diff --git "a/trivial/test.jsonl" "b/trivial/test.jsonl"
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+++ "b/trivial/test.jsonl"
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+[
+    {
+        "id": "2063/A",
+        "aliases": null,
+        "contest_id": "2063",
+        "contest_name": "Codeforces Round 1000 (Div. 2)",
+        "contest_type": "CF",
+        "contest_start": 1737547500,
+        "contest_start_year": 2025,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Minimal Coprime",
+        "description": "Today, Little John used all his savings to buy a segment. He wants to build a house on this segment.\n\nA segment of positive integers $$$[l,r]$$$ is called coprime if $$$l$$$ and $$$r$$$ are coprime$$$^{\\text{∗}}$$$.\n\nA coprime segment $$$[l,r]$$$ is called minimal coprime if it does not contain$$$^{\\text{†}}$$$ any coprime segment not equal to itself. To better understand this statement, you can refer to the notes.\n\nGiven $$$[l,r]$$$, a segment of positive integers, find the number of minimal coprime segments contained in $$$[l,r]$$$.",
+        "input_format": "Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \\le t \\le 100$$$). The description of the test cases follows.\n\nThe only line of each test case consists of two integers $$$l$$$ and $$$r$$$ ($$$1 \\le l \\le r \\le 10^9$$$).",
+        "output_format": "For each test case, output the number of minimal coprime segments contained in $$$[l,r]$$$, on a separate line.",
+        "interaction_format": null,
+        "note": "On the first test case, the given segment is $$$[1,2]$$$. The segments contained in $$$[1,2]$$$ are as follows.\n\n- $$$[1,1]$$$: This segment is coprime, since the numbers $$$1$$$ and $$$1$$$ are coprime, and this segment does not contain any other segment inside. Thus, $$$[1,1]$$$ is minimal coprime.\n- $$$[1,2]$$$: This segment is coprime. However, as it contains $$$[1,1]$$$, which is also coprime, $$$[1,2]$$$ is not minimal coprime.\n- $$$[2,2]$$$: This segment is not coprime because $$$2$$$ and $$$2$$$ share $$$2$$$ positive common divisors: $$$1$$$ and $$$2$$$.\n\nTherefore, the segment $$$[1,2]$$$ contains $$$1$$$ minimal coprime segment.",
+        "examples": [
+            {
+                "input": "6\n1 2\n1 10\n49 49\n69 420\n1 1\n9982 44353",
+                "output": "1\n9\n0\n351\n1\n34371"
+            }
+        ],
+        "editorial": "It is easy to see that two integers $$$l$$$ and $$$l+1$$$ are always coprime, due to $$$\\gcd(l,l+1)=\\gcd(1,l)=\\gcd(0,1)=1$$$. Thus, all segments in the form $$$[l,l+1]$$$ are coprime, and all minimal coprime segments must have a size of at most $$$2$$$ (otherwise, it will contain another segment of size $$$2$$$).\nIt can be observed that the only coprime segment with size $$$1$$$ is $$$[1,1]$$$ as $$$\\gcd(a,a)=a$$$. The set of minimal coprime segments is as follows:\n• $$$[1,1]$$$ is minimal coprime.\n• For all $$$l>1$$$, $$$[l,l+1]$$$ is minimal coprime.\nAnd there is no other segment that is minimal coprime.\nThe solution is thus as follows:\n• If $$$x=y=1$$$, it contains $$$[1,1]$$$ and the answer is $$$1$$$.\n• Otherwise, $$$y>1$$$. There exists exactly one $$$l$$$ for all $$$x \\le l < y$$$ such that $$$[l,*]$$$ is minimal coprime and is contained in $$$[x,y]$$$.   In other words, the answer is $$$(y-1)-x+1=y-x$$$ in this case.\n• In other words, the answer is $$$(y-1)-x+1=y-x$$$ in this case.\nThe problem has been solved with time complexity $$$\\mathcal{O}(1)$$$ per test case.",
+        "rating": 800,
+        "tags": [
+            "math",
+            "number theory"
+        ],
+        "testset_size": 4,
+        "official_tests": [
+            {
+                "input": "6\r\n1 2\r\n1 10\r\n49 49\r\n69 420\r\n1 1\r\n9982 44353\r\n",
+                "output": "1\r\n9\r\n0\r\n351\r\n1\r\n34371\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2019/A",
+        "aliases": null,
+        "contest_id": "2019",
+        "contest_name": "Codeforces Round 975 (Div. 2)",
+        "contest_type": "CF",
+        "contest_start": 1727444100,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Max Plus Size",
+        "description": "You are given an array $$$a_1, a_2, \\ldots, a_n$$$ of positive integers.\n\nYou can color some elements of the array red, but there cannot be two adjacent red elements (i.e., for $$$1 \\leq i \\leq n-1$$$, at least one of $$$a_i$$$ and $$$a_{i+1}$$$ must not be red).\n\nYour score is the maximum value of a red element plus the number of red elements. Find the maximum score you can get.",
+        "input_format": "Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \\le t \\le 500$$$). The description of the test cases follows.\n\nThe first line of each test case contains a single integer $$$n$$$ ($$$1 \\le n \\le 100$$$) — the length of the array.\n\nThe second line of each test case contains $$$n$$$ integers $$$a_1, a_2, \\ldots, a_n$$$ ($$$1 \\le a_i \\le 1000$$$) — the given array.",
+        "output_format": "For each test case, output a single integer: the maximum possible score you can get after coloring some elements red according to the statement.",
+        "interaction_format": null,
+        "note": "In the first test case, you can color the array as follows: $$$[\\color{red}{5}, 4, \\color{red}{5}]$$$. Your score is $$$\\max([5, 5]) + \\text{size}([5, 5]) = 5+2 = 7$$$. This is the maximum score you can get.\n\nIn the second test case, you can color the array as follows: $$$[\\color{red}{4}, 5, \\color{red}{4}]$$$. Your score is $$$\\max([4, 4]) + \\text{size}([4, 4]) = 4+2 = 6$$$. This is the maximum score you can get.\n\nIn the third test case, you can color the array as follows: $$$[\\color{red}{3}, 3, \\color{red}{3}, 3, \\color{red}{4}, 1, 2, \\color{red}{3}, 4, \\color{red}{5}]$$$. Your score is $$$\\max([3, 3, 4, 3, 5]) + \\text{size}([3, 3, 4, 3, 5]) = 5+5 = 10$$$. This is the maximum score you can get.",
+        "examples": [
+            {
+                "input": "4\n3\n5 4 5\n3\n4 5 4\n10\n3 3 3 3 4 1 2 3 4 5\n9\n17 89 92 42 29 92 14 70 45",
+                "output": "7\n6\n10\n97"
+            }
+        ],
+        "editorial": null,
+        "rating": 800,
+        "tags": [
+            "brute force",
+            "dp",
+            "greedy"
+        ],
+        "testset_size": 4,
+        "official_tests": [
+            {
+                "input": "4\r\n3\r\n5 4 5\r\n3\r\n4 5 4\r\n10\r\n3 3 3 3 4 1 2 3 4 5\r\n9\r\n17 89 92 42 29 92 14 70 45\r\n",
+                "output": "7\r\n6\r\n10\r\n97\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "1998/A",
+        "aliases": null,
+        "contest_id": "1998",
+        "contest_name": "Codeforces Round 965 (Div. 2)",
+        "contest_type": "CF",
+        "contest_start": 1723300500,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Find K Distinct Points with Fixed Center",
+        "description": "You are given three integers $$$x_c$$$, $$$y_c$$$, and $$$k$$$ ($$$-100 \\leq x_c, y_c \\leq 100$$$, $$$1 \\leq k \\leq 1000$$$).\n\nYou need to find $$$k$$$ distinct points ($$$x_1, y_1$$$), ($$$x_2, y_2$$$), $$$\\ldots$$$, ($$$x_k, y_k$$$), having integer coordinates, on the 2D coordinate plane such that:\n\n- their center$$$^{\\text{∗}}$$$ is ($$$x_c, y_c$$$)\n- $$$-10^9 \\leq x_i, y_i \\leq 10^9$$$ for all $$$i$$$ from $$$1$$$ to $$$k$$$\n\nIt can be proven that at least one set of $$$k$$$ distinct points always exists that satisfies these conditions.",
+        "input_format": "The first line contains $$$t$$$ ($$$1 \\leq t \\leq 100$$$) — the number of test cases.\n\nEach test case contains three integers $$$x_c$$$, $$$y_c$$$, and $$$k$$$ ($$$-100 \\leq x_c, y_c \\leq 100$$$, $$$1 \\leq k \\leq 1000$$$) — the coordinates of the center and the number of distinct points you must output.\n\nIt is guaranteed that the sum of $$$k$$$ over all test cases does not exceed $$$1000$$$.",
+        "output_format": "For each test case, output $$$k$$$ lines, the $$$i$$$-th line containing two space separated integers, $$$x_i$$$ and $$$y_i$$$, ($$$-10^9 \\leq x_i, y_i \\leq 10^9$$$) — denoting the position of the $$$i$$$-th point.\n\nIf there are multiple answers, print any of them. It can be shown that a solution always exists under the given constraints.",
+        "interaction_format": null,
+        "note": "For the first test case, $$$\\left( \\frac{10}{1}, \\frac{10}{1} \\right) = (10, 10)$$$.\n\nFor the second test case, $$$\\left( \\frac{-1 + 5 - 4}{3}, \\frac{-1 -1 + 2}{3} \\right) = (0, 0)$$$.",
+        "examples": [
+            {
+                "input": "4\n10 10 1\n0 0 3\n-5 -8 8\n4 -5 3",
+                "output": "10 10\n-1 -1\n5 -1\n-4 2\n-6 -7\n-5 -7\n-4 -7\n-4 -8\n-4 -9\n-5 -9\n-6 -9\n-6 -8\n1000 -1000\n-996 995\n8 -10"
+            }
+        ],
+        "editorial": null,
+        "rating": 800,
+        "tags": [
