = {L1, L2};k1: Number;k2: Number;Slope(L1) = k1;Slope(L2) = k2;k1*k2 = -3",
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+    {
+      "index": 7429,
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+        "b": 2.8284271247461903,
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+      "optimization": {
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7429.png",
+      "answer": "8*sqrt(3)",
+      "fact_expr": "G: Hyperbola;F1: Point;A: Point;F2: Point;B: Point;L:Line;Expression(G) = (x^2/4 - y^2/8 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2,L);Intersection(F2,RightPart(G))={A,B};AngleOf(F1, A, B) = pi/3",
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+    {
+      "index": 7430,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 1.4142135623730951,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7430.png",
+      "answer": "x^2-2*y^2=1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/2 = 1);P: Point;PointOnCurve(P, G);Z: Line;PointOnCurve(P, Z);IsPerpendicular(Z, xAxis);Q: Point;FootPoint(Z, xAxis) = Q;M: Point;MidPoint(LineSegmentOf(P, Q)) = M",
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+    {
+      "index": 7431,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 2.0,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7431.png",
+      "answer": "3*sqrt(6)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/4 = 1);A: Point;LeftVertex(G) = A;B: Point;C: Point;PointOnCurve(B, LeftPart(G));PointOnCurve(C, RightPart(G));IsParallel(LineSegmentOf(B, C), xAxis);L1: Line;L2: Line;PointOnCurve(B, L1);PointOnCurve(C, L2);IsPerpendicular(L1, LineOf(A, B));IsPerpendicular(L2, LineOf(A, C));D: Point;Intersection(L1, L2) = D;Area(TriangleOf(B, C, D)) = (27*sqrt(3))/4",
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+    {
+      "index": 7432,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7432.png",
+      "answer": "4/3",
+      "fact_expr": "C: Parabola;p: Number;G: Line;F: Point;B: Point;A: Point;Expression(C) = (y^2 = 2*(p*x));Coordinate(A) = (-2, 3);PointOnCurve(A, Directrix(C));PointOnCurve(A, G);TangentPoint(G, C) = B;Quadrant(B) = 1;Focus(C) = F",
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+        "y_coef": 4.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7433.png",
+      "answer": "2 - 2*sqrt(2)",
+      "fact_expr": "G: Ellipse;H: Line;t: Number;A: Point;M: Point;N: Point;Expression(G) = (x^2/16 + y^2/4 = 1);Expression(H) = (x = t*y + 4);LowerVertex(G) = A;Intersection(H, G) = {M, N};Negation(M = N);WhenMax(XCoordinate(Circumcenter(TriangleOf(A, M, N))))",
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+    {
+      "index": 7434,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7434.png",
+      "answer": "(4/3, 0)",
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+    {
+      "index": 7435,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 7436,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7436.png",
+      "answer": "12",
+      "fact_expr": "C: Parabola;G: Point;P: Point;Expression(C) = (y^2 = 4*x);Coordinate(G) = (6, 12);PointOnCurve(P, C)",
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+      "index": 7437,
+      "success": true,
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+      "error": null,
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+        "b": 1.5,
+        "orientation": "vertical"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7437.png",
+      "answer": "(0,15)",
+      "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (y^2/5 - x^2/m = 1);Eccentricity(G) = e;e:Number;In(e,(1,2))",
+      "coords": {}
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+    {
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+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.4142135623730951,
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+        "x_coef": 4.0,
+        "y_coef": 2.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7438.png",
+      "answer": "x+2*y-3=0",
+      "fact_expr": "G: Ellipse;H: LineSegment;Expression(G) = (x^2/4 + y^2/2 = 1);Coordinate(MidPoint(H)) = (1,1);IsChordOf(H,G)",
+      "coords": {}
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+    {
+      "index": 7439,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7439.png",
+      "answer": "sqrt(3)",
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+    },
+    {
+      "index": 7440,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "left"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7440.png",
+      "answer": "(0, 0)",
+      "fact_expr": "G: Parabola;A: Point;B: Point;P: Point;Expression(G) = (y^2 = -4*x);Coordinate(A) = (2, 0);Coordinate(B) = (4, 0);PointOnCurve(P, G);WhenMin(DotProduct(VectorOf(A, P), VectorOf(B, P)))",
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+          0.0
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+        "B": [
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+          0.0
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+    {
+      "index": 7441,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7441.png",
+      "answer": "[1, 5)+(5, +\\infty)",
+      "fact_expr": "k: Real;H: Line;Expression(H) = (-k*x + y - 1 = 0);G: Ellipse;Expression(G) = (x^2/5 + y^2/m = 1);m: Real;IsIntersect(H, G)",
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+    {
+      "index": 7442,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7443,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7443.png",
+      "answer": "sqrt(7)/14",
+      "fact_expr": "G: Ellipse;B: Point;A: Point;F: Point;C: Point;D: Point;Expression(G) = (x^2/4 + y^2/3 = 1);UpperVertex(G)=A;LowerVertex(G)=C;LeftVertex(G)=B;LeftFocus(G) = F;Intersection(LineOf(A,B), LineOf(F, C)) = D",
+      "coords": {}
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+    {
+      "index": 7444,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7444.png",
+      "answer": "25",
+      "fact_expr": "C: Ellipse;Expression(C)=(x^2/25+y^2/16=1);P: Point;F1: Point;F2: Point;LeftFocus(C)=F1;RightFocus(C)=F2;PointOnCurve(P,C);Abs(LineSegmentOf(P,F1))*Abs(LineSegmentOf(P,F2))=m;m: Number",
+      "coords": {}
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+    {
+      "index": 7445,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7445.png",
+      "answer": "{0,1/2,-1}",
+      "fact_expr": "H: Line;Expression(H) = (y = k*(x + 2) + 1);G: Parabola;Expression(G) = (y^2 = 4*x);NumIntersection(H, G) = 1;k: Number",
+      "coords": {}
+    },
+    {
+      "index": 7446,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 2.23606797749979,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7446.png",
+      "answer": "8/3",
+      "fact_expr": "G: Hyperbola;P: Point;Expression(G) = (x^2/4 - y^2/5 = 1);PointOnCurve(P, LeftPart(G));Distance(P, RightFocus(G)) = 8",
+      "coords": {}
+    },
+    {
+      "index": 7447,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7447.png",
+      "answer": "{2,8}",
+      "fact_expr": "C: Parabola;p: Number;G: Circle;H: Point;M: Point;F: Point;p>0;Expression(C) = (y^2 = 2*p*x);Coordinate(H) = (0, 2);Focus(C) = F;PointOnCurve(M, C);Abs(LineSegmentOf(M, F)) = 5;IsDiameter(LineSegmentOf(M,F),G);PointOnCurve(H,G)",
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+        "H": [
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+          2.0
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+    {
+      "index": 7448,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7448.png",
+      "answer": "(3*sqrt(2)-sqrt(10))/2",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;LeftFocus(C) = F;l: Line;O: Origin;PointOnCurve(O, l);Inclination(l) = pi/3;M: Point;N: Point;Intersection(l, C) = {M, N};AngleOf(M, F, N) = (2*pi)/3",
+      "coords": {}
+    },
+    {
+      "index": 7449,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7449.png",
+      "answer": "sqrt(3)/2",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;H: Line;PointOnCurve(Focus(G), H);IsPerpendicular(H, xAxis);Length(InterceptChord(H, G)) = a/2",
+      "coords": {}
+    },
+    {
+      "index": 7450,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7451,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 4.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7451.png",
+      "answer": "2*sqrt(2)+2*sqrt(5)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/4 = 1);P:Point;PointOnCurve(P,G);H: Line;Expression(H) = (2*x - y + 10 = 0)",
+      "coords": {}
+    },
+    {
+      "index": 7452,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7452.png",
+      "answer": "2*sqrt(10)/5",
+      "fact_expr": "l: Line;G: Ellipse;P: Point;Expression(G) = (x^2/4 + y^2 = 1);Expression(l) = (x + 2*y = 0);PointOnCurve(P, G)",
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+      "answer": "16/3",
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+      "answer": "7/2",
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+      "answer": "(pm*2,0)",
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+      "answer": "5",
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+      "success": false,
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+      "success": true,
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+      "answer": "2",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7468.png",
+      "answer": "10",
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7469.png",
+      "answer": "x^2 = -12 \\cdot y",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7470.png",
+      "answer": "{5,3}",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7471.png",
+      "answer": "pm*8",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = m*x);m: Number;H: Ellipse;Expression(H) = (x^2/9 + y^2/5 = 1);Focus(G) = OneOf(Focus(H))",
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+        "iterations": 106
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7472.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Hyperbola;a: Real;a>0;Expression(G) = (-y^2/4 + x^2/a^2 = 1);Eccentricity(G) = sqrt(3)",
+      "coords": {}
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+      "index": 7473,
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+        "direction": "left"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7473.png",
+      "answer": "-2",
+      "fact_expr": "M: Point;Coordinate(M) = (2, 2);C: Parabola;Expression(C) = (y^2 = -8*x);G: Line;k: Number;Slope(G)=k;PointOnCurve(Focus(C),G);A: Point;B: Point;Intersection(G,C)={A,B};IsPerpendicular(VectorOf(M,A),VectorOf(M,B))",