+            "constructive algorithms",
+            "implementation",
+            "math"
+        ],
+        "testset_size": 16,
+        "official_tests": [
+            {
+                "input": "4\r\n10 10 1\r\n0 0 3\r\n-5 -8 8\r\n4 -5 3\r\n",
+                "output": "10 10\r\n1 0\r\n-1 0\r\n0 0\r\n-4 -8\r\n-6 -8\r\n-3 -8\r\n-7 -8\r\n-2 -8\r\n-8 -8\r\n-1 -8\r\n-9 -8\r\n5 -5\r\n3 -5\r\n4 -5\r\n"
+            },
+            {
+                "input": "30\r\n-84 -60 1\r\n-41 -100 2\r\n8 -8 3\r\n-52 -62 4\r\n-61 -76 5\r\n-52 -52 6\r\n14 -11 7\r\n-2 -54 8\r\n46 8 9\r\n100 100 10\r\n-35 34 11\r\n87 5 12\r\n42 -44 13\r\n66 -32 14\r\n71 -31 15\r\n9 -51 16\r\n90 -62 17\r\n86 -52 18\r\n7 88 19\r\n25 -14 20\r\n-7 37 21\r\n-55 98 22\r\n64 51 23\r\n34 -85 24\r\n8 -18 25\r\n34 64 26\r\n93 -37 27\r\n16 80 28\r\n-70 -28 29\r\n63 31 30\r\n",
+                "output": "-84 -60\n-40 -99\n-42 -101\n9 -7\n7 -9\n8 -8\n-51 -61\n-50 -60\n-53 -63\n-54 -64\n-60 -75\n-59 -74\n-62 -77\n-63 -78\n-61 -76\n-51 -51\n-50 -50\n-49 -49\n-53 -53\n-54 -54\n-55 -55\n15 -10\n16 -9\n17 -8\n13 -12\n12 -13\n11 -14\n14 -11\n-1 -53\n0 -52\n1 -51\n2 -50\n-3 -55\n-4 -56\n-5 -57\n-6 -58\n47 9\n48 10\n49 11\n50 12\n45 7\n44 6\n43 5\n42 4\n46 8\n101 101\n102 102\n103 103\n104 104\n105 105\n99 99\n98 98\n97 97\n96 96\n95 95\n-34 35\n-33 36\n-32 37\n-31 38\n-30 39\n-36 33\n-37 32\n-38 31\n-39 30\n-40 29\n-35 34\n88 6\n89 7\n90 8\n91 9\n92 10\n93 11\n86 4\n85 3\n84 2\n83 1\n82 0\n81 -1\n43 -43\n44 -42\n45 -41\n46 -40\n47 -39\n48 -38\n41 -45\n40 -46\n39 -47\n38 -48\n37 -49\n36 -50\n42 -44\n67 -31\n68 -30\n69 -29\n70 -28\n71 -27\n72 -26\n73 -25\n65 -33\n64 -34\n63 -35\n62 -36\n61 -37\n60 -38\n59 -39\n72 -30\n73 -29\n74 -28\n75 -27\n76 -26\n77 -25\n78 -24\n70 -32\n69 -33\n68 -34\n67 -35\n66 -36\n65 -37\n64 -38\n71 -31\n10 -50\n11 -49\n12 -48\n13 -47\n14 -46\n15 -45\n16 -44\n17 -43\n8 -52\n7 -53\n6 -54\n5 -55\n4 -56\n3 -57\n2 -58\n1 -59\n91 -61\n92 -60\n93 -59\n94 -58\n95 -57\n96 -56\n97 -55\n98 -54\n89 -63\n88 -64\n87 -65\n86 -66\n85 -67\n84 -68\n83 -69\n82 -70\n90 -62\n87 -51\n88 -50\n89 -49\n90 -48\n91 -47\n92 -46\n93 -45\n94 -44\n95 -43\n85 -53\n84 -54\n83 -55\n82 -56\n81 -57\n80 -58\n79 -59\n78 -60\n77 -61\n8 89\n9 90\n10 91\n11 92\n12 93\n13 94\n14 95\n15 96\n16 97\n6 87\n5 86\n4 85\n3 84\n2 83\n1 82\n0 81\n-1 80\n-2 79\n7 88\n26 -13\n27 -12\n28 -11\n29 -10\n30 -9\n31 -8\n32 -7\n33 -6\n34 -5\n35 -4\n24 -15\n23 -16\n22 -17\n21 -18\n20 -19\n19 -20\n18 -21\n17 -22\n16 -23\n15 -24\n-6 38\n-5 39\n-4 40\n-3 41\n-2 42\n-1 43\n0 44\n1 45\n2 46\n3 47\n-8 36\n-9 35\n-10 34\n-11 33\n-12 32\n-13 31\n-14 30\n-15 29\n-16 28\n-17 27\n-7 37\n-54 99\n-53 100\n-52 101\n-51 102\n-50 103\n-49 104\n-48 105\n-47 106\n-46 107\n-45 108\n-44 109\n-56 97\n-57 96\n-58 95\n-59 94\n-60 93\n-61 92\n-62 91\n-63 90\n-64 89\n-65 88\n-66 87\n65 52\n66 53\n67 54\n68 55\n69 56\n70 57\n71 58\n72 59\n73 60\n74 61\n75 62\n63 50\n62 49\n61 48\n60 47\n59 46\n58 45\n57 44\n56 43\n55 42\n54 41\n53 40\n64 51\n35 -84\n36 -83\n37 -82\n38 -81\n39 -80\n40 -79\n41 -78\n42 -77\n43 -76\n44 -75\n45 -74\n46 -73\n33 -86\n32 -87\n31 -88\n30 -89\n29 -90\n28 -91\n27 -92\n26 -93\n25 -94\n24 -95\n23 -96\n22 -97\n9 -17\n10 -16\n11 -15\n12 -14\n13 -13\n14 -12\n15 -11\n16 -10\n17 -9\n18 -8\n19 -7\n20 -6\n7 -19\n6 -20\n5 -21\n4 -22\n3 -23\n2 -24\n1 -25\n0 -26\n-1 -27\n-2 -28\n-3 -29\n-4 -30\n8 -18\n35 65\n36 66\n37 67\n38 68\n39 69\n40 70\n41 71\n42 72\n43 73\n44 74\n45 75\n46 76\n47 77\n33 63\n32 62\n31 61\n30 60\n29 59\n28 58\n27 57\n26 56\n25 55\n24 54\n23 53\n22 52\n21 51\n94 -36\n95 -35\n96 -34\n97 -33\n98 -32\n99 -31\n100 -30\n101 -29\n102 -28\n103 -27\n104 -26\n105 -25\n106 -24\n92 -38\n91 -39\n90 -40\n89 -41\n88 -42\n87 -43\n86 -44\n85 -45\n84 -46\n83 -47\n82 -48\n81 -49\n80 -50\n93 -37\n17 81\n18 82\n19 83\n20 84\n21 85\n22 86\n23 87\n24 88\n25 89\n26 90\n27 91\n28 92\n29 93\n30 94\n15 79\n14 78\n13 77\n12 76\n11 75\n10 74\n9 73\n8 72\n7 71\n6 70\n5 69\n4 68\n3 67\n2 66\n-69 -27\n-68 -26\n-67 -25\n-66 -24\n-65 -23\n-64 -22\n-63 -21\n-62 -20\n-61 -19\n-60 -18\n-59 -17\n-58 -16\n-57 -15\n-56 -14\n-71 -29\n-72 -30\n-73 -31\n-74 -32\n-75 -33\n-76 -34\n-77 -35\n-78 -36\n-79 -37\n-80 -38\n-81 -39\n-82 -40\n-83 -41\n-84 -42\n-70 -28\n64 32\n65 33\n66 34\n67 35\n68 36\n69 37\n70 38\n71 39\n72 40\n73 41\n74 42\n75 43\n76 44\n77 45\n78 46\n62 30\n61 29\n60 28\n59 27\n58 26\n57 25\n56 24\n55 23\n54 22\n53 21\n52 20\n51 19\n50 18\n49 17\n48 16\n"
+            },
+            {
+                "input": "1\r\n-100 -100 1000\r\n",
+                "output": "-99 -99\n-98 -98\n-97 -97\n-96 -96\n-95 -95\n-94 -94\n-93 -93\n-92 -92\n-91 -91\n-90 -90\n-89 -89\n-88 -88\n-87 -87\n-86 -86\n-85 -85\n-84 -84\n-83 -83\n-82 -82\n-81 -81\n-80 -80\n-79 -79\n-78 -78\n-77 -77\n-76 -76\n-75 -75\n-74 -74\n-73 -73\n-72 -72\n-71 -71\n-70 -70\n-69 -69\n-68 -68\n-67 -67\n-66 -66\n-65 -65\n-64 -64\n-63 -63\n-62 -62\n-61 -61\n-60 -60\n-59 -59\n-58 -58\n-57 -57\n-56 -56\n-55 -55\n-54 -54\n-53 -53\n-52 -52\n-51 -51\n-50 -50\n-49 -49\n-48 -48\n-47 -47\n-46 -46\n-45 -45\n-44 -44\n-43 -43\n-42 -42\n-41 -41\n-40 -40\n-39 -39\n-38 -38\n-37 -37\n-36 -36\n-35 -35\n-34 -34\n-33 -33\n-32 -32\n-31 -31\n-30 -30\n-29 -29\n-28 -28\n-27 -27\n-26 -26\n-25 -25\n-24 -24\n-23 -23\n-22 -22\n-21 -21\n-20 -20\n-19 -19\n-18 -18\n-17 -17\n-16 -16\n-15 -15\n-14 -14\n-13 -13\n-12 -12\n-11 -11\n-10 -10\n-9 -9\n-8 -8\n-7 -7\n-6 -6\n-5 -5\n-4 -4\n-3 -3\n-2 -2\n-1 -1\n0 0\n1 1\n2 2\n3 3\n4 4\n5 5\n6 6\n7 7\n8 8\n9 9\n10 10\n11 11\n12 12\n13 13\n14 14\n15 15\n16 16\n17 17\n18 18\n19 19\n20 20\n21 21\n22 22\n23 23\n24 24\n25 25\n26 26\n27 27\n28 28\n29 29\n30 30\n31 31\n32 32\n33 33\n34 34\n35 35\n36 36\n37 37\n38 38\n39 39\n40 40\n41 41\n42 42\n43 43\n44 44\n45 45\n46 46\n47 47\n48 48\n49 49\n50 50\n51 51\n52 52\n53 53\n54 54\n55 55\n56 56\n57 57\n58 58\n59 59\n60 60\n61 61\n62 62\n63 63\n64 64\n65 65\n66 66\n67 67\n68 68\n69 69\n70 70\n71 71\n72 72\n73 73\n74 74\n75 75\n76 76\n77 77\n78 78\n79 79\n80 80\n81 81\n82 82\n83 83\n84 84\n85 85\n86 86\n87 87\n88 88\n89 89\n90 90\n91 91\n92 92\n93 93\n94 94\n95 95\n96 96\n97 97\n98 98\n99 99\n100 100\n101 101\n102 102\n103 103\n104 104\n105 105\n106 106\n107 107\n108 108\n109 109\n110 110\n111 111\n112 112\n113 113\n114 114\n115 115\n116 116\n117 117\n118 118\n119 119\n120 120\n121 121\n122 122\n123 123\n124 124\n125 125\n126 126\n127 127\n128 128\n129 129\n130 130\n131 131\n132 132\n133 133\n134 134\n135 135\n136 136\n137 137\n138 138\n139 139\n140 140\n141 141\n142 142\n143 143\n144 144\n145 145\n146 146\n147 147\n148 148\n149 149\n150 150\n151 151\n152 152\n153 153\n154 154\n155 155\n156 156\n157 157\n158 158\n159 159\n160 160\n161 161\n162 162\n163 163\n164 164\n165 165\n166 166\n167 167\n168 168\n169 169\n170 170\n171 171\n172 172\n173 173\n174 174\n175 175\n176 176\n177 177\n178 178\n179 179\n180 180\n181 181\n182 182\n183 183\n184 184\n185 185\n186 186\n187 187\n188 188\n189 189\n190 190\n191 191\n192 192\n193 193\n194 194\n195 195\n196 196\n197 197\n198 198\n199 199\n200 200\n201 201\n202 202\n203 203\n204 204\n205 205\n206 206\n207 207\n208 208\n209 209\n210 210\n211 211\n212 212\n213 213\n214 214\n215 215\n216 216\n217 217\n218 218\n219 219\n220 220\n221 221\n222 222\n223 223\n224 224\n225 225\n226 226\n227 227\n228 228\n229 229\n230 230\n231 231\n232 232\n233 233\n234 234\n235 235\n236 236\n237 237\n238 238\n239 239\n240 240\n241 241\n242 242\n243 243\n244 244\n245 245\n246 246\n247 247\n248 248\n249 249\n250 250\n251 251\n252 252\n253 253\n254 254\n255 255\n256 256\n257 257\n258 258\n259 259\n260 260\n261 261\n262 262\n263 263\n264 264\n265 265\n266 266\n267 267\n268 268\n269 269\n270 270\n271 271\n272 272\n273 273\n274 274\n275 275\n276 276\n277 277\n278 278\n279 279\n280 280\n281 281\n282 282\n283 283\n284 284\n285 285\n286 286\n287 287\n288 288\n289 289\n290 290\n291 291\n292 292\n293 293\n294 294\n295 295\n296 296\n297 297\n298 298\n299 299\n300 300\n301 301\n302 302\n303 303\n304 304\n305 305\n306 306\n307 307\n308 308\n309 309\n310 310\n311 311\n312 312\n313 313\n314 314\n315 315\n316 316\n317 317\n318 318\n319 319\n320 320\n321 321\n322 322\n323 323\n324 324\n325 325\n326 326\n327 327\n328 328\n329 329\n330 330\n331 331\n332 332\n333 333\n334 334\n335 335\n336 336\n337 337\n338 338\n339 339\n340 340\n341 341\n342 342\n343 343\n344 344\n345 345\n346 346\n347 347\n348 348\n349 349\n350 350\n351 351\n352 352\n353 353\n354 354\n355 355\n356 356\n357 357\n358 358\n359 359\n360 360\n361 361\n362 362\n363 363\n364 364\n365 365\n366 366\n367 367\n368 368\n369 369\n370 370\n371 371\n372 372\n373 373\n374 374\n375 375\n376 376\n377 377\n378 378\n379 379\n380 380\n381 381\n382 382\n383 383\n384 384\n385 385\n386 386\n387 387\n388 388\n389 389\n390 390\n391 391\n392 392\n393 393\n394 394\n395 395\n396 396\n397 397\n398 398\n399 399\n400 400\n-101 -101\n-102 -102\n-103 -103\n-104 -104\n-105 -105\n-106 -106\n-107 -107\n-108 -108\n-109 -109\n-110 -110\n-111 -111\n-112 -112\n-113 -113\n-114 -114\n-115 -115\n-116 -116\n-117 -117\n-118 -118\n-119 -119\n-120 -120\n-121 -121\n-122 -122\n-123 -123\n-124 -124\n-125 -125\n-126 -126\n-127 -127\n-128 -128\n-129 -129\n-130 -130\n-131 -131\n-132 -132\n-133 -133\n-134 -134\n-135 -135\n-136 -136\n-137 -137\n-138 -138\n-139 -139\n-140 -140\n-141 -141\n-142 -142\n-143 -143\n-144 -144\n-145 -145\n-146 -146\n-147 -147\n-148 -148\n-149 -149\n-150 -150\n-151 -151\n-152 -152\n-153 -153\n-154 -154\n-155 -155\n-156 -156\n-157 -157\n-158 -158\n-159 -159\n-160 -160\n-161 -161\n-162 -162\n-163 -163\n-164 -164\n-165 -165\n-166 -166\n-167 -167\n-168 -168\n-169 -169\n-170 -170\n-171 -171\n-172 -172\n-173 -173\n-174 -174\n-175 -175\n-176 -176\n-177 -177\n-178 -178\n-179 -179\n-180 -180\n-181 -181\n-182 -182\n-183 -183\n-184 -184\n-185 -185\n-186 -186\n-187 -187\n-188 -188\n-189 -189\n-190 -190\n-191 -191\n-192 -192\n-193 -193\n-194 -194\n-195 -195\n-196 -196\n-197 -197\n-198 -198\n-199 -199\n-200 -200\n-201 -201\n-202 -202\n-203 -203\n-204 -204\n-205 -205\n-206 -206\n-207 -207\n-208 -208\n-209 -209\n-210 -210\n-211 -211\n-212 -212\n-213 -213\n-214 -214\n-215 -215\n-216 -216\n-217 -217\n-218 -218\n-219 -219\n-220 -220\n-221 -221\n-222 -222\n-223 -223\n-224 -224\n-225 -225\n-226 -226\n-227 -227\n-228 -228\n-229 -229\n-230 -230\n-231 -231\n-232 -232\n-233 -233\n-234 -234\n-235 -235\n-236 -236\n-237 -237\n-238 -238\n-239 -239\n-240 -240\n-241 -241\n-242 -242\n-243 -243\n-244 -244\n-245 -245\n-246 -246\n-247 -247\n-248 -248\n-249 -249\n-250 -250\n-251 -251\n-252 -252\n-253 -253\n-254 -254\n-255 -255\n-256 -256\n-257 -257\n-258 -258\n-259 -259\n-260 -260\n-261 -261\n-262 -262\n-263 -263\n-264 -264\n-265 -265\n-266 -266\n-267 -267\n-268 -268\n-269 -269\n-270 -270\n-271 -271\n-272 -272\n-273 -273\n-274 -274\n-275 -275\n-276 -276\n-277 -277\n-278 -278\n-279 -279\n-280 -280\n-281 -281\n-282 -282\n-283 -283\n-284 -284\n-285 -285\n-286 -286\n-287 -287\n-288 -288\n-289 -289\n-290 -290\n-291 -291\n-292 -292\n-293 -293\n-294 -294\n-295 -295\n-296 -296\n-297 -297\n-298 -298\n-299 -299\n-300 -300\n-301 -301\n-302 -302\n-303 -303\n-304 -304\n-305 -305\n-306 -306\n-307 -307\n-308 -308\n-309 -309\n-310 -310\n-311 -311\n-312 -312\n-313 -313\n-314 -314\n-315 -315\n-316 -316\n-317 -317\n-318 -318\n-319 -319\n-320 -320\n-321 -321\n-322 -322\n-323 -323\n-324 -324\n-325 -325\n-326 -326\n-327 -327\n-328 -328\n-329 -329\n-330 -330\n-331 -331\n-332 -332\n-333 -333\n-334 -334\n-335 -335\n-336 -336\n-337 -337\n-338 -338\n-339 -339\n-340 -340\n-341 -341\n-342 -342\n-343 -343\n-344 -344\n-345 -345\n-346 -346\n-347 -347\n-348 -348\n-349 -349\n-350 -350\n-351 -351\n-352 -352\n-353 -353\n-354 -354\n-355 -355\n-356 -356\n-357 -357\n-358 -358\n-359 -359\n-360 -360\n-361 -361\n-362 -362\n-363 -363\n-364 -364\n-365 -365\n-366 -366\n-367 -367\n-368 -368\n-369 -369\n-370 -370\n-371 -371\n-372 -372\n-373 -373\n-374 -374\n-375 -375\n-376 -376\n-377 -377\n-378 -378\n-379 -379\n-380 -380\n-381 -381\n-382 -382\n-383 -383\n-384 -384\n-385 -385\n-386 -386\n-387 -387\n-388 -388\n-389 -389\n-390 -390\n-391 -391\n-392 -392\n-393 -393\n-394 -394\n-395 -395\n-396 -396\n-397 -397\n-398 -398\n-399 -399\n-400 -400\n-401 -401\n-402 -402\n-403 -403\n-404 -404\n-405 -405\n-406 -406\n-407 -407\n-408 -408\n-409 -409\n-410 -410\n-411 -411\n-412 -412\n-413 -413\n-414 -414\n-415 -415\n-416 -416\n-417 -417\n-418 -418\n-419 -419\n-420 -420\n-421 -421\n-422 -422\n-423 -423\n-424 -424\n-425 -425\n-426 -426\n-427 -427\n-428 -428\n-429 -429\n-430 -430\n-431 -431\n-432 -432\n-433 -433\n-434 -434\n-435 -435\n-436 -436\n-437 -437\n-438 -438\n-439 -439\n-440 -440\n-441 -441\n-442 -442\n-443 -443\n-444 -444\n-445 -445\n-446 -446\n-447 -447\n-448 -448\n-449 -449\n-450 -450\n-451 -451\n-452 -452\n-453 -453\n-454 -454\n-455 -455\n-456 -456\n-457 -457\n-458 -458\n-459 -459\n-460 -460\n-461 -461\n-462 -462\n-463 -463\n-464 -464\n-465 -465\n-466 -466\n-467 -467\n-468 -468\n-469 -469\n-470 -470\n-471 -471\n-472 -472\n-473 -473\n-474 -474\n-475 -475\n-476 -476\n-477 -477\n-478 -478\n-479 -479\n-480 -480\n-481 -481\n-482 -482\n-483 -483\n-484 -484\n-485 -485\n-486 -486\n-487 -487\n-488 -488\n-489 -489\n-490 -490\n-491 -491\n-492 -492\n-493 -493\n-494 -494\n-495 -495\n-496 -496\n-497 -497\n-498 -498\n-499 -499\n-500 -500\n-501 -501\n-502 -502\n-503 -503\n-504 -504\n-505 -505\n-506 -506\n-507 -507\n-508 -508\n-509 -509\n-510 -510\n-511 -511\n-512 -512\n-513 -513\n-514 -514\n-515 -515\n-516 -516\n-517 -517\n-518 -518\n-519 -519\n-520 -520\n-521 -521\n-522 -522\n-523 -523\n-524 -524\n-525 -525\n-526 -526\n-527 -527\n-528 -528\n-529 -529\n-530 -530\n-531 -531\n-532 -532\n-533 -533\n-534 -534\n-535 -535\n-536 -536\n-537 -537\n-538 -538\n-539 -539\n-540 -540\n-541 -541\n-542 -542\n-543 -543\n-544 -544\n-545 -545\n-546 -546\n-547 -547\n-548 -548\n-549 -549\n-550 -550\n-551 -551\n-552 -552\n-553 -553\n-554 -554\n-555 -555\n-556 -556\n-557 -557\n-558 -558\n-559 -559\n-560 -560\n-561 -561\n-562 -562\n-563 -563\n-564 -564\n-565 -565\n-566 -566\n-567 -567\n-568 -568\n-569 -569\n-570 -570\n-571 -571\n-572 -572\n-573 -573\n-574 -574\n-575 -575\n-576 -576\n-577 -577\n-578 -578\n-579 -579\n-580 -580\n-581 -581\n-582 -582\n-583 -583\n-584 -584\n-585 -585\n-586 -586\n-587 -587\n-588 -588\n-589 -589\n-590 -590\n-591 -591\n-592 -592\n-593 -593\n-594 -594\n-595 -595\n-596 -596\n-597 -597\n-598 -598\n-599 -599\n-600 -600\n"
+            },
+            {
+                "input": "1\r\n2 2 4\r\n",
+                "output": "3 2\r\n1 2\r\n4 2\r\n0 2\r\n"
+            },
+            {
+                "input": "1\r\n1 1 2\r\n",
+                "output": "2 1\r\n0 1\r\n"
+            },
+            {
+                "input": "1\r\n1 2 4\r\n",
+                "output": "2 2\r\n0 2\r\n3 2\r\n-1 2\r\n"
+            },
+            {
+                "input": "1\r\n2 3 6\r\n",
+                "output": "3 3\r\n1 3\r\n4 3\r\n0 3\r\n5 3\r\n-1 3\r\n"
+            },
+            {
+                "input": "1\r\n3 3 5\r\n",
+                "output": "4 3\r\n2 3\r\n5 3\r\n1 3\r\n3 3\r\n"
+            },
+            {
+                "input": "1\r\n1 3 4\r\n",
+                "output": "2 3\r\n0 3\r\n3 3\r\n-1 3\r\n"
+            },
+            {
+                "input": "1\r\n-5 20 2\r\n",
+                "output": "-4 20\r\n-6 20\r\n"
+            },
+            {
+                "input": "1\r\n3 3 6\r\n",
+                "output": "4 3\r\n2 3\r\n5 3\r\n1 3\r\n6 3\r\n0 3\r\n"
+            },
+            {
+                "input": "1\r\n1 1 4\r\n",
+                "output": "2 1\r\n0 1\r\n3 1\r\n-1 1\r\n"
+            },
+            {
+                "input": "1\r\n2 -2 4\r\n",
+                "output": "3 -2\r\n1 -2\r\n4 -2\r\n0 -2\r\n"
+            },
+            {
+                "input": "1\r\n1 2 2\r\n",
+                "output": "2 2\r\n0 2\r\n"
+            },
+            {
+                "input": "1\r\n1 1 6\r\n",
+                "output": "2 1\r\n0 1\r\n3 1\r\n-1 1\r\n4 1\r\n-2 1\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": "import sys\n\ndef main():\n    input_path = sys.argv[1]\n    ref_output_path = sys.argv[2]  # Not used\n    submission_output_path = sys.argv[3]\n\n    with open(input_path) as f:\n        input_lines = f.read().splitlines()\n    with open(submission_output_path) as f:\n        sub_lines = [line.strip() for line in f]\n\n    t = int(input_lines[0])\n    ptr = 1\n    sub_ptr = 0\n\n    for _ in range(t):\n        if ptr >= len(input_lines):\n            print(0)\n            return\n        xc, yc, k = map(int, input_lines[ptr].split())\n        ptr += 1\n\n        points = []\n        for __ in range(k):\n            if sub_ptr >= len(sub_lines):\n                print(0)\n                return\n            line = sub_lines[sub_ptr]\n            sub_ptr +=1\n            parts = line.split()\n            if len(parts)!=2:\n                print(0)\n                return\n            try:\n                x = int(parts[0])\n                y = int(parts[1])\n            except:\n                print(0)\n                return\n            if not (-1e9 <= x <= 1e9 and -1e9 <= y <= 1e9):\n                print(0)\n                return\n            points.append((x, y))\n\n        if len(set(points)) != k:\n            print(0)\n            return\n\n        sum_x = sum(x for x, y in points)\n        sum_y = sum(y for x, y in points)\n        if sum_x != xc * k or sum_y != yc * k:\n            print(0)\n            return\n\n    if sub_ptr != len(sub_lines):\n        print(0)\n        return\n\n    print(1)\n\nif __name__ == \"__main__\":\n    main()",
+        "executable": true
+    },
+    {
+        "id": "2030/A",
+        "aliases": null,
+        "contest_id": "2030",
+        "contest_name": "Codeforces Round 979 (Div. 2)",
+        "contest_type": "CF",