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+          2.0
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+      "index": 7474,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 7475,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 7476,
+      "success": true,
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "answer": "12",
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+          2.0
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+        "orientation": "horizontal"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7477.png",
+      "answer": "1",
+      "fact_expr": "G: Hyperbola;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 - y^2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(90, degree)",
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+    {
+      "index": 7478,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7478.png",
+      "answer": "(18, \\pm 12)",
+      "fact_expr": "G: Parabola;P: Point;Expression(G) = (y^2 = 8*x);PointOnCurve(P, G);Distance(P, Focus(G)) = 20",
+      "coords": {}
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+    {
+      "index": 7479,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7479.png",
+      "answer": "8",
+      "fact_expr": "l: Line;G: Parabola;p: Number;A: Point;B: Point;p>0;Expression(G) = (y^2 = 2*p*x);Expression(Directrix(G)) = (x = -1);Expression(l)=(y=x-1);Intersection(l,G)={A,B};IsChordOf(LineSegmentOf(A,B),G)",
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+    {
+      "index": 7480,
+      "success": true,
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+      "error": null,
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+        "b": 1.5,
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7480.png",
+      "answer": "1+sqrt(2)",
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+    {
+      "index": 7481,
+      "success": true,
+      "conic_type": "parabola",
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7481.png",
+      "answer": "13",
+      "fact_expr": "G: Parabola;P: Point;F: Point;Expression(G) = (y^2 = 16*x);PointOnCurve(P, G);Distance(P,SymmetryAxis(G))=12;Focus(G) = F",
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+      "index": 7482,
+      "success": true,
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+      "error": null,
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+        "y_coef": 3.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7482.png",
+      "answer": "sqrt(5)/3",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;Abs(LineSegmentOf(F1, F2)) = 4*sqrt(5);P: Point;PointOnCurve(P, C);Perimeter(TriangleOf(P, F1, F2)) = 4*sqrt(5) + 12",
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+    {
+      "index": 7483,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 3.0,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7483.png",
+      "answer": "4",
+      "fact_expr": "G: Hyperbola;A: Circle;M: Point;P: Point;F1: Point;F2: Point;m:Number;Expression(G) = (x^2/16 - y^2/9 = 1);Coordinate(M) = (m, 0);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, RightPart(G));IsInscribedCircle(A, TriangleOf(P,F1,F2));TangentPoint(xAxis,A)=M",
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+    {
+      "index": 7484,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7484.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;OneOf(Focus(G)) = F;F: Point;L: Line;PointOnCurve(F, L) ;IsPerpendicular(L, Asymptote(G)) ;FootPoint(L, Asymptote(G)) = M;M: Point;Intersection(OverlappingLine(LineSegmentOf(F, M)), yAxis) = E;E: Point;Abs(LineSegmentOf(F, M)) = Abs(LineSegmentOf(M, E))",
+      "coords": {}
+    },
+    {
+      "index": 7485,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7485.png",
+      "answer": "2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);M: Point;PointOnCurve(M, G);H: Line;Expression(H) = (y = x);Distance(M, H) = 4*sqrt(2);m: Number;n: Number;Coordinate(M) = (m, n);m>0;n>0",
+      "coords": {}
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+    {
+      "index": 7486,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 4.47213595499958,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7486.png",
+      "answer": "17",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/20 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 9",
+      "coords": {}
+    },
+    {
+      "index": 7487,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.4142135623730951,
+        "major_axis": "y",
+        "x_coef": 2.0,
+        "y_coef": 4.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7487.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "G: Ellipse;Expression(G)=(y^2/4+x^2/2=1)",
+      "coords": {}
+    },
+    {
+      "index": 7488,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.4142135623730951,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7488.png",
+      "answer": "y=pm*sqrt(2)*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/2 = 1)",
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+    },
+    {
+      "index": 7489,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "error": null,
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+        "b": 1.7320508075688772,
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+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7490.png",
+      "answer": "(0,2]+[8,+oo)",
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+      "coords": {}
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+    {
+      "index": 7491,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7491.png",
+      "answer": "ApplyUnit(60,degree)",
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+      "coords": {}
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+    {
+      "index": 7492,
+      "success": true,
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+      "error": null,
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+        "b": 1.5,
+        "orientation": "vertical"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7492.png",
+      "answer": "y=pm*(2/3)*sqrt(3)*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/3 + y^2/4 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 7493,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7494,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7494.png",
+      "answer": "(4+sqrt(6))/2",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;M: Point;P: Point;F2: Point;F1: Point;a > b;b > 0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, LeftPart(C));LeftFocus(C) = F1;RightFocus(C) = F2;OneOf(Endpoint(ImageinaryAxis(C)))=M;Min(Abs(LineSegmentOf(M, P)) + Abs(LineSegmentOf(P, F2))) = Abs(LineSegmentOf(F1, F2))",
+      "coords": {}
+    },
+    {
+      "index": 7495,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7495.png",
+      "answer": "5",
+      "fact_expr": "G: Parabola;p: Number;A: Point;B: Point;F: Point;P: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;IsChordOf(LineSegmentOf(A, B), G);Intersection(PerpendicularBisector(LineSegmentOf(A, B)), xAxis) = P;Abs(LineSegmentOf(A, B)) = 10;PointOnCurve(F, LineSegmentOf(A, B))",
+      "coords": {}
+    },
+    {
+      "index": 7496,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7496.png",
+      "answer": "8",
+      "fact_expr": "G: Ellipse;H: Line;F2: Point;A: Point;B: Point;F1: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Focus(G) = {F1, F2};PointOnCurve(F1, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(F2, A)) + Abs(LineSegmentOf(F2, B)) = 12",
+      "coords": {}
+    },
+    {
+      "index": 7497,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 7498,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7498.png",
+      "answer": "[2*sqrt(2), 2*sqrt(6)]",
+      "fact_expr": "l: Line;C: Parabola;M: Point;A: Point;F: Point;B: Point;Expression(C) = (y^2 = 4*x);Coordinate(M) = (-2, 0);Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};Quadrant(A) = 1;3/2 <= Area(TriangleOf(M, A, F))/Area(TriangleOf(M, B, F));Area(TriangleOf(M, A, F))/Area(TriangleOf(M, B, F)) <= 2",
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+        "M": [
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+    {
+      "index": 7499,
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7499.png",
+      "answer": "1/2",
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+          2.0
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+    {
+      "index": 7500,
+      "success": true,
+      "conic_type": "parabola",
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7500.png",
+      "answer": "2",
+      "fact_expr": "P: Point;Coordinate(P) = (-1, -1);F: Point;Focus(C) = F;C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;l: Line;PointOnCurve(F,l) = True;Slope(l) = -2;A: Point;B: Point;Intersection(C,l) = {A,B};DotProduct(VectorOf(A, P), VectorOf(B, P)) = 0",
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+        "P": [
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+          -1.0
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+    {
+      "index": 7501,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 7502,
+      "success": true,
+      "conic_type": "parabola",
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7502.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "C: Parabola;p: Number;F: Point;P: Point;Q: Point;M: Point;l: Line;p>0;Expression(C) = (y^2 = 2*(p*x));Coordinate(P) = (0, 2);Focus(C) = F;Intersection(LineSegmentOf(P, F), C) = M;PointOnCurve(M, l);IsPerpendicular(Directrix(C), l);FootPoint(Directrix(C), l) = Q;AngleOf(P, Q, F) = ApplyUnit(90, degree)",
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+          2.0
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+      "index": 7503,
+      "success": true,
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+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7503.png",
+      "answer": "2/3",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;F1: Point;M: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(M, RightPart(C));Theta: Number;AngleOf(F1, M, F2) = Theta;DotProduct(VectorOf(M, F1), VectorOf(M, F2)) = 4*b^2",
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+    {
+      "index": 7504,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7504.png",
+      "answer": "y^2=pm*8*x",
+      "fact_expr": "l: Line;G: Parabola;a: Number;O: Origin;A: Point;F: Point;Expression(G) = (y^2 = a*x);Slope(l)=2;Focus(G)=F;PointOnCurve(F,l);Intersection(l,yAxis)=A;Area(TriangleOf(O,A,F))=4",
+      "coords": {}
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+    {
+      "index": 7505,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.0,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7505.png",
+      "answer": "4*sqrt(5)/7",
+      "fact_expr": "H: Line;Expression(H) = (y = 2*x - 3);G: Hyperbola;Expression(G) = (x^2/2 - y^2 = 1);A: Point;B: Point;Intersection(H, G) = {A, B}",
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+    {
+      "index": 7506,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 7507,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 7508,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7508.png",
+      "answer": "(\\sqrt{5}-1)/2",
+      "fact_expr": "M: Ellipse;a: Number;b: Number;A: Point;B: Point;F: Point;a > b;b > 0;Expression(M) = (y^2/b^2 + x^2/a^2 = 1);OneOf(Vertex(MajorAxis(M)))=A;OneOf(Vertex(MinorAxis(M)))=B;RightFocus(M) = F;IsPerpendicular(LineSegmentOf(A, B), LineSegmentOf(B, F))",
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+    {
+      "index": 7509,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 7510,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 5.0,
+        "y_coef": 1.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7510.png",
+      "answer": "-10",
+      "fact_expr": "l: Line;G: Ellipse;O: Origin;A: Point;E: Point;M: Point;F: Point;N: Point;lambda1:Number;lambda2:Number;Expression(G) = (x^2/5 + y^2 = 1);Center(G)=O;RightFocus(G) = F;OneOf(Endpoint(MinorAxis(G)))=A;PointOnCurve(F, l);Intersection(l, G) = {M, N};Intersection(OverlappingLine(LineSegmentOf(O, A)),l) = E;VectorOf(E, M) = lambda1*VectorOf(M, F);VectorOf(E, N) = lambda2*VectorOf(N, F)",
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+    {
+      "index": 7511,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 7512,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7512.png",
+      "answer": "2",
+      "fact_expr": "C:Ellipse;Expression(C)=(x^2/2+y^2=1)",
+      "coords": {}
+    },
+    {
+      "index": 7513,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "major_axis": "x",
+        "x_coef": 5.0,
+        "y_coef": 4.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7513.png",
+      "answer": "sqrt(5)-1",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/5 + y^2/4 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;M: Point;PointOnCurve(M,G);Negation(M=Endpoint(MajorAxis(G)));theta: Number;AngleOf(F1, M, F2) = 2*theta;I: Point;Incenter(TriangleOf(M, F1, F2)) = I",
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+    },
+    {
+      "index": 7514,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7514.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = x + 2);PointOnCurve(OneOf(Focus(G)),H);PointOnCurve(OneOf(Vertex(G)),H)",
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+    {
+      "index": 7515,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7515.png",
+      "answer": "9",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;G: Line;PointOnCurve(F, G);A: Point;B: Point;Intersection(G, C) = {A, B};N: Point;MidPoint(LineSegmentOf(A, B)) = N;P: Point;Coordinate(P) = (12, 4);WhenMin(Abs(LineSegmentOf(N, A))+Abs(LineSegmentOf(N, P)))",
+      "coords": {
+        "P": [
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+          4.0
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+    {
+      "index": 7516,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7516.png",
+      "answer": "2",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);P: Point;PointOnCurve(P, G);O: Origin;F: Point;LeftFocus(G) = F;M: Point;LineSegmentOf(O, M) = (1/2)*(LineSegmentOf(O, F) + LineSegmentOf(O, P))",
+      "coords": {}
+    },
+    {
+      "index": 7517,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "up"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7517.png",
+      "answer": "y=-1",
+      "fact_expr": "l: Line;G: Parabola;H: Circle;A: Point;Expression(G) = (x^2 = 4*y);Coordinate(A) = (0, 1);PointOnCurve(A, H);PointOnCurve(Center(H),G);IsTangent(l,H)",
+      "coords": {
+        "A": [
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+          1.0
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+    },
+    {
+      "index": 7518,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7519,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7519.png",
+      "answer": "(-1, -\\sqrt{3}/2)",
+      "fact_expr": "G: Hyperbola;H: Line;k: Real;Expression(G) = (x^2/4 - y^2/3 = 1);Expression(H) = (y = k*x + 1);NumIntersection(H, RightPart(G)) = 2",
+      "coords": {}
+    },
+    {
+      "index": 7520,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.242640687119285,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 18.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7520.png",
+      "answer": "pi/3",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/25 + y^2/18 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 6",
+      "coords": {}
+    },
+    {
+      "index": 7521,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 3.4641016151377544,
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+        "x_coef": 16.0,
+        "y_coef": 12.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7521.png",
+      "answer": "(-2,3)",
+      "fact_expr": "G: Ellipse;A: Point;M: Point;F: Point;Expression(G) = (x^2/16 + y^2/12 = 1);Coordinate(A) = (-2, sqrt(3));RightFocus(G) = F;PointOnCurve(M, G);WhenMin(Abs(LineSegmentOf(A,M))+Abs(LineSegmentOf(M,F)))",
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+          1.7320508075688772
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+    {
+      "index": 7522,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7522.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Parabola;H: Line;A: Point;P: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (x = -1);Coordinate(A) = (0, -1);PointOnCurve(P, G)",
+      "coords": {
+        "A": [
+          0.0,
+          -1.0
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+    {
+      "index": 7523,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 4.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7523.png",
+      "answer": "2*sqrt(41)-6",
+      "fact_expr": "A: Point;Coordinate(A) = (6, 0);P: Point;PointOnCurve(P, G);G: Parabola;Expression(G) = (y^2 = 16*x);B: Point;PointOnCurve(B, H);H: Curve;Expression(H) = (y^2 + (x - 4)^2 = 1)",
+      "coords": {
+        "A": [
+          6.0,
+          0.0
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+    {
+      "index": 7524,
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+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 2.8647773265838623,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 1.6550993677810766e-05,
+        "converged": true,
+        "iterations": 144
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7524.png",
+      "answer": "3",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(C) = sqrt(10)",
+      "coords": {}
+    },
+    {
+      "index": 7525,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 7526,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7526.png",
+      "answer": "3",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(C) = F;G: Line;Expression(G) = (y = k*x);k: Number;Negation(k=0);Intersection(G, C) = {A, B};A: Point;B: Point;AngleOf(A, F, B) = ApplyUnit(90, degree);O: Origin;Area(TriangleOf(O, A, F)) = 4*a^2",
+      "coords": {}
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+    {
+      "index": 7527,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 4.242640687119285,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 18.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7527.png",
+      "answer": "pi/3",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/18 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 6",
+      "coords": {}
+    },
+    {
+      "index": 7528,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
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+        "radius": 4.0
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_7528.png",
+      "answer": "(x-1)^2+(y-3)^2=2",
+      "fact_expr": "l: Line;C: Circle;B: Point;A: Point;P: Point;M: Point;O: Origin;Expression(C) = (-8*y + x^2 + y^2 = 0);Coordinate(P) = (2, 2);PointOnCurve(P,l);Intersection(l,C)={A,B};MidPoint(LineSegmentOf(A,B))=M",
+      "coords": {}
+    },
+    {
+      "index": 7529,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7529.png",
+      "answer": "sqrt(3)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2 + y^2 = 2*a^2);OneOf(Intersection(H,G))=P;LeftFocus(G) = F1;RightFocus(G) = F2;LineSegmentOf(P, F1) = 3*LineSegmentOf(P, F2)",
+      "coords": {}
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+    {
+      "index": 7530,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7530.png",
+      "answer": "[3, +\\infty)",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;M: Point;PointOnCurve(M, C);N: Point;Coordinate(N) = (2, 2)",
+      "coords": {
+        "N": [
+          2.0,
+          2.0
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+    },
+    {
+      "index": 7531,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7531.png",
+      "answer": "sqrt(10)",
+      "fact_expr": "P: Point;x0: Number;y0: Number;Coordinate(P) = (x0, y0);G: Parabola;Expression(G) = (y^2 = 4*x);PointOnCurve(P, G);H: Point;Coordinate(H) = (0, 3)",
+      "coords": {
+        "H": [
+          0.0,
+          3.0
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+    },
+    {
+      "index": 7532,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7532.png",
+      "answer": "4*sqrt(13)/3",
+      "fact_expr": "l: Line;Expression(l) = (y = m + 2*sqrt(3)*x);m: Number;C: Hyperbola;Expression(C) = (x^2 - y^2/3 = 1);PointOnCurve(RightVertex(C),l) = True;L1: Line;L2: Line;Asymptote(C) = {L1, L2};Intersection(l, L1) = A;Intersection(l, L2) = B;A: Point;B: Point",
+      "coords": {}
+    },
+    {
+      "index": 7533,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7533.png",
+      "answer": "5/3",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G);PointOnCurve(F1, LineSegmentOf(A, B));Perimeter(InscribedCircle(TriangleOf(A, B, F2))) = pi;x1: Number;y1: Number;x2: Number;y2: Number;Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2)",
+      "coords": {}
+    },
+    {
+      "index": 7534,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7534.png",