+        "contest_start": 1729346700,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "A Gift From Orangutan",
+        "description": "While exploring the jungle, you have bumped into a rare orangutan with a bow tie! You shake hands with the orangutan and offer him some food and water. In return...\n\nThe orangutan has gifted you an array $$$a$$$ of length $$$n$$$. Using $$$a$$$, you will construct two arrays $$$b$$$ and $$$c$$$, both containing $$$n$$$ elements, in the following manner:\n\n- $$$b_i = \\min(a_1, a_2, \\ldots, a_i)$$$ for each $$$1 \\leq i \\leq n$$$.\n- $$$c_i = \\max(a_1, a_2, \\ldots, a_i)$$$ for each $$$1 \\leq i \\leq n$$$.\n\nDefine the score of $$$a$$$ as $$$\\sum_{i=1}^n c_i - b_i$$$ (i.e. the sum of $$$c_i - b_i$$$ over all $$$1 \\leq i \\leq n$$$). Before you calculate the score, you can shuffle the elements of $$$a$$$ however you want.\n\nFind the maximum score that you can get if you shuffle the elements of $$$a$$$ optimally.",
+        "input_format": "The first line contains $$$t$$$ ($$$1 \\leq t \\leq 100$$$) — the number of test cases.\n\nThe first line of each test case contains an integer $$$n$$$ ($$$1 \\leq n \\leq 1000$$$) — the number of elements in $$$a$$$.\n\nThe following line contains $$$n$$$ integers $$$a_1, a_2, \\ldots, a_n$$$ ($$$1 \\leq a_i \\leq 1000$$$) — the elements of the array $$$a$$$.\n\nIt is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$1000$$$.",
+        "output_format": "For each test case, output the maximum score that you can get.",
+        "interaction_format": null,
+        "note": "In the first test case, there is no other way to rearrange $$$a$$$. So, $$$b = [69]$$$ and $$$c = [69]$$$. The only possible score is $$$69 - 69 = 0$$$.\n\nIn the second test case, you can rearrange $$$a$$$ as $$$[7, 5, 6]$$$. Here, $$$b = [7, 5, 5]$$$ and $$$c = [7, 7, 7]$$$. The score in this case is $$$(7 - 7) + (7 - 5) + (7 - 5) = 4$$$. It can be shown this is the maximum possible score.",
+        "examples": [
+            {
+                "input": "3\n1\n69\n3\n7 6 5\n5\n1 1 1 2 2",
+                "output": "0\n4\n4"
+            }
+        ],
+        "editorial": null,
+        "rating": 800,
+        "tags": [
+            "constructive algorithms",
+            "greedy",
+            "math",
+            "sortings"
+        ],
+        "testset_size": 5,
+        "official_tests": [
+            {
+                "input": "3\r\n1\r\n69\r\n3\r\n7 6 5\r\n5\r\n1 1 1 2 2\r\n",
+                "output": "0\r\n4\r\n4\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2030/B",
+        "aliases": null,
+        "contest_id": "2030",
+        "contest_name": "Codeforces Round 979 (Div. 2)",
+        "contest_type": "CF",
+        "contest_start": 1729346700,
+        "contest_start_year": 2024,
+        "index": "B",
+        "time_limit": 1.5,
+        "memory_limit": 256.0,
+        "title": "Minimise Oneness",
+        "description": "For an arbitrary binary string $$$t$$$$$$^{\\text{∗}}$$$, let $$$f(t)$$$ be the number of non-empty subsequences$$$^{\\text{†}}$$$ of $$$t$$$ that contain only $$$\\mathtt{0}$$$, and let $$$g(t)$$$ be the number of non-empty subsequences of $$$t$$$ that contain at least one $$$\\mathtt{1}$$$.\n\nNote that for $$$f(t)$$$ and for $$$g(t)$$$, each subsequence is counted as many times as it appears in $$$t$$$. E.g., $$$f(\\mathtt{000}) = 7, g(\\mathtt{100}) = 4$$$.\n\nWe define the oneness of the binary string $$$t$$$ to be $$$|f(t)-g(t)|$$$, where for an arbitrary integer $$$z$$$, $$$|z|$$$ represents the absolute value of $$$z$$$.\n\nYou are given a positive integer $$$n$$$. Find a binary string $$$s$$$ of length $$$n$$$ such that its oneness is as small as possible. If there are multiple strings, you can print any of them.",
+        "input_format": "The first line contains an integer $$$t$$$ ($$$1 \\leq t \\leq 10^4$$$) — the number of test cases.\n\nThe only line of each test case contains an integer $$$n$$$ ($$$1 \\leq n \\leq 2\\cdot10^5$$$) — the length of $$$s$$$.\n\nIt is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2\\cdot10^5$$$.",
+        "output_format": "For each test case, output $$$s$$$ on a new line. If multiple answers exist, output any.",
+        "interaction_format": null,
+        "note": "In the first test case, for the example output, $$$f(t)=1$$$ because there is one subsequence that contains only $$$\\mathtt{0}$$$ ($$$\\mathtt{0}$$$), and $$$g(t)=0$$$ because there are no subsequences that contain at least one $$$1$$$. The oneness is $$$|1-0|=1$$$. The output $$$\\mathtt{1}$$$ is correct as well because its oneness is $$$|0-1|=1$$$.\n\nFor the example output of the second test case, $$$f(t)=1$$$ because there is one non-empty subsequence that contains only $$$\\mathtt{0}$$$, and $$$g(t)=2$$$ because there are two non-empty subsequences that contain at least one $$$\\mathtt{1}$$$ ($$$\\mathtt{01}$$$ and $$$\\mathtt{1}$$$). The oneness is thus $$$|1-2|=1$$$. It can be shown that $$$1$$$ is the minimum possible value of its oneness over all possible binary strings of size $$$2$$$.",
+        "examples": [
+            {
+                "input": "3\n1\n2\n3",
+                "output": "0\n01\n010"
+            }
+        ],
+        "editorial": null,
+        "rating": 800,
+        "tags": [
+            "combinatorics",
+            "constructive algorithms",
+            "games",
+            "math"
+        ],
+        "testset_size": 5,
+        "official_tests": [
+            {
+                "input": "3\r\n1\r\n2\r\n3\r\n",
+                "output": "1\r\n10\r\n100\r\n"
+            },
+            {
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1\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": "import sys\n\ndef main(input_path, output_path, submission_path):\n    with open(input_path) as f:\n        input_lines = f.readlines()\n    with open(submission_path) as f:\n        submission_lines = [line.strip() for line in f.readlines()]\n\n    t = int(input_lines[0])\n    cases = input_lines[1:t+1]\n\n    if len(submission_lines) != t:\n        print(0)\n        return\n\n    for i in range(t):\n        n = int(cases[i].strip())\n        s = submission_lines[i]\n\n        if len(s) != n:\n            print(0)\n            return\n\n        if any(c not in {'0', '1'} for c in s):\n            print(0)\n            return\n\n        c0 = s.count('0')\n\n        if n == 1:\n            if c0 not in {0, 1}:\n                print(0)\n                return\n        else:\n            if c0 != n - 1:\n                print(0)\n                return\n\n    print(1)\n\nif __name__ == \"__main__\":\n    main(sys.argv[1], sys.argv[2], sys.argv[3])",
+        "executable": true
+    },
+    {
+        "id": "2001/A",
+        "aliases": null,
+        "contest_id": "2001",
+        "contest_name": "Codeforces Round 967 (Div. 2)",
+        "contest_type": "CF",
+        "contest_start": 1724164500,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Make All Equal",
+        "description": "You are given a cyclic array $$$a_1, a_2, \\ldots, a_n$$$.\n\nYou can perform the following operation on $$$a$$$ at most $$$n - 1$$$ times:\n\n- Let $$$m$$$ be the current size of $$$a$$$, you can choose any two adjacent elements where the previous one is no greater than the latter one (In particular, $$$a_m$$$ and $$$a_1$$$ are adjacent and $$$a_m$$$ is the previous one), and delete exactly one of them. In other words, choose an integer $$$i$$$ ($$$1 \\leq i \\leq m$$$) where $$$a_i \\leq a_{(i \\bmod m) + 1}$$$ holds, and delete exactly one of $$$a_i$$$ or $$$a_{(i \\bmod m) + 1}$$$ from $$$a$$$.\n\nYour goal is to find the minimum number of operations needed to make all elements in $$$a$$$ equal.",
+        "input_format": "Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \\le t \\le 500$$$). The description of the test cases follows.\n\nThe first line of each test case contains a single integer $$$n$$$ ($$$1 \\le n \\le 100$$$) — the length of the array $$$a$$$.\n\nThe second line of each test case contains $$$n$$$ integers $$$a_1, a_2, \\ldots, a_n$$$ ($$$1 \\le a_i \\le n$$$) — the elements of array $$$a$$$.",
+        "output_format": "For each test case, output a single line containing an integer: the minimum number of operations needed to make all elements in $$$a$$$ equal.",
+        "interaction_format": null,
+        "note": "In the first test case, there is only one element in $$$a$$$, so we can't do any operation.\n\nIn the second test case, we can perform the following operations to make all elements in $$$a$$$ equal:\n\n- choose $$$i = 2$$$, delete $$$a_3$$$, then $$$a$$$ would become $$$[1, 2]$$$.\n- choose $$$i = 1$$$, delete $$$a_1$$$, then $$$a$$$ would become $$$[2]$$$.\n\nIt can be proven that we can't make all elements in $$$a$$$ equal using fewer than $$$2$$$ operations, so the answer is $$$2$$$.",