+      "answer": "x^2/6+y^2/3=1",
+      "fact_expr": "G: Hyperbola;H: Ellipse;M: Point;Expression(G) = (x^2 - 2*y^2 = 2);Coordinate(M) = (-sqrt(6), 0);PointOnCurve(M, H);Focus(G)=Focus(H)",
+      "coords": {
+        "M": [
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+          0.0
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+    },
+    {
+      "index": 7535,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 3.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 9.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7535.png",
+      "answer": "[-6,6]",
+      "fact_expr": "H: Circle;Expression(H) = (x^2 + (-a + y)^2 = 9);G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);IsIntersect(H, G) = True;a: Real",
+      "coords": {}
+    },
+    {
+      "index": 7536,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7536.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;e: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P,F1),LineSegmentOf(P,F2));Area(TriangleOf(P, F1, F2)) = 2*a*b;Eccentricity(G) = e",
+      "coords": {}
+    },
+    {
+      "index": 7537,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7537.png",
+      "answer": "8",
+      "fact_expr": "G: Parabola;A: Point;F: Point;P: Point;l: Line;Expression(G) = (y^2 = 8*x);Focus(G) = F;Directrix(G)=l;PointOnCurve(P,G);IsPerpendicular(LineSegmentOf(P,A),l);FootPoint(LineSegmentOf(P,A),l)=A;Slope(LineOf(A,F)) = -sqrt(3)",
+      "coords": {}
+    },
+    {
+      "index": 7538,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 2.8284271247461903,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7538.png",
+      "answer": "49*pi",
+      "fact_expr": "G: Hyperbola;H: Circle;O: Origin;N: Point;F1: Point;P: Point;M: Point;F2: Point;Expression(G) = (x^2/16 - y^2/8 = 1);PointOnCurve(P, RightPart(G));LeftFocus(G) = F1;RightFocus(G) = F2;lambda: Number;mu: Number;lambda > 0;VectorOf(F1, P) = lambda*VectorOf(P, M);VectorOf(P, N) = mu*((VectorOf(P, F2)/Abs(VectorOf(P, F2))) + (VectorOf(P, M)/Abs(VectorOf(P, M))));DotProduct(VectorOf(P, N), VectorOf(F2, N)) = 0;VectorOf(P, F2) = 3;Center(H) = O;Radius(H) = LineSegmentOf(O, N)",
+      "coords": {}
+    },
+    {
+      "index": 7539,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7540,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7541,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7541.png",
+      "answer": "4/5",
+      "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;I: Point;Expression(G) = (x^2/16 - y^2/9 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, RightPart(G));Incenter(TriangleOf(P, F1, F2)) = I;Area(TriangleOf(I,P,F2))=Area(TriangleOf(I,P,F1))-lambda*Area(TriangleOf(I,F1,F2));lambda:Number",
+      "coords": {}
+    },
+    {
+      "index": 7542,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7542.png",
+      "answer": "x=-1",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x)",
+      "coords": {}
+    },
+    {
+      "index": 7543,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "vertical"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7543.png",
+      "answer": "sqrt(3)*x + pm*y = 0",
+      "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/4 + y^2/12 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 7544,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7545,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7545.png",
+      "answer": "sqrt(7)/4",
+      "fact_expr": "C: Ellipse;a: Number;M: Point;F: Point;O: Origin;N: Point;a>0;Expression(C) = (y^2/9 + x^2/a = 1);RightFocus(C) = F;PointOnCurve(M, C);MidPoint(LineSegmentOf(M,F)) = N;Abs(LineSegmentOf(M,F)) = 2*Abs(LineSegmentOf(O,N));2*Abs(LineSegmentOf(O,N)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 7546,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7546.png",
+      "answer": "sqrt(5)",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;B: Point;F: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);OneOf(Focus(C))=F;OneOf(Endpoint(ImageinaryAxis(C)))=B;PointOnCurve(MidPoint(LineSegmentOf(F,B)),C)",
+      "coords": {}
+    },
+    {
+      "index": 7547,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7547.png",
+      "answer": "6/5",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;A: Point;F: Point;B: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;PointOnCurve(F, G);Slope(G)=sqrt(3);Intersection(G, C) = {A,B};VectorOf(A, F) = 4*VectorOf(F, B)",
+      "coords": {}
+    },
+    {
+      "index": 7548,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.8284271247461903,
+        "major_axis": "x",
+        "x_coef": 9.0,
+        "y_coef": 8.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7548.png",
+      "answer": "1/3",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/8 = 1)",
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+      "answer": "3/2",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7550.png",
+      "answer": "(-sqrt(15)/3,-1)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7551.png",
+      "answer": "4*sqrt(3)/3",
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+      "answer": "sqrt(3)",
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+      "answer": "8*sqrt(2)/3",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/2 + y^2 = 1);F1: Point;LeftFocus(C) = F1;l: Line;Expression(l) = (y = x - 1);A: Point;B: Point;Intersection(l, C) = {A, B}",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7554.png",
+      "answer": "(\\sqrt{5}+1)/2",
+      "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;A: Point;RightVertex(C) = A;B1: Point;B2: Point;Endpoint(ImageinaryAxis(C)) = {B1, B2};F: Point;RightFocus(C) = F;IsPerpendicular(LineSegmentOf(B2, F), LineSegmentOf(A, B1))",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7555.png",
+      "answer": "4*sqrt(2)",
+      "fact_expr": "G: Hyperbola;H: Line;P: Point;Q: Point;F:Point;Expression(G) = (x^2 - y^2 = 2);RightFocus(G)=F;PointOnCurve(F,H);Inclination(H)=ApplyUnit(30,degree);Intersection(H,G)={P,Q}",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7556.png",
+      "answer": "143/12",
+      "fact_expr": "G: Hyperbola;H: Point;Expression(G) = (x^2/4 - y^2 = 1);Coordinate(H) = (x1, y1);PointOnCurve(H, G);x1:Number;y1:Number",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7558.png",
+      "answer": "3",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Focus(C)={F1,F2};PointOnCurve(P, C);IsPerpendicular(VectorOf(P,F1),VectorOf(P,F2));Area(TriangleOf(P,F1,F2))=9",
+      "coords": {}
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+    {
+      "index": 7559,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "y_coef": 2.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7559.png",
+      "answer": "1",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/8 + y^2/2 = 1);M: Circle;Expression(M) = (-r^2 + 2*(sqrt(2)*x) + x^2 + y^2 + 2 = 0);r: Number;0 < r;r < sqrt(2);P: Point;UpperVertex(C) = P;Z1: Line;Z2: Line;TangentOfPoint(P, M) = {Z1, Z2};A: Point;Intersection(Z1, C) = A;B: Point;Intersection(Z2, C) = B;Negation(P=A);Negation(P=B)",
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+    {
+      "index": 7560,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7560.png",
+      "answer": "10",
+      "fact_expr": "G: Parabola;M: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(M, G);Distance(M, yAxis) = 9",
+      "coords": {}
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+    {
+      "index": 7561,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7561.png",
+      "answer": "sqrt(3)-1",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(P,G);AngleOf(P, F1, F2) = pi/6;AngleOf(P,F2,F1)=pi/3;Eccentricity(G)=e;e:Number",
+      "coords": {}
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+    {
+      "index": 7562,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_point_off_curve",
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+        "b": 2.0,
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+        "x_coef": 9.0,
+        "y_coef": 4.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7562.png",
+      "answer": "x^2/15+y^2/10=1",
+      "fact_expr": "M: Point;Coordinate(M) = (-3, 2);Z: Curve;Expression(Z) = (x^2/9 + y^2/4 = 1);G: Ellipse;PointOnCurve(M, G);Focus(Z) = Focus(G)",
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+          2.0
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+      "validation_reasons": [
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+      "index": 7563,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.0,
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+        "y_coef": 1.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7563.png",
+      "answer": "8",
+      "fact_expr": "M: Point;Coordinate(M) = (sqrt(3), 0);G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);H: Line;Expression(H) = (y = k*(x + sqrt(3)));Intersection(G, H) = {A, B};A: Point;B: Point;k:Number",
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+      "index": 7564,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.7320508075688772,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7564.png",
+      "answer": "(\\sqrt{5}-1)/2",
+      "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;A: Point;LeftVertex(C) = A;B: Point;UpperVertex(C) = B;F: Point;RightFocus(C) = F;AngleOf(A, B, F) = ApplyUnit(90, degree)",
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+    {
+      "index": 7565,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 4.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7565.png",
+      "answer": "y=pm*sqrt(2)*x",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/8 - y^2/16 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 7566,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7566.png",
+      "answer": "12",
+      "fact_expr": "N: Parabola;Expression(N) = (y^2 = 2*p*x);p: Number;p>0;G: Line;Expression(G) = (x = m);m: Number;m>0;Intersection(G, N) = {A, B};A: Point;B: Point;O: Origin;IsPerpendicular(LineSegmentOf(O, A), LineSegmentOf(O, B)) = True;Coordinate(Focus(N))=(3,0)",
+      "coords": {}
+    },
+    {
+      "index": 7567,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7568,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7568.png",
+      "answer": "2*sqrt(3)/3",
+      "fact_expr": "C1:Parabola;C2:Hyperbola;a: Number;a>0;Expression(C2) = (x^2/3 - y^2 = 1);Expression(C1) = (y = x^2/(4*a));Intersection(LineSegmentOf(Focus(C1),RightFocus(C2)),C1)=M;Quadrant(M)=1;IsParallel(TangentOnPoint(M,C1),OneOf(Asymptote(C2)));M:Point",
+      "coords": {}
+    },
+    {
+      "index": 7569,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7569.png",
+      "answer": "(3*sqrt(2))/7",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2/3 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;l: Line;A: Point;B: Point;PointOnCurve(F2,l);Inclination(l) = pi/4;Intersection(l, C) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 7570,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 3.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7570.png",
+      "answer": "(4,-4*sqrt(3))",
+      "fact_expr": "G: Parabola;M: Point;F: Point;Expression(G) = (y^2 = 12*x);Focus(G) = F;PointOnCurve(M,G);Quadrant(M)=4;Abs(LineSegmentOf(M, F)) = 7",
+      "coords": {}
+    },
+    {
+      "index": 7571,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7571.png",
+      "answer": "11",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2/4 - y^2/3 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;l: Line;PointOnCurve(F1, l);A: Point;B: Point;Intersection(l, LeftPart(C)) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 7572,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7573,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7574,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 0.5,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7574.png",
+      "answer": "sqrt(14)/2",
+      "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*x);F: Point;Focus(C) = F;l: Line;PointOnCurve(F, l);Intersection(l, C) = {A, B};B: Point;A: Point;Abs(LineSegmentOf(A, F)) = 2*Abs(LineSegmentOf(B, F));IsDiameter(LineSegmentOf(A, B), G) = True;G: Circle",
+      "coords": {}
+    },
+    {
+      "index": 7575,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7575.png",
+      "answer": "1/5",
+      "fact_expr": "Gamma:Ellipse;A: Point;F: Point;B: Point;C: Point;D: Point;Expression(Gamma) = (x^2/a^2+y^2/b^2=1);LeftVertex(Gamma)=C;LeftFocus(Gamma)=F;LowerVertex(Gamma)=A;UpperVertex(Gamma)=B;Intersection(LineSegmentOf(A,F),LineSegmentOf(B,C))=D;VectorOf(C, D) = 2*VectorOf(D, B)",
+      "coords": {}
+    },
+    {
+      "index": 7576,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7576.png",
+      "answer": "4+2*sqrt(3)",
+      "fact_expr": "G: Ellipse;P: Point;F1: Point;O: Origin;Expression(G) = (x^2/4 + y^2 = 1);LeftFocus(G) = F1;PointOnCurve(P, G)",
+      "coords": {}
+    },
+    {
+      "index": 7577,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7577.png",
+      "answer": "(\\sqrt{3} - 1)/2",
+      "fact_expr": "E: Ellipse;b: Number;a: Number;G: Circle;F1: Point;F2: Point;P: Point;Q: Point;c: Number;a > b;b > 0;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);LeftFocus(E) = F1;RightFocus(E) = F2;IsDiameter(LineSegmentOf(F1, F2), G);Intersection(G, E) = {P, Q};Quadrant(P) = 1;Quadrant(Q) = 3;Area(TriangleOf(P, Q, F2)) = c^2*(sqrt(3)/2)",
+      "coords": {}
+    },
+    {
+      "index": 7578,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.4142135623730951,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7578.png",
+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
+      "fact_expr": "G: Ellipse;H: Line;A: Point;B: Point;Expression(G) = (x^2/2 + y^2 = 1);Expression(H) = (y = x + 1);Intersection(H, G) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 7579,
+      "success": true,
+      "conic_type": "circle",
+      "error": null,
+      "params": {
+        "center": [
+          0.0,
+          0.0
+        ],
+        "radius": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/circle/problem_7579.png",
+      "answer": "4*x^2+y^2=1",
+      "fact_expr": "G: Circle;H: LineSegment;Expression(G) = (x^2 + y^2 = 1);PointOnCurve(P,G);IsPerpendicular(H,yAxis);M:Point;MidPoint(H)=M;P:Point",
+      "coords": {}
+    },
+    {
+      "index": 7580,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.4142135623730951,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 2.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7580.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/2 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 7581,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7581.png",
+      "answer": "(2*sqrt(2),3*sqrt(2))",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2/2 - y^2/b^2 = 1);a: Number;b: Number;a>0;b>0;e: Number;Eccentricity(C) = e;In(e, (sqrt(5), sqrt(10)))",
+      "coords": {}
+    },
+    {
+      "index": 7582,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7583,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7583.png",
+      "answer": "12/5",
+      "fact_expr": "C: Parabola;p: Number;F: Point;l: Line;P: Point;p>0;Expression(C) = (y^2 = 2*p*x);Focus(C) = F;Directrix(C) = l;PointOnCurve(P, l);L1: Line;L2: Line;TangentOfPoint(P, C) = {L1, L2};A: Point;B: Point;TangentPoint(L1, C) = A;TangentPoint(L2, C) = B;Abs(LineSegmentOf(P, A)) = 3;Abs(LineSegmentOf(P, B)) = 4",
+      "coords": {}
+    },
+    {
+      "index": 7584,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7584.png",
+      "answer": "8",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);H: Line;PointOnCurve(Focus(G), H) = True;Intersection(H, G) = {A, B};A: Point;B: Point;Coordinate(A) = (x1, y1);x1: Number;y1: Number;Coordinate(B) = (x2, y2);x2: Number;y2: Number;x1 + x2 = 6",
+      "coords": {}
+    },
+    {
+      "index": 7585,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7585.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, G);Focus(G) = {F1,F2};AngleOf(P,F1,F2)=ApplyUnit(75,degree);AngleOf(P, F2, F1) = ApplyUnit(15, degree)",
+      "coords": {}
+    },
+    {
+      "index": 7586,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7587,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7587.png",
+      "answer": "y^2/3 - x^2/3 = 1",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 1);Z: Hyperbola;Asymptote(G) = Asymptote(Z);H: Point;Coordinate(H) = (1, 2);PointOnCurve(H, Z)",
+      "coords": {
+        "H": [
+          1.0,
+          2.0
+        ]
+      }
+    },
+    {
+      "index": 7588,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7588.png",
+      "answer": "12",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);H: Line;PointOnCurve(Focus(G), H);A: Point;B: Point;x1: Number;x2: Number;y1: Number;y2: Number;Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);Intersection(H, G) = {A, B};x1 + x2 = 10",
+      "coords": {}
+    },
+    {
+      "index": 7589,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7590,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7591,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.23606797749979,
+        "b": 2.0,
+        "major_axis": "x",
+        "x_coef": 5.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7591.png",
+      "answer": "(\\sqrt{15}/2, 1),(\\sqrt{15}/2, -1),(-\\sqrt{15}/2, 1),(-\\sqrt{15}/2, -1)",
+      "fact_expr": "G: Ellipse;P: Point;F1:Point;F2:Point;Expression(G) = (x^2/5 + y^2/4 = 1);PointOnCurve(P,G);Focus(G)={F1,F2};Area(TriangleOf(P,F1,F2))=1",
+      "coords": {}
+    },
+    {
+      "index": 7592,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7592.png",
+      "answer": "[\\sqrt{2}/2,1)",
+      "fact_expr": "G: Ellipse;b: Number;a: Number;O: Circle;A: Point;P: Point;B: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(O) = (x^2 + y^2 = b^2);L1:Line;L2:Line;TangentOfPoint(P,O)={L1,L2};TangentPoint(L1,O)=A;TangentPoint(L2,O)=B;AngleOf(A, P, B) = ApplyUnit(90, degree);PointOnCurve(P,G)",
+      "coords": {}
+    },
+    {
+      "index": 7593,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.23606797749979,
+        "b": 2.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7593.png",
+      "answer": "2*x + pm*sqrt(5)*y = 0",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2/5 - y^2/4 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 7594,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "left"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7594.png",
+      "answer": "9/2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = -4*x);F: Point;Focus(G) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, G) = {A, B};C: Point;Intersection(l, Directrix(G)) = C;VectorOf(F, C) = 4*VectorOf(F, B)",
+      "coords": {}
+    },
+    {
+      "index": 7595,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7596,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7597,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 3.4641016151377544,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7597.png",
+      "answer": "x^2/16+y^2/12=1",
+      "fact_expr": "G: Hyperbola;H: Ellipse;Expression(G) = (x^2/4 - y^2/12 = 1);Focus(G) = Vertex(H);Vertex(G) = Focus(H)",
+      "coords": {}
+    },
+    {
+      "index": 7598,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7599,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7600,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7600.png",
+      "answer": "sqrt(2)/2",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;A: Point;B: Point;PointOnCurve(A, G) = True;PointOnCurve(B, G) = True;IsDiameter(LineSegmentOf(A, B), H) = True;H: Circle;PointOnCurve(F, H) = True;MidPoint(LineSegmentOf(A, B)) = M;M: Point;PointOnCurve(M, LineOf(M, N)) = True;IsPerpendicular(LineOf(M, N), Directrix(G)) = True;FootPoint(LineOf(M, N), Directrix(G)) = N;N: Point",
+      "coords": {}
+    },
+    {
+      "index": 7601,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7601.png",
+      "answer": "5",
+      "fact_expr": "G: Parabola;C: Circle;M: Point;A: Point;F: Point;Expression(G) = (y^2 = 8*x);Expression(C) = (-2*y - 8*x + x^2 + y^2 + 16 = 0);PointOnCurve(M, G);Focus(G) = F;PointOnCurve(A, C)",
+      "coords": {}
+    },
+    {
+      "index": 7602,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 3.0,
+        "b": 2.23606797749979,
+        "major_axis": "y",
+        "x_coef": 5.0,
+        "y_coef": 9.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7602.png",
+      "answer": "12",
+      "fact_expr": "l: Line;C: Ellipse;A: Point;F1: Point;B: Point;F2: Point;Expression(C) = (x^2/5 + y^2/9 = 1);Focus(C) = {F1,F2};PointOnCurve(F2, l);Intersection(l, C) = {A, B}",
+      "coords": {}
+    },
+    {
+      "index": 7603,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7603.png",
+      "answer": "8",
+      "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;l:Number;Expression(G) = (y^2 = 4*x);Slope(H)=l;Intersection(H,G)={A,B};PointOnCurve(Focus(G),H)",