+        "examples": [
+            {
+                "input": "7\n1\n1\n3\n1 2 3\n3\n1 2 2\n5\n5 4 3 2 1\n6\n1 1 2 2 3 3\n8\n8 7 6 3 8 7 6 3\n6\n1 1 4 5 1 4",
+                "output": "0\n2\n1\n4\n4\n6\n3"
+            }
+        ],
+        "editorial": "Let $$$x$$$ be one of the most frequent elements in $$$a$$$, then the answer must be at least $$$(n - \\text{frequency of } x)$$$, and this lower bound is always achievable by keep erasing a non-$$$x$$$ element from $$$a$$$, and we can prove it's always possible to do so.\nproof:\nLet $$$y$$$ denote any value that is not $$$x$$$, then $$$a$$$ would have the form $$$x \\ldots x, y \\ldots y, x \\ldots x, \\ldots$$$. Let's focus on the subarray $$$x, y \\ldots y, x$$$. In this subarray, there must exist adjacent elements that satisfy $$$a_i \\leq a_{(i \\bmod m) + 1}$$$, otherwise we would have $$$x > y > \\ldots > y > x$$$ implying $$$x > x$$$ which leads to a contradiction.",
+        "rating": 800,
+        "tags": [
+            "greedy",
+            "implementation"
+        ],
+        "testset_size": 5,
+        "official_tests": [
+            {
+                "input": "7\r\n1\r\n1\r\n3\r\n1 2 3\r\n3\r\n1 2 2\r\n5\r\n5 4 3 2 1\r\n6\r\n1 1 2 2 3 3\r\n8\r\n8 7 6 3 8 7 6 3\r\n6\r\n1 1 4 5 1 4\r\n",
+                "output": "0\r\n2\r\n1\r\n4\r\n4\r\n6\r\n3\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2001/B",
+        "aliases": null,
+        "contest_id": "2001",
+        "contest_name": "Codeforces Round 967 (Div. 2)",
+        "contest_type": "CF",
+        "contest_start": 1724164500,
+        "contest_start_year": 2024,
+        "index": "B",
+        "time_limit": 1.5,
+        "memory_limit": 256.0,
+        "title": "Generate Permutation",
+        "description": "There is an integer sequence $$$a$$$ of length $$$n$$$, where each element is initially $$$-1$$$.\n\nMisuki has two typewriters where the first one writes letters from left to right, with a pointer initially pointing to $$$1$$$, and another writes letters from right to left with a pointer initially pointing to $$$n$$$.\n\nMisuki would choose one of the typewriters and use it to perform the following operations until $$$a$$$ becomes a permutation of $$$[1, 2, \\ldots, n]$$$\n\n- write number: write the minimum positive integer that isn't present in the array $$$a$$$ to the element $$$a_i$$$, $$$i$$$ is the position where the pointer points at. Such operation can be performed only when $$$a_i = -1$$$.\n- carriage return: return the pointer to its initial position (i.e. $$$1$$$ for the first typewriter, $$$n$$$ for the second)\n- move pointer: move the pointer to the next position, let $$$i$$$ be the position the pointer points at before this operation, if Misuki is using the first typewriter, $$$i := i + 1$$$ would happen, and $$$i := i - 1$$$ otherwise. Such operation can be performed only if after the operation, $$$1 \\le i \\le n$$$ holds.\n\nYour task is to construct any permutation $$$p$$$ of length $$$n$$$, such that the minimum number of carriage return operations needed to make $$$a = p$$$ is the same no matter which typewriter Misuki is using.",
+        "input_format": "Each test contains multiple test cases. The first line of input contains a single integer $$$t$$$ ($$$1 \\le t \\le 500$$$) — the number of test cases. The description of the test cases follows.\n\nThe first line of each test case contains a single integer $$$n$$$ ($$$1 \\le n \\le 2 \\cdot 10^5$$$) — the length of the permutation.\n\nIt is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \\cdot 10^5$$$.",
+        "output_format": "For each test case, output a line of $$$n$$$ integers, representing the permutation $$$p$$$ of length $$$n$$$ such that the minimum number of carriage return operations needed to make $$$a = p$$$ is the same no matter which typewriter Misuki is using, or $$$-1$$$ if it is impossible to do so.\n\nIf there are multiple valid permutations, you can output any of them.",
+        "interaction_format": null,
+        "note": "In the first testcase, it's possible to make $$$a = p = [1]$$$ using $$$0$$$ carriage return operations.\n\nIn the second testcase, it is possible to make $$$a = p = [1, 2]$$$ with the minimal number of carriage returns as follows:\n\nIf Misuki is using the first typewriter:\n\n- Write number: write $$$1$$$ to $$$a_1$$$, $$$a$$$ becomes $$$[1, -1]$$$\n- Move pointer: move the pointer to the next position. (i.e. $$$2$$$)\n- Write number: write $$$2$$$ to $$$a_2$$$, $$$a$$$ becomes $$$[1, 2]$$$\n\nIf Misuki is using the second typewriter:\n\n- Move pointer: move the pointer to the next position. (i.e. $$$1$$$)\n- Write number: write $$$1$$$ to $$$a_1$$$, $$$a$$$ becomes $$$[1, -1]$$$\n- Carriage return: return the pointer to $$$2$$$.\n- Write number: write $$$2$$$ to $$$a_2$$$, $$$a$$$ becomes $$$[1, 2]$$$\n\nIt can be proven that the minimum number of carriage returns needed to transform $$$a$$$ into $$$p$$$ when using the first typewriter is $$$0$$$ and it is $$$1$$$ when using the second one, so this permutation is not valid.\n\nSimilarly, $$$p = [2, 1]$$$ is also not valid, so there is no solution for $$$n = 2$$$.\n\nIn the third testcase, it is possibile to make $$$a = p = [3, 1, 2]$$$ with $$$1$$$ carriage return with both the first and the second typewriter. It can be proven that, for both typewriters, it is impossible to write $$$p$$$ with $$$0$$$ carriage returns.\n\nWith the first typewriter it is possible to:\n\n- Move pointer: move the pointer to the next position. (i.e. $$$2$$$)\n- Write number: write $$$1$$$ to $$$a_2$$$, $$$a$$$ becomes $$$[-1, 1, -1]$$$\n- Move pointer: move the pointer to the next position. (i.e. $$$3$$$)\n- Write number: write $$$2$$$ to $$$a_3$$$, $$$a$$$ becomes $$$[-1, 1, 2]$$$\n- Carriage return: return the pointer to $$$1$$$.\n- Write number: write $$$3$$$ to $$$a_1$$$, $$$a$$$ becomes $$$[3,1,2]$$$\n\nWith the second typewriter it is possible to:\n\n- Move pointer: move the pointer to the next position. (i.e. $$$2$$$)\n- Write number: write $$$1$$$ to $$$a_2$$$, $$$a$$$ becomes $$$[-1, 1, -1]$$$\n- Carriage return: return the pointer to $$$3$$$.\n- Write number: write $$$2$$$ to $$$a_3$$$, $$$a$$$ becomes $$$[-1, 1, 2]$$$\n- Move pointer: move the pointer to the next position. (i.e. $$$2$$$)\n- Move pointer: move the pointer to the next position. (i.e. $$$1$$$)\n- Write number: write $$$3$$$ to $$$a_1$$$, $$$a$$$ becomes $$$[3, 1, 2]$$$",
+        "examples": [
+            {
+                "input": "3\n1\n2\n3",
+                "output": "1\n-1\n3 1 2"
+            }
+        ],
+        "editorial": "Consider the cost function:\nLet $$$c_{cw}, c_{ccw}$$$ denote the minimum number of operations needed to produce $$$p_n$$$ if the roulette is rotating clockwise/counter-clockwise. Let's consider how to calculate them.\nFor $$$c_{cw}$$$, obviously we need to do the operation at least once, and we need to do one extra operation whenever the position of value $$$x + 1$$$ is before $$$x$$$, for all $$$x \\in [1, n - 1]$$$, so $$$c_{cw} = 1 + (\\text{#}x \\in [1, n - 1] \\text{ such that } \\mathrm{pos}_x > \\mathrm{pos}_{x + 1})$$$.\nSimilarly, $$$c_{ccw} = 1 + (\\text{#}x \\in [1, n - 1] \\text{ such that } \\mathrm{pos}_x < \\mathrm{pos}_{x + 1})$$$.\nSince for $$$x \\in [1, n - 1]$$$, either $$$\\mathrm{pos}_x < \\mathrm{pos}_{x + 1}$$$ or $$$\\mathrm{pos}_x > \\mathrm{pos}_{x + 1}$$$ would be true, $$$c_{cw} + c_{ccw} = n + 1$$$, which is a constant, so $$$c_{cw} = c_{ccw} \\leftrightarrow c_{cw} = \\frac{n + 1}{2}$$$, which only has solution when $$$n$$$ is odd, and such $$$p_n$$$ can be constructed easily (for example, $$$\\{N - 1, N, N - 3, N - 2, ..., 4, 5, 2, 3, 1\\}$$$).",
+        "rating": 800,
+        "tags": [
+            "constructive algorithms"
+        ],
+        "testset_size": 10,
+        "official_tests": [
+            {
+                "input": "3\r\n1\r\n2\r\n3\r\n",
+                "output": "1\r\n-1\r\n1 3 2\r\n"
+            },
+            {
+                "input": "3\r\n13\r\n123456\r\n76531\r\n",