+      "coords": {}
+    },
+    {
+      "index": 7604,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7604.png",
+      "answer": "2",
+      "fact_expr": "E: Hyperbola;b: Number;a: Number;G: Line;P: Point;A: Point;F: Point;B: Point;a>0;b>0;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(E) = F;PointOnCurve(F, G);Intersection(G, E) = {A, B};Intersection(G, yAxis) = P;VectorOf(P, A) = 3*VectorOf(A, F);VectorOf(P, B) = (-7/3)*VectorOf(B, F)",
+      "coords": {}
+    },
+    {
+      "index": 7605,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7605.png",
+      "answer": "sqrt(3)/3",
+      "fact_expr": "C: Ellipse;b: Number;a: Number;G: Line;A: Point;F1: Point;B: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, G);Intersection(G, C) = {A, B};Abs(LineSegmentOf(A, F1)) = 2*Abs(LineSegmentOf(B, F1));Abs(LineSegmentOf(A, B)) = Abs(LineSegmentOf(B, F2))",
+      "coords": {}
+    },
+    {
+      "index": 7606,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7607,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7607.png",
+      "answer": "11",
+      "fact_expr": "G: Hyperbola;m: Real;Expression(G) = (-y^2/5 + x^2/m = 1);FocalLength(G) = 8",
+      "coords": {}
+    },
+    {
+      "index": 7608,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7609,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 4.0,
+        "b": 3.0,
+        "major_axis": "y",
+        "x_coef": 9.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7609.png",
+      "answer": "11",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/16 = 1);H: Line;A: Point;B: Point;F1: Point;F2: Point;Focus(G) = {F1, F2};PointOnCurve(F2, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = 5",
+      "coords": {}
+    },
+    {
+      "index": 7610,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
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+      "answer": "4*x+y+3=0",
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+      "answer": "4*sqrt(3)",
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+      "answer": "sqrt(5)",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "answer": "6",
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+      "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers).  \nDo not add explanations.",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7618.png",
+      "answer": "-1/3",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "success": true,
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+        "direction": "up"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7620.png",
+      "answer": "-1",
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+      "index": 7621,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 7622,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "success": true,
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+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7623.png",
+      "answer": "[\\sqrt{3}/3, 1)",
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+    {
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+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7624.png",
+      "answer": "2/3",
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+      "coords": {}
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+    {
+      "index": 7625,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7625.png",
+      "answer": "10",
+      "fact_expr": "G: Parabola;f: LineSegment;Expression(G) = (y^2 = 4*x);PointOnCurve(Focus(G), f);IsChordOf(f, G);XCoordinate(MidPoint(f)) = 4",
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+    {
+      "index": 7626,
+      "success": true,
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+      "error": null,
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+        "direction": "up"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7626.png",
+      "answer": "(0,1/4)",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = y)",
+      "coords": {}
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+    {
+      "index": 7627,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 3.0,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7627.png",
+      "answer": "8",
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+        "F": [
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+          0.0
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+      "index": 7628,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 7629,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_point_off_curve",
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+        "b": 2.298015331604694,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 1.93011531933813e-06,
+        "converged": true,
+        "iterations": 91
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7629.png",
+      "answer": "y^2/2 - x^2 = 1",
+      "fact_expr": "G: Hyperbola;H: Point;Coordinate(H) = (1, 2);Expression(Asymptote(G)) = (y = pm*(sqrt(2)*x));PointOnCurve(H, G)",
+      "coords": {
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+          1.0,
+          2.0
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+        "hyperbola_point_off_curve"
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+    {
+      "index": 7630,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7630.png",
+      "answer": "9",
+      "fact_expr": "C: Parabola;P: Point;F: Point;A: Point;B: Point;l: Line;Expression(C) = (y^2 = 8*x);Focus(C) = F;Directrix(C) = l;PointOnCurve(P, l);Intersection(LineOf(P,F), C) = {A, B};VectorOf(P, A) = 3*VectorOf(A, F)",
+      "coords": {}
+    },
+    {
+      "index": 7631,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 1.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7631.png",
+      "answer": "2",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2 + x^2/a^2 = 1);a: Number;a>0;PointOnCurve(Focus(G), xAxis) = True;Length(MajorAxis(G)) = 2*Length(MinorAxis(G))",
+      "coords": {}
+    },
+    {
+      "index": 7632,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7632.png",
+      "answer": "{-1/2, 0, 1}",
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+        "P": [
+          -2.0,
+          -1.0
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+      "index": 7633,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 7634,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7635,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7636,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7636.png",
+      "answer": "(1, \\sqrt{2}+1]",
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+      "coords": {}
+    },
+    {
+      "index": 7637,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7638,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 1.0,
+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7638.png",
+      "answer": "7",
+      "fact_expr": "F1: Point;LeftFocus(C) = F1;C: Hyperbola;Expression(C) = (x^2/4 - y^2 = 1);l: Line;O: Origin;PointOnCurve(O, l);P: Point;Q: Point;Intersection(l, C) = {P, Q};Abs(LineSegmentOf(P, F1)) = 3",
+      "coords": {}
+    },
+    {
+      "index": 7639,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7639.png",
+      "answer": "sqrt(2)/2",
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+          2.0
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+    {
+      "index": 7640,
+      "success": true,
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+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7640.png",
+      "answer": "x^2/2 + y^2 = 1",
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+      "coords": {
+        "M": [
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+          0.5
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+    {
+      "index": 7641,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 1.0,
+        "major_axis": "x",
+        "x_coef": 2.0,
+        "y_coef": 1.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7641.png",
+      "answer": "[2,2*sqrt(2))\n0",
+      "fact_expr": "C: Ellipse;G: Line;y0: Number;x0: Number;P: Point;F1: Point;F2: Point;Expression(C)=(x^2/2 + y^2 = 1);Expression(G)=((x0*x)/2+y0*y=1);Focus(C)={F1,F2};Coordinate(P)=(x0,y0);0<x0^2/2+y0^2;x0^2/2+y0^2<1",
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+    {
+      "index": 7642,
+      "success": true,
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+      "error": null,
+      "params": {
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+        "b": 4.47213595499958,
+        "major_axis": "x",
+        "x_coef": 45.0,
+        "y_coef": 20.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7642.png",
+      "answer": "40",
+      "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/45 + y^2/20 = 1);AngleOf(F1, P, F2) = ApplyUnit(90, degree);Focus(G)={F1,F2};PointOnCurve(P,G)",
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+    {
+      "index": 7643,
+      "success": false,
+      "conic_type": "ellipse",
+      "error": "validation: ellipse_point_off_curve",
+      "params": {
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+        "major_axis": "x",
+        "x_coef": 3.999999999999999,
+        "y_coef": 2.9999999999999996
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+      "success": false,
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+      "answer": "{2,8}",
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+      "answer": "1/16",
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+      "index": 7666,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 7667,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "orientation": "horizontal"
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+        "converged": true,
+        "note": "using explicit params"
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+      "answer": "sqrt(3)",
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+      "success": true,
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+      "error": null,
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+      "answer": "1/4",
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+      "index": 7669,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 7670,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7670.png",
+      "answer": "sqrt(7)/3",
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+    {
+      "index": 7671,
+      "success": false,
+      "conic_type": "hyperbola",
+      "error": "validation: hyperbola_point_off_curve",