+                "output": "1 3 5 7 9 11 13 12 10 8 6 4 2 \n-1\n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 101 103 105 107 109 111 113 115 117 119 121 123 125 127 129 131 133 135 137 139 141 143 145 147 149 151 153 155 157 159 161 163 165 167 169 171 173 175 177 179 181 183 185 187 189 191 193 195 197 199 201 203 205 207 209 211 213 215 217 219 221 223 225 227 229 231 233 235 237 239 241 243 245 247 249 251 253 255 257 259 261 263 265 267 269 271 273 275 277 279 281 283 285 287 289 291 293 295 297 299 301 303 305 307 309 311 313 315 317 319 321 323 325 327 329 331 333 335 337 339 341 343 345 347 349 351 353 355 357 359 361 363 365 367 369 371 373 375 377 379 381 383 385 387 389 391 393 395 397 399 401 403 405 407 409 411 413 415 417 419 421 423 425 427 429 431 433 435 437 439 441 443 445 447 449 451 453 455 457 459 461 463 465 467 469 471 473 475 477 479 481 483 485 487 489 491 493 495 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1320 1318 1316 1314 1312 1310 1308 1306 1304 1302 1300 1298 1296 1294 1292 1290 1288 1286 1284 1282 1280 1278 1276 1274 1272 1270 1268 1266 1264 1262 1260 1258 1256 1254 1252 1250 1248 1246 1244 1242 1240 1238 1236 1234 1232 1230 1228 1226 1224 1222 1220 1218 1216 1214 1212 1210 1208 1206 1204 1202 1200 1198 1196 1194 1192 1190 1188 1186 1184 1182 1180 1178 1176 1174 1172 1170 1168 1166 1164 1162 1160 1158 1156 1154 1152 1150 1148 1146 1144 1142 1140 1138 1136 1134 1132 1130 1128 1126 1124 1122 1120 1118 1116 1114 1112 1110 1108 1106 1104 1102 1100 1098 1096 1094 1092 1090 1088 1086 1084 1082 1080 1078 1076 1074 1072 1070 1068 1066 1064 1062 1060 1058 1056 1054 1052 1050 1048 1046 1044 1042 1040 1038 1036 1034 1032 1030 1028 1026 1024 1022 1020 1018 1016 1014 1012 1010 1008 1006 1004 1002 1000 998 996 994 992 990 988 986 984 982 980 978 976 974 972 970 968 966 964 962 960 958 956 954 952 950 948 946 944 942 940 938 936 934 932 930 928 926 924 922 920 918 916 914 912 910 908 906 904 902 900 898 896 894 892 890 888 886 884 882 880 878 876 874 872 870 868 866 864 862 860 858 856 854 852 850 848 846 844 842 840 838 836 834 832 830 828 826 824 822 820 818 816 814 812 810 808 806 804 802 800 798 796 794 792 790 788 786 784 782 780 778 776 774 772 770 768 766 764 762 760 758 756 754 752 750 748 746 744 742 740 738 736 734 732 730 728 726 724 722 720 718 716 714 712 710 708 706 704 702 700 698 696 694 692 690 688 686 684 682 680 678 676 674 672 670 668 666 664 662 660 658 656 654 652 650 648 646 644 642 640 638 636 634 632 630 628 626 624 622 620 618 616 614 612 610 608 606 604 602 600 598 596 594 592 590 588 586 584 582 580 578 576 574 572 570 568 566 564 562 560 558 556 554 552 550 548 546 544 542 540 538 536 534 532 530 528 526 524 522 520 518 516 514 512 510 508 506 504 502 500 498 496 494 492 490 488 486 484 482 480 478 476 474 472 470 468 466 464 462 460 458 456 454 452 450 448 446 444 442 440 438 436 434 432 430 428 426 424 422 420 418 416 414 412 410 408 406 404 402 400 398 396 394 392 390 388 386 384 382 380 378 376 374 372 370 368 366 364 362 360 358 356 354 352 350 348 346 344 342 340 338 336 334 332 330 328 326 324 322 320 318 316 314 312 310 308 306 304 302 300 298 296 294 292 290 288 286 284 282 280 278 276 274 272 270 268 266 264 262 260 258 256 254 252 250 248 246 244 242 240 238 236 234 232 230 228 226 224 222 220 218 216 214 212 210 208 206 204 202 200 198 196 194 192 190 188 186 184 182 180 178 176 174 172 170 168 166 164 162 160 158 156 154 152 150 148 146 144 142 140 138 136 134 132 130 128 126 124 122 120 118 116 114 112 110 108 106 104 102 100 98 96 94 92 90 88 86 84 82 80 78 76 74 72 70 68 66 64 62 60 58 56 54 52 50 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 \n"
+            },
+            {
+                "input": "1\r\n200000\r\n",
+                "output": "-1\r\n"
+            },
+            {
+                "input": "1\r\n199999\r\n",
+                "output": "1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 101 103 105 107 109 111 113 115 117 119 121 123 125 127 129 131 133 135 137 139 141 143 145 147 149 151 153 155 157 159 161 163 165 167 169 171 173 175 177 179 181 183 185 187 189 191 193 195 197 199 201 203 205 207 209 211 213 215 217 219 221 223 225 227 229 231 233 235 237 239 241 243 245 247 249 251 253 255 257 259 261 263 265 267 269 271 273 275 277 279 281 283 285 287 289 291 293 295 297 299 301 303 305 307 309 311 313 315 317 319 321 323 325 327 329 331 333 335 337 339 341 343 345 347 349 351 353 355 357 359 361 363 365 367 369 371 373 375 377 379 381 383 385 387 389 391 393 395 397 399 401 403 405 407 409 411 413 415 417 419 421 423 425 427 429 431 433 435 437 439 441 443 445 447 449 451 453 455 457 459 461 463 465 467 469 471 473 475 477 479 481 483 485 487 489 491 493 495 497 499 501 503 505 507 509 511 513 515 517 519 521 523 525 527 529 531 533 535 537 539 541 543 545 547 549 551 553 555 557 559 561 563 565 567 569 571 573 575 577 579 581 583 585 587 589 591 593 595 597 599 601 603 605 607 609 611 613 615 617 619 621 623 625 627 629 631 633 635 637 639 641 643 645 647 649 651 653 655 657 659 661 663 665 667 669 671 673 675 677 679 681 683 685 687 689 691 693 695 697 699 701 703 705 707 709 711 713 715 717 719 721 723 725 727 729 731 733 735 737 739 741 743 745 747 749 751 753 755 757 759 761 763 765 767 769 771 773 775 777 779 781 783 785 787 789 791 793 795 797 799 801 803 805 807 809 811 813 815 817 819 821 823 825 827 829 831 833 835 837 839 841 843 845 847 849 851 853 855 857 859 861 863 865 867 869 871 873 875 877 879 881 883 885 887 889 891 893 895 897 899 901 903 905 907 909 911 913 915 917 919 921 923 925 927 929 931 933 935 937 939 941 943 945 947 949 951 953 955 957 959 961 963 965 967 969 971 973 975 977 979 981 983 985 987 989 991 993 995 997 999 1001 1003 1005 1007 1009 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3011 3013 3015 3017 3019 3021 3023 3025 3027 3029 3031 3033 3035 3037 3039 3041 3043 3045 3047 3049 3051 3053 3055 3057 3059 3061 3063 3065 3067 3069 3071 3073 3075 3077 3079 3081 3083 3085 3087 3089 3091 3093 3095 3097 3099 3101 3103 3105 3107 3109 3111 3113 3115 3117 3119 3121 3123 3125 3127 3129 3131 3133 3135 3137 3139 3141 3143 3145 3147 3149 3151 3153 3155 3157 3159 3161 3163 3165 3167 3169 3171 3173 3175 3177 3179 3181 3183 3185 3187 3189 3191 3193 3195 3197 3199 3201 3203 3205 3207 3209 3211 3213 3215 3217 3219 3221 3223 3225 3227 3229 3231 3233 3235 3237 3239 3241 3243 3245 3247 3249 3251 3253 3255 3257 3259 3261 3263 3265 3267 3269 3271 3273 3275 3277 3279 3281 3283 3285 3287 3289 3291 3293 3295 3297 3299 3301 3303 3305 3307 3309 3311 3313 3315 3317 3319 3321 3323 3325 3327 3329 3331 3333 3335 3337 3339 3341 3343 3345 3347 3349 3351 3353 3355 3357 3359 3361 3363 3365 3367 3369 3371 3373 3375 3377 3379 3381 3383 3385 3387 3389 3391 3393 3395 3397 3399 3401 3403 3405 3407 3409 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3976 3974 3972 3970 3968 3966 3964 3962 3960 3958 3956 3954 3952 3950 3948 3946 3944 3942 3940 3938 3936 3934 3932 3930 3928 3926 3924 3922 3920 3918 3916 3914 3912 3910 3908 3906 3904 3902 3900 3898 3896 3894 3892 3890 3888 3886 3884 3882 3880 3878 3876 3874 3872 3870 3868 3866 3864 3862 3860 3858 3856 3854 3852 3850 3848 3846 3844 3842 3840 3838 3836 3834 3832 3830 3828 3826 3824 3822 3820 3818 3816 3814 3812 3810 3808 3806 3804 3802 3800 3798 3796 3794 3792 3790 3788 3786 3784 3782 3780 3778 3776 3774 3772 3770 3768 3766 3764 3762 3760 3758 3756 3754 3752 3750 3748 3746 3744 3742 3740 3738 3736 3734 3732 3730 3728 3726 3724 3722 3720 3718 3716 3714 3712 3710 3708 3706 3704 3702 3700 3698 3696 3694 3692 3690 3688 3686 3684 3682 3680 3678 3676 3674 3672 3670 3668 3666 3664 3662 3660 3658 3656 3654 3652 3650 3648 3646 3644 3642 3640 3638 3636 3634 3632 3630 3628 3626 3624 3622 3620 3618 3616 3614 3612 3610 3608 3606 3604 3602 3600 3598 3596 3594 3592 3590 3588 3586 3584 3582 3580 3578 3576 3574 3572 3570 3568 3566 3564 3562 3560 3558 3556 3554 3552 3550 3548 3546 3544 3542 3540 3538 3536 3534 3532 3530 3528 3526 3524 3522 3520 3518 3516 3514 3512 3510 3508 3506 3504 3502 3500 3498 3496 3494 3492 3490 3488 3486 3484 3482 3480 3478 3476 3474 3472 3470 3468 3466 3464 3462 3460 3458 3456 3454 3452 3450 3448 3446 3444 3442 3440 3438 3436 3434 3432 3430 3428 3426 3424 3422 3420 3418 3416 3414 3412 3410 3408 3406 3404 3402 3400 3398 3396 3394 3392 3390 3388 3386 3384 3382 3380 3378 3376 3374 3372 3370 3368 3366 3364 3362 3360 3358 3356 3354 3352 3350 3348 3346 3344 3342 3340 3338 3336 3334 3332 3330 3328 3326 3324 3322 3320 3318 3316 3314 3312 3310 3308 3306 3304 3302 3300 3298 3296 3294 3292 3290 3288 3286 3284 3282 3280 3278 3276 3274 3272 3270 3268 3266 3264 3262 3260 3258 3256 3254 3252 3250 3248 3246 3244 3242 3240 3238 3236 3234 3232 3230 3228 3226 3224 3222 3220 3218 3216 3214 3212 3210 3208 3206 3204 3202 3200 3198 3196 3194 3192 3190 3188 3186 3184 3182 3180 3178 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2776 2774 2772 2770 2768 2766 2764 2762 2760 2758 2756 2754 2752 2750 2748 2746 2744 2742 2740 2738 2736 2734 2732 2730 2728 2726 2724 2722 2720 2718 2716 2714 2712 2710 2708 2706 2704 2702 2700 2698 2696 2694 2692 2690 2688 2686 2684 2682 2680 2678 2676 2674 2672 2670 2668 2666 2664 2662 2660 2658 2656 2654 2652 2650 2648 2646 2644 2642 2640 2638 2636 2634 2632 2630 2628 2626 2624 2622 2620 2618 2616 2614 2612 2610 2608 2606 2604 2602 2600 2598 2596 2594 2592 2590 2588 2586 2584 2582 2580 2578 2576 2574 2572 2570 2568 2566 2564 2562 2560 2558 2556 2554 2552 2550 2548 2546 2544 2542 2540 2538 2536 2534 2532 2530 2528 2526 2524 2522 2520 2518 2516 2514 2512 2510 