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+        "orientation": "vertical"
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+        "converged": true,
+        "iterations": 2
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7671.png",
+      "answer": "sqrt(3)",
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+          2.0
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+      "validation_reasons": [
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+      "index": 7672,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "up"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7672.png",
+      "answer": "sqrt(7)/7",
+      "fact_expr": "l: Line;C: Parabola;G: Point;A: Point;B: Point;Expression(C) = (x^2 = 12*y);Coordinate(G) = (0, 3);PointOnCurve(G,l);Intersection(l, C) = {A, B};Abs(LineSegmentOf(A, B)) = 14;Inclination(l)=alpha;alpha:Number",
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+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 7674,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 7675,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "x_coef": 4.0,
+        "y_coef": 3.0
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+        "converged": true,
+        "note": "using explicit params"
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+      "answer": "sqrt(3)/3",
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+      "answer": "10",
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+      "answer": "[1, 5) + (5, +oo)",
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+      "answer": "8",
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+      "answer": "y = pm*2*x",
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+      "answer": "sqrt(2)/2",
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+      "answer": "-1/2",
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+      "answer": "(\\sqrt{3}+1)/2",
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+      "answer": "(\\sqrt{2}/2,1)",
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7690.png",
+      "answer": "x^2 - y^2/4 = 1",
+      "fact_expr": "G: Hyperbola;H: Point;Coordinate(H) = (sqrt(5), 0);Expression(Asymptote(G)) = (y = pm*2*x);OneOf(Focus(G))=H",
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+      "answer": "24",
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+      "index": 7693,
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+      "answer": "x^2/16 - y^2/64 = 1",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7694.png",
+      "answer": "1/2",
+      "fact_expr": "G: Parabola;Expression(G) = (x^2 = y)",
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+        "iterations": 116
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7695.png",
+      "answer": "4*sqrt(3)",
+      "fact_expr": "C: Hyperbola;m: Number;F1: Point;F2: Point;M: Point;m>0;Expression(C) = (-y^2/4 + x^2/m = 1);Coordinate(F1) = (-4, 0);Coordinate(F2) = (4, 0);Focus(C) = {F1, F2};PointOnCurve(M, C);AngleOf(F1, M, F2) = ApplyUnit(60, degree)",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7696.png",
+      "answer": "(2,pm*2*sqrt(2))",
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+      "index": 7697,
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+      "answer": "x^2/10 + y^2/8 = 1 (the answer is not unique)",
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+      "answer": "3",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1)",
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+    {
+      "index": 7699,
+      "success": true,
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+      "error": null,
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+        "direction": "up"
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7699.png",
+      "answer": "4",
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+      "index": 7700,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7700.png",
+      "answer": "y^2=4*x",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;H: Circle;p>0;Expression(H) = (-6*x + x^2 + y^2 - 7 = 0);IsTangent(H,Directrix(G))",
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+    {
+      "index": 7701,
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7701.png",
+      "answer": "9*x + 8*y - 26 = 0",
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+      "index": 7702,
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+      "answer": "2",
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+      "index": 7703,
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+        "y_coef": 75.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7703.png",
+      "answer": "(1/2, -1/2)",
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+      "index": 7704,
+      "success": true,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7704.png",
+      "answer": "sqrt(3)/4",
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+      "index": 7706,
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+      "answer": "2",
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+      "index": 7707,
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+      "error": "Unsupported or unparseable expression"
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+      "index": 7708,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7708.png",
+      "answer": "9",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7709.png",
+      "answer": "sqrt(2)",
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+        "converged": true
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7711.png",
+      "answer": "x=-1",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;Coordinate(Focus(G)) = (1, 0)",
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7712.png",
+      "answer": "sqrt(3)",
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+    {
+      "index": 7713,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7713.png",
+      "answer": "8",
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+    {
+      "index": 7714,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7714.png",
+      "answer": "y=sqrt(2)*x+1",
+      "fact_expr": "C: Hyperbola;G: Line;Expression(C) = (x^2 - y^2/2 = 1);IsParallel(Asymptote(C),G);Slope(G)>0",
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+      "index": 7715,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7715.png",
+      "answer": "sqrt(2)/2",
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+      "index": 7716,
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7716.png",
+      "answer": "(1/2,1)",
+      "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;Expression(G) = (y^2 = 2*x);Coordinate(A) = (3, 1);PointOnCurve(P, G);Focus(G)=F;WhenMin(Abs(LineSegmentOf(P,F))+Abs(LineSegmentOf(P,A)))",
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+      "index": 7717,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 7718,
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+        "iterations": 122
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7718.png",
+      "answer": "x^2/9 - y^2/27 = 1",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 24*x);Expression(OneOf(Asymptote(G))) = (y = sqrt(3)*x);PointOnCurve(OneOf(Focus(G)), Directrix(H))",
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+      "index": 7719,
+      "success": true,
+      "conic_type": "ellipse",
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+        "y_coef": 3.0
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+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7719.png",
+      "answer": "-3/4",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);Endpoint(MajorAxis(G)) = {M,N};M: Point;N: Point;P: Point;PointOnCurve(P, G) = True",
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+      "index": 7720,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 7721,
+      "success": true,
+      "conic_type": "hyperbola",
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+        "b": 6.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7721.png",
+      "answer": "33",
+      "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/64 - y^2/36 = 1);PointOnCurve(P, G);Focus(G) = {F1, F2};Abs(LineSegmentOf(P, F1)) = 17",
+      "coords": {}
+    },
+    {
+      "index": 7722,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.5,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7722.png",
+      "answer": "2",
+      "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Curve;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (-4*x + x^2 + y^2 + 1 = 0);IsTangent(Asymptote(G), H)",
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+    },
+    {
+      "index": 7723,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7723.png",
+      "answer": "y^0 = -16 * x",
+      "fact_expr": "C: Hyperbola;G: Parabola;O: Origin;Expression(C) = (x^2/10 - y^2/6 = 1);Vertex(G)=O;SymmetryAxis(G)=xAxis;Focus(G)=LeftFocus(C)",
+      "coords": {}
+    },
+    {
+      "index": 7724,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "orientation": "horizontal"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7724.png",
+      "answer": "y = pm*(1/2)*x",
+      "fact_expr": "E: Hyperbola;Expression(E) = (x^2/4 - y^2 = 1)",
+      "coords": {}
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+    {
+      "index": 7725,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+      "index": 7726,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "major_axis": "x",
+        "x_coef": 6.0,
+        "y_coef": 4.0
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7726.png",
+      "answer": "2*sqrt(2)",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2/6 + y^2/4 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 7727,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 7728,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7728.png",
+      "answer": "x^2=sqrt(2)*y",
+      "fact_expr": "C1: Parabola;Expression(C1) = (y^2=4*x);C2: Parabola;Expression(C2) = (x^2=2*p*y);p: Number;p>0;O: Origin;Intersection(C1,C2) = A;A: Point;Negation(A=O);Distance(A,Focus(C1)) = 3",
+      "coords": {}
+    },
+    {
+      "index": 7729,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
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+        "b": 2.0,
+        "major_axis": "y",
+        "x_coef": 4.0,
+        "y_coef": 9.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7729.png",
+      "answer": "4",