2508 2506 2504 2502 2500 2498 2496 2494 2492 2490 2488 2486 2484 2482 2480 2478 2476 2474 2472 2470 2468 2466 2464 2462 2460 2458 2456 2454 2452 2450 2448 2446 2444 2442 2440 2438 2436 2434 2432 2430 2428 2426 2424 2422 2420 2418 2416 2414 2412 2410 2408 2406 2404 2402 2400 2398 2396 2394 2392 2390 2388 2386 2384 2382 2380 2378 2376 2374 2372 2370 2368 2366 2364 2362 2360 2358 2356 2354 2352 2350 2348 2346 2344 2342 2340 2338 2336 2334 2332 2330 2328 2326 2324 2322 2320 2318 2316 2314 2312 2310 2308 2306 2304 2302 2300 2298 2296 2294 2292 2290 2288 2286 2284 2282 2280 2278 2276 2274 2272 2270 2268 2266 2264 2262 2260 2258 2256 2254 2252 2250 2248 2246 2244 2242 2240 2238 2236 2234 2232 2230 2228 2226 2224 2222 2220 2218 2216 2214 2212 2210 2208 2206 2204 2202 2200 2198 2196 2194 2192 2190 2188 2186 2184 2182 2180 2178 2176 2174 2172 2170 2168 2166 2164 2162 2160 2158 2156 2154 2152 2150 2148 2146 2144 2142 2140 2138 2136 2134 2132 2130 2128 2126 2124 2122 2120 2118 2116 2114 2112 2110 2108 2106 2104 2102 2100 2098 2096 2094 2092 2090 2088 2086 2084 2082 2080 2078 2076 2074 2072 2070 2068 2066 2064 2062 2060 2058 2056 2054 2052 2050 2048 2046 2044 2042 2040 2038 2036 2034 2032 2030 2028 2026 2024 2022 2020 2018 2016 2014 2012 2010 2008 2006 2004 2002 2000 1998 1996 1994 1992 1990 1988 1986 1984 1982 1980 1978 1976 1974 1972 1970 1968 1966 1964 1962 1960 1958 1956 1954 1952 1950 1948 1946 1944 1942 1940 1938 1936 1934 1932 1930 1928 1926 1924 1922 1920 1918 1916 1914 1912 1910 1908 1906 1904 1902 1900 1898 1896 1894 1892 1890 1888 1886 1884 1882 1880 1878 1876 1874 1872 1870 1868 1866 1864 1862 1860 1858 1856 1854 1852 1850 1848 1846 1844 1842 1840 1838 1836 1834 1832 1830 1828 1826 1824 1822 1820 1818 1816 1814 1812 1810 1808 1806 1804 1802 1800 1798 1796 1794 1792 1790 1788 1786 1784 1782 1780 1778 1776 1774 1772 1770 1768 1766 1764 1762 1760 1758 1756 1754 1752 1750 1748 1746 1744 1742 1740 1738 1736 1734 1732 1730 1728 1726 1724 1722 1720 1718 1716 1714 1712 1710 1708 1706 1704 1702 1700 1698 1696 1694 1692 1690 1688 1686 1684 1682 1680 1678 1676 1674 1672 1670 1668 1666 1664 1662 1660 1658 1656 1654 1652 1650 1648 1646 1644 1642 1640 1638 1636 1634 1632 1630 1628 1626 1624 1622 1620 1618 1616 1614 1612 1610 1608 1606 1604 1602 1600 1598 1596 1594 1592 1590 1588 1586 1584 1582 1580 1578 1576 1574 1572 1570 1568 1566 1564 1562 1560 1558 1556 1554 1552 1550 1548 1546 1544 1542 1540 1538 1536 1534 1532 1530 1528 1526 1524 1522 1520 1518 1516 1514 1512 1510 1508 1506 1504 1502 1500 1498 1496 1494 1492 1490 1488 1486 1484 1482 1480 1478 1476 1474 1472 1470 1468 1466 1464 1462 1460 1458 1456 1454 1452 1450 1448 1446 1444 1442 1440 1438 1436 1434 1432 1430 1428 1426 1424 1422 1420 1418 1416 1414 1412 1410 1408 1406 1404 1402 1400 1398 1396 1394 1392 1390 1388 1386 1384 1382 1380 1378 1376 1374 1372 1370 1368 1366 1364 1362 1360 1358 1356 1354 1352 1350 1348 1346 1344 1342 1340 1338 1336 1334 1332 1330 1328 1326 1324 1322 1320 1318 1316 1314 1312 1310 1308 1306 1304 1302 1300 1298 1296 1294 1292 1290 1288 1286 1284 1282 1280 1278 1276 1274 1272 1270 1268 1266 1264 1262 1260 1258 1256 1254 1252 1250 1248 1246 1244 1242 1240 1238 1236 1234 1232 1230 1228 1226 1224 1222 1220 1218 1216 1214 1212 1210 1208 1206 1204 1202 1200 1198 1196 1194 1192 1190 1188 1186 1184 1182 1180 1178 1176 1174 1172 1170 1168 1166 1164 1162 1160 1158 1156 1154 1152 1150 1148 1146 1144 1142 1140 1138 1136 1134 1132 1130 1128 1126 1124 1122 1120 1118 1116 1114 1112 1110 1108 1106 1104 1102 1100 1098 1096 1094 1092 1090 1088 1086 1084 1082 1080 1078 1076 1074 1072 1070 1068 1066 1064 1062 1060 1058 1056 1054 1052 1050 1048 1046 1044 1042 1040 1038 1036 1034 1032 1030 1028 1026 1024 1022 1020 1018 1016 1014 1012 1010 1008 1006 1004 1002 1000 998 996 994 992 990 988 986 984 982 980 978 976 974 972 970 968 966 964 962 960 958 956 954 952 950 948 946 944 942 940 938 936 934 932 930 928 926 924 922 920 918 916 914 912 910 908 906 904 902 900 898 896 894 892 890 888 886 884 882 880 878 876 874 872 870 868 866 864 862 860 858 856 854 852 850 848 846 844 842 840 838 836 834 832 830 828 826 824 822 820 818 816 814 812 810 808 806 804 802 800 798 796 794 792 790 788 786 784 782 780 778 776 774 772 770 768 766 764 762 760 758 756 754 752 750 748 746 744 742 740 738 736 734 732 730 728 726 724 722 720 718 716 714 712 710 708 706 704 702 700 698 696 694 692 690 688 686 684 682 680 678 676 674 672 670 668 666 664 662 660 658 656 654 652 650 648 646 644 642 640 638 636 634 632 630 628 626 624 622 620 618 616 614 612 610 608 606 604 602 600 598 596 594 592 590 588 586 584 582 580 578 576 574 572 570 568 566 564 562 560 558 556 554 552 550 548 546 544 542 540 538 536 534 532 530 528 526 524 522 520 518 516 514 512 510 508 506 504 502 500 498 496 494 492 490 488 486 484 482 480 478 476 474 472 470 468 466 464 462 460 458 456 454 452 450 448 446 444 442 440 438 436 434 432 430 428 426 424 422 420 418 416 414 412 410 408 406 404 402 400 398 396 394 392 390 388 386 384 382 380 378 376 374 372 370 368 366 364 362 360 358 356 354 352 350 348 346 344 342 340 338 336 334 332 330 328 326 324 322 320 318 316 314 312 310 308 306 304 302 300 298 296 294 292 290 288 286 284 282 280 278 276 274 272 270 268 266 264 262 260 258 256 254 252 250 248 246 244 242 240 238 236 234 232 230 228 226 224 222 220 218 216 214 212 210 208 206 204 202 200 198 196 194 192 190 188 186 184 182 180 178 176 174 172 170 168 166 164 162 160 158 156 154 152 150 148 146 144 142 140 138 136 134 132 130 128 126 124 122 120 118 116 114 112 110 108 106 104 102 100 98 96 94 92 90 88 86 84 82 80 78 76 74 72 70 68 66 64 62 60 58 56 54 52 50 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 \n"
+            },
+            {
+                "input": "10\r\n19407\r\n9640\r\n4540\r\n13445\r\n19574\r\n2723\r\n2489\r\n6331\r\n12036\r\n7285\r\n",
+                "output": "1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 101 103 105 107 109 111 113 115 117 119 121 123 125 127 129 131 133 135 137 139 141 143 145 147 149 151 153 155 157 159 161 163 165 167 169 171 173 175 177 179 181 183 185 187 189 191 193 195 197 199 201 203 205 207 209 211 213 215 217 219 221 223 225 227 229 231 233 235 237 239 241 243 245 247 249 251 253 255 257 259 261 263 265 267 269 271 273 275 277 279 281 283 285 287 289 291 293 295 297 299 301 303 305 307 309 311 313 315 317 319 321 323 325 327 329 331 333 335 337 339 341 343 345 347 349 351 353 355 357 359 361 363 365 367 369 371 373 375 377 379 381 383 385 387 389 391 393 395 397 399 401 403 405 407 409 411 413 415 417 419 421 423 425 427 429 431 433 435 437 439 441 443 445 447 449 451 453 455 457 459 461 463 465 467 469 471 473 475 477 479 481 483 485 487 489 491 493 495 497 499 501 503 505 507 509 511 513 515 517 519 521 523 525 527 529 531 533 535 537 539 541 543 545 547 549 551 553 555 557 559 561 563 565 567 569 571 573 575 577 579 581 583 585 587 589 591 593 595 597 599 601 603 605 607 609 611 613 615 617 619 621 623 625 627 629 631 633 635 637 639 641 643 645 647 649 651 653 655 657 659 661 663 665 667 669 671 673 675 677 679 681 683 685 687 689 691 693 695 697 699 701 703 705 707 709 711 713 715 717 719 721 723 725 727 729 731 733 735 737 739 741 743 745 747 749 751 753 755 757 759 761 763 765 767 769 771 773 775 777 779 781 783 785 787 789 791 793 795 797 799 801 803 805 807 809 811 813 815 817 819 821 823 825 827 829 831 833 835 837 839 841 843 845 847 849 851 853 855 857 859 861 863 865 867 869 871 873 875 877 879 881 883 885 887 889 891 893 895 897 899 901 903 905 907 909 911 913 915 917 919 921 923 925 927 929 931 933 935 937 939 941 943 945 947 949 951 953 955 957 959 961 963 965 967 969 971 973 975 977 979 981 983 985 987 989 991 993 995 997 999 1001 1003 1005 1007 1009 1011 1013 1015 1017 1019 1021 1023 1025 1027 1029 1031 1033 1035 1037 1039 1041 1043 1045 1047 1049 1051 1053 1055 1057 1059 1061 1063 1065 1067 1069 1071 1073 1075 1077 1079 1081 1083 1085 1087 1089 1091 1093 1095 1097 1099 1101 1103 1105 1107 1109 1111 1113 1115 1117 1119 1121 1123 1125 1127 1129 1131 1133 1135 1137 1139 1141 1143 1145 1147 1149 1151 1153 1155 1157 1159 1161 1163 1165 1167 1169 1171 1173 1175 1177 1179 1181 1183 1185 1187 1189 1191 1193 1195 1197 1199 1201 1203 1205 1207 1209 1211 1213 1215 1217 1219 1221 1223 1225 1227 1229 1231 1233 1235 1237 1239 1241 1243 1245 1247 1249 1251 1253 1255 1257 1259 1261 1263 1265 1267 1269 1271 1273 1275 1277 1279 1281 1283 1285 1287 1289 1291 1293 1295 1297 1299 1301 1303 1305 1307 1309 1311 1313 1315 1317 1319 1321 1323 1325 1327 1329 1331 1333 1335 1337 1339 1341 1343 1345 1347 1349 1351 1353 1355 1357 1359 1361 1363 1365 1367 1369 1371 1373 1375 1377 1379 1381 1383 1385 1387 1389 1391 1393 1395 1397 1399 1401 1403 1405 1407 1409 1411 1413 1415 1417 1419 1421 1423 1425 1427 1429 1431 1433 1435 1437 1439 1441 1443 1445 1447 1449 1451 1453 1455 