+      "fact_expr": "G: Ellipse;P: Point;F2: Point;F1: Point;Expression(G)=(x^2/4+y^2/9=1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(P,G)",
+      "coords": {}
+    },
+    {
+      "index": 7730,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 7731,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
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+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7731.png",
+      "answer": "3",
+      "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (2, 2);Focus(G) = F;PointOnCurve(P, G)",
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+          2.0
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+    {
+      "index": 7732,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
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+    {
+      "index": 7733,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
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+        "b": 1.0,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
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+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7733.png",
+      "answer": "(-1, -\\sqrt{2}/2) + (-\\sqrt{2}/2, \\sqrt{2}/2) + (\\sqrt{2}/2, 1)",
+      "fact_expr": "C: Hyperbola;l: Line;k: Number;A: Point;B: Point;Expression(l) = (y = k*x + 1);Expression(C) = (x^2/2 - y^2 = 1);Intersection(l, C) = {A, B};Negation(A = B)",
+      "coords": {}
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+    {
+      "index": 7734,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7735,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
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+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7735.png",
+      "answer": "3",
+      "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2/3 = 1);RightFocus(C) = F;F: Point;O: Origin;l: Line;PointOnCurve(O,l) = True;Intersection(l,C) = {A,B};B: Point;A: Point;LineSegmentOf(A, F) = 1",
+      "coords": {}
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+    {
+      "index": 7736,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7736.png",
+      "answer": "16",
+      "fact_expr": "l: Line;G: Parabola;M: Point;N: Point;A: Point;B: Point;Expression(G) = (y^2 = 8*x);Coordinate(M) = (2, 0);Coordinate(N) = (3, 1);PointOnCurve(M, l);PointOnCurve(N,l);Intersection(l,G)={A,B}",
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+        "M": [
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+        "N": [
+          3.0,
+          1.0
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+    {
+      "index": 7737,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
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+        "b": 0.7071067811865476,
+        "orientation": "horizontal"
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+      "optimization": {
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+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7737.png",
+      "answer": "(\\sqrt{6}/2,0)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - 2*y^2 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 7738,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
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+        "direction": "right"
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+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7738.png",
+      "answer": "100",
+      "fact_expr": "T: Point;Coordinate(T) = (-2, 2);TangentOfPoint(T,C) = {l1,l2};l1: Line;l2: Line;C: Parabola;Expression(C) = (y^2 = 2*p*x);p: Number;TangentPoint(l1,C) = A;TangentPoint(l2,C) = B;A: Point;B: Point;Focus(C) = F;F: Point;PointOnCurve(F,LineOf(A,B)) = True",
+      "coords": {
+        "T": [
+          -2.0,
+          2.0
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+      }
+    },
+    {
+      "index": 7739,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7739.png",
+      "answer": "6",
+      "fact_expr": "G: Hyperbola;m: Number;H: Parabola;Expression(G) = (-y^2/3 + x^2/m = 1);Expression(H) = (y^2 = 12*x);RightFocus(G) = Focus(H)",
+      "coords": {}
+    },
+    {
+      "index": 7740,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7740.png",
+      "answer": "(2,2*sqrt(2))",
+      "fact_expr": "C: Parabola;p: Number;G: Circle;P: Point;F: Point;Expression(C) = (y^2 = 2*(p*x));Expression(G) = (-2*x + x^2 + y^2 = 0);Quadrant(P) = 1;PointOnCurve(P, C);LineSegmentOf(P, F) = 3;Focus(C)=F;Center(G)=F",
+      "coords": {}
+    },
+    {
+      "index": 7741,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.7320508075688772,
+        "major_axis": "x",
+        "x_coef": 4.0,
+        "y_coef": 3.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7741.png",
+      "answer": "sqrt(6)/3",
+      "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;O: Origin;PointOnCurve(O, L) = True;Inclination(L) = pi/6;Intersection(L, G) = {P, Q};Q: Point;P: Point;L: Line;IsDiameter(LineSegmentOf(P, Q), H) = True;H: Circle;F: Point;RightFocus(G) = F;PointOnCurve(F, H) = True",
+      "coords": {}
+    },
+    {
+      "index": 7742,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7742.png",
+      "answer": "1/5",
+      "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);H: Line;Expression(H) = (y = 2*x - 4);A: Point;B: Point;Intersection(G, H) = {A, B};C: Point;PointOnCurve(C, G);O: Origin;lambda: Real;VectorOf(O, A) + VectorOf(O, B) = lambda*VectorOf(O, C)",
+      "coords": {}
+    },
+    {
+      "index": 7743,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 5.0,
+        "b": 4.0,
+        "major_axis": "x",
+        "x_coef": 25.0,
+        "y_coef": 16.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7743.png",
+      "answer": "12/5",
+      "fact_expr": "Gamma: Ellipse;Expression(Gamma) = (x^2/25 + y^2/16 = 1);P: Point;l1: Line;l2: Line;PointOnCurve(P,l1) = True;PointOnCurve(P,l2) = True;IsPerpendicular(l1,l2) = True;Intersection(l1,Gamma) = {A,B};A: Point;B: Point;Intersection(l2,Gamma) = {C,D};C: Point;D: Point;O: Origin;1/(Abs(LineSegmentOf(C, P))*Abs(LineSegmentOf(D, P))) + 1/(Abs(LineSegmentOf(A, P))*Abs(LineSegmentOf(B, P))) = 41/256",
+      "coords": {}
+    },
+    {
+      "index": 7744,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7745,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7745.png",
+      "answer": "4/3",
+      "fact_expr": "G: Hyperbola;a: Number;H: Ellipse;Expression(G) = (-y^2/7 + x^2/a = 1);Expression(H) = (x^2/25 + y^2/9 = 1);Focus(G) = Focus(H)",
+      "coords": {}
+    },
+    {
+      "index": 7746,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7747,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 1.0,
+        "direction": "right"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7747.png",
+      "answer": "3",
+      "fact_expr": "C: Curve;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;G: Parabola;P: Point;PointOnCurve(P, G) = True;A: Point;Coordinate(A) = (2, 1)",
+      "coords": {
+        "A": [
+          2.0,
+          1.0
+        ]
+      }
+    },
+    {
+      "index": 7748,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7749,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 1.0,
+        "b": 0.7071067811865476,
+        "major_axis": "y",
+        "x_coef": 0.5,
+        "y_coef": 1.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7749.png",
+      "answer": "sqrt(2)",
+      "fact_expr": "G: Ellipse;Expression(G) = (2*x^2 + y^2 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 7750,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.8284271247461903,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7750.png",
+      "answer": "16/7",
+      "fact_expr": "C: Hyperbola;Expression(C) = (x^2/8 - y^2 = 1);RightFocus(C) = F;F: Point;Asymptote(C) = {l1,l2};l1: Line;l2: Line;PointOnCurve(F,l) = True;l: Line;Intersection(l,l1) = A;Intersection(l,l2) = B;A: Point;B: Point;IsPerpendicular(LineSegmentOf(A,B),l2) = True",
+      "coords": {}
+    },
+    {
+      "index": 7751,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "vertical"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7751.png",
+      "answer": "3",
+      "fact_expr": "H: Ellipse;Expression(H) = (x^2/4 + y^2/a^2 = 1);a: Number;G: Hyperbola;Expression(G) = (y^2/2 - x^2/a = 1);Focus(H) = Focus(G)",
+      "coords": {}
+    },
+    {
+      "index": 7752,
+      "success": false,
+      "error": "Unsupported or unparseable expression"
+    },
+    {
+      "index": 7753,
+      "success": true,
+      "conic_type": "ellipse",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.0,
+        "major_axis": "y",
+        "x_coef": 1.0,
+        "y_coef": 4.0
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/ellipse/problem_7753.png",
+      "answer": "sqrt(6)/3",
+      "fact_expr": "G: Ellipse;Expression(G) = (x^2 + y^2/3 = 1)",
+      "coords": {}
+    },
+    {
+      "index": 7754,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 2.0,
+        "b": 1.5,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7754.png",
+      "answer": "(1,\\sqrt{5}]",
+      "fact_expr": "C: Hyperbola;b: Number;a: Number;F1: Point;F2: Point;O: Origin;M: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) =F1;RightFocus(C)=F2;PointOnCurve(M, RightPart(C));Abs(LineSegmentOf(F1, F2)) = 2*Abs(LineSegmentOf(O, M));Tan(AngleOf(M, F2, F1)) >= 2",
+      "coords": {}
+    },
+    {
+      "index": 7755,
+      "success": true,
+      "conic_type": "parabola",
+      "error": null,
+      "params": {
+        "p": 2.0,
+        "direction": "up"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/parabola/problem_7755.png",
+      "answer": "10",
+      "fact_expr": "G: Parabola;P: Point;Expression(G) = (x^2 = 8*y);PointOnCurve(P, G);Distance(P, Focus(G)) = 12",
+      "coords": {}
+    },
+    {
+      "index": 7756,
+      "success": true,
+      "conic_type": "hyperbola",
+      "error": null,
+      "params": {
+        "a": 1.7320508075688772,
+        "b": 1.0,
+        "orientation": "horizontal"
+      },
+      "optimization": {
+        "final_loss": 0.0,
+        "converged": true,
+        "note": "using explicit params"
+      },
+      "output_path": "/Users/wendinggao/Analytic_SDF/results/train/hyperbola/problem_7756.png",
+      "answer": "5+2*sqrt(3)",
+      "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, RightPart(G));Q: Point;Coordinate(Q) = (-2, 3)",
+      "coords": {
+        "Q": [
+          -2.0,
+          3.0
+        ]
+      }
+    }
+  ]
+}
\ No newline at end of file