1457 1459 1461 1463 1465 1467 1469 1471 1473 1475 1477 1479 1481 1483 1485 1487 1489 1491 1493 1495 1497 1499 1501 1503 1505 1507 1509 1511 1513 1515 1517 1519 1521 1523 1525 1527 1529 1531 1533 1535 1537 1539 1541 1543 1545 1547 1549 1551 1553 1555 1557 1559 1561 1563 1565 1567 1569 1571 1573 1575 1577 1579 1581 1583 1585 1587 1589 1591 1593 1595 1597 1599 1601 1603 1605 1607 1609 1611 1613 1615 1617 1619 1621 1623 1625 1627 1629 1631 1633 1635 1637 1639 1641 1643 1645 1647 1649 1651 1653 1655 1657 1659 1661 1663 1665 1667 1669 1671 1673 1675 1677 1679 1681 1683 1685 1687 1689 1691 1693 1695 1697 1699 1701 1703 1705 1707 1709 1711 1713 1715 1717 1719 1721 1723 1725 1727 1729 1731 1733 1735 1737 1739 1741 1743 1745 1747 1749 1751 1753 1755 1757 1759 1761 1763 1765 1767 1769 1771 1773 1775 1777 1779 1781 1783 1785 1787 1789 1791 1793 1795 1797 1799 1801 1803 1805 1807 1809 1811 1813 1815 1817 1819 1821 1823 1825 1827 1829 1831 1833 1835 1837 1839 1841 1843 1845 1847 1849 1851 1853 1855 1857 1859 1861 1863 1865 1867 1869 1871 1873 1875 1877 1879 1881 1883 1885 1887 1889 1891 1893 1895 1897 1899 1901 1903 1905 1907 1909 1911 1913 1915 1917 1919 1921 1923 1925 1927 1929 1931 1933 1935 1937 1939 1941 1943 1945 1947 1949 1951 1953 1955 1957 1959 1961 1963 1965 1967 1969 1971 1973 1975 1977 1979 1981 1983 1985 1987 1989 1991 1993 1995 1997 1999 2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 2021 2023 2025 2027 2029 2031 2033 2035 2037 2039 2041 2043 2045 2047 2049 2051 2053 2055 2057 2059 2061 2063 2065 2067 2069 2071 2073 2075 2077 2079 2081 2083 2085 2087 2089 2091 2093 2095 2097 2099 2101 2103 2105 2107 2109 2111 2113 2115 2117 2119 2121 2123 2125 2127 2129 2131 2133 2135 2137 2139 2141 2143 2145 2147 2149 2151 2153 2155 2157 2159 2161 2163 2165 2167 2169 2171 2173 2175 2177 2179 2181 2183 2185 2187 2189 2191 2193 2195 2197 2199 2201 2203 2205 2207 2209 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528 526 524 522 520 518 516 514 512 510 508 506 504 502 500 498 496 494 492 490 488 486 484 482 480 478 476 474 472 470 468 466 464 462 460 458 456 454 452 450 448 446 444 442 440 438 436 434 432 430 428 426 424 422 420 418 416 414 412 410 408 406 404 402 400 398 396 394 392 390 388 386 384 382 380 378 376 374 372 370 368 366 364 362 360 358 356 354 352 350 348 346 344 342 340 338 336 334 332 330 328 326 324 322 320 318 316 314 312 310 308 306 304 302 300 298 296 294 292 290 288 286 284 282 280 278 276 274 272 270 268 266 264 262 260 258 256 254 252 250 248 246 244 242 240 238 236 234 232 230 228 226 224 222 220 218 216 214 212 210 208 206 204 202 200 198 196 194 192 190 188 186 184 182 180 178 176 174 172 170 168 166 164 162 160 158 156 154 152 150 148 146 144 142 140 138 136 134 132 130 128 126 124 122 120 118 116 114 112 110 108 106 104 102 100 98 96 94 92 90 88 86 84 82 80 78 76 74 72 70 68 66 64 62 60 58 56 54 52 50 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 \n"
+            },
+            {
+                "input": "10\r\n5531\r\n5208\r\n18107\r\n7984\r\n8142\r\n8407\r\n15293\r\n16196\r\n1346\r\n18888\r\n",
+                "output": "1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 101 103 105 107 109 111 113 115 117 119 121 123 125 127 129 131 133 135 137 139 141 143 145 147 149 151 153 155 157 159 161 163 165 167 169 171 173 175 177 179 181 183 185 187 189 191 193 195 197 199 201 203 205 207 209 211 213 215 217 219 221 223 225 227 229 231 233 235 237 239 241 243 245 247 249 251 253 255 257 259 261 263 265 267 269 271 273 275 277 279 281 283 285 287 289 291 293 295 297 299 301 303 305 307 309 311 313 315 317 319 321 323 325 327 329 331 333 335 337 339 341 343 345 347 349 351 353 355 357 359 361 363 365 367 369 371 373 375 377 379 381 383 385 387 389 391 393 395 397 399 401 403 405 407 409 411 413 415 417 419 421 423 425 427 429 431 433 435 437 439 441 443 445 447 449 451 453 455 457 459 461 463 465 467 469 471 473 475 477 479 481 483 485 487 489 491 493 495 497 499 501 503 505 507 509 511 513 515 517 519 521 523 525 527 529 531 533 535 537 539 541 543 545 547 549 551 553 555 557 559 561 563 565 567 569 571 573 575 577 579 581 583 585 587 589 591 593 595 597 599 601 603 605 607 609 611 613 615 617 619 621 623 625 627 629 631 633 635 637 639 641 643 645 647 649 651 653 655 657 659 661 663 665 667 669 671 673 675 677 679 681 683 685 687 689 691 693 695 697 699 701 703 705 707 709 711 713 715 717 719 721 723 725 727 729 731 733 735 737 739 741 743 745 747 749 751 753 755 757 759 761 763 765 767 769 771 773 775 777 779 781 783 785 787 789 791 793 795 797 799 801 803 805 807 809 811 813 815 817 819 821 823 825 827 829 831 833 835 837 839 841 843 845 847 849 851 853 855 857 859 861 863 865 867 869 871 873 875 877 879 881 883 885 887 889 891 893 895 897 899 901 903 905 907 909 911 913 915 917 919 921 923 925 927 929 931 933 935 937 939 941 943 945 947 949 951 953 955 957 959 961 963 965 967 969 971 973 975 977 979 981 983 985 987 989 991 993 995 997 999 1001 1003 1005 1007 1009 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548 546 544 542 540 538 536 534 532 530 528 526 524 522 520 518 516 514 512 510 508 506 504 502 500 498 496 494 492 490 488 486 484 482 480 478 476 474 472 470 468 466 464 462 460 458 456 454 452 450 448 446 444 442 440 438 436 434 432 430 428 426 424 422 420 418 416 414 412 410 408 406 404 402 400 398 396 394 392 390 388 386 384 382 380 378 376 374 372 370 368 366 364 362 360 358 356 354 352 350 348 346 344 342 340 338 336 334 332 330 328 326 324 322 320 318 316 314 312 310 308 306 304 302 300 298 296 294 292 290 288 286 284 282 280 278 276 274 272 270 268 266 264 262 260 258 256 254 252 250 248 246 244 242 240 238 236 234 232 230 228 226 224 222 220 218 216 214 212 210 208 206 204 202 200 198 196 194 192 190 188 186 184 182 180 178 176 174 172 170 168 166 164 162 160 158 156 154 152 150 148 146 144 142 140 138 136 134 132 130 128 126 124 122 120 118 116 114 112 110 108 106 104 102 100 98 96 94 92 90 88 86 84 82 80 78 76 74 72 70 68 66 64 62 60 58 56 54 52 50 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 \n-1\n-1\n-1\n"
+            },
+            {
+                "input": "10\r\n19709\r\n11723\r\n12174\r\n11077\r\n7657\r\n2759\r\n16764\r\n8956\r\n1604\r\n8596\r\n",
+                "output": "1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 101 103 105 107 109 111 113 115 117 119 121 123 125 127 129 131 133 135 137 139 141 143 145 147 149 151 153 155 157 159 161 163 165 167 169 171 173 175 177 179 181 183 185 187 189 191 193 195 197 199 201 203 205 207 209 211 213 215 217 219 221 223 225 227 229 231 233 235 237 239 241 243 245 247 249 251 253 255 257 259 261 263 265 267 269 271 273 275 277 279 281 283 285 287 289 291 293 295 297 299 301 303 305 307 309 311 313 315 317 319 321 323 325 327 329 331 333 335 337 339 341 343 345 347 349 351 353 355 357 359 361 363 365 367 369 371 373 375 377 379 381 383 385 387 389 391 393 395 397 399 401 403 405 407 409 411 413 415 417 419 421 423 425 427 429 431 433 435 437 439 441 443 445 447 449 451 453 455 457 459 461 463 465 467 469 471 473 475 477 479 481 483 485 487 489 491 493 495 497 499 501 503 505 507 509 511 513 515 517 519 521 523 525 527 529 531 533 535 537 539 541 543 545 547 549 551 553 555 557 559 561 563 565 567 569 571 573 575 577 579 581 583 585 587 589 591 593 595 597 599 601 603 605 607 609 611 613 615 617 619 621 623 625 627 629 631 633 635 637 639 641 643 645 647 649 651 653 655 657 659 661 663 665 667 669 671 673 675 677 679 681 683 685 687 689 691 693 695 697 699 701 703 705 707 709 711 713 715 717 719 721 723 725 727 729 731 733 735 737 739 741 743 745 747 749 751 753 755 757 759 761 763 765 767 769 771 773 775 777 779 781 783 785 787 789 791 793 795 797 799 801 803 805 807 809 811 813 815 817 819 821 823 825 827 829 831 833 835 837 839 841 843 845 847 849 851 853 855 857 859 861 863 865 867 869 871 873 875 877 879 881 883 885 887 889 891 893 895 897 899 901 903 905 907 909 911 913 915 917 919 921 923 925 927 929 931 933 935 937 939 941 943 945 947 949 951 953 955 957 959 961 963 965 967 969 971 973 975 977 979 981 983 985 987 989 991 993 995 997 999 1001 1003 1005 1007 1009 1011 1013 1015 1017 1019 1021 1023 1025 1027 1029 1031 1033 1035 1037 1039 1041 1043 1045 1047 1049 1051 1053 1055 1057 1059 1061 1063 1065 1067 1069 1071 1073 1075 1077 1079 1081 1083 1085 1087 1089 1091 1093 1095 1097 1099 1101 1103 1105 1107 1109 1111 1113 1115 1117 1119 1121 1123 1125 1127 1129 1131 1133 1135 1137 1139 1141 1143 1145 1147 1149 1151 1153 1155 1157 1159 1161 1163 1165 1167 1169 1171 1173 1175 1177 1179 1181 1183 1185 1187 1189 1191 1193 1195 1197 1199 1201 1203 1205 1207 1209 1211 1213 1215 1217 1219 1221 1223 1225 1227 1229 1231 1233 1235 1237 1239 1241 1243 1245 1247 1249 1251 1253 1255 1257 1259 1261 1263 1265 1267 1269 1271 1273 1275 1277 1279 1281 1283 1285 1287 1289 1291 1293 1295 1297 1299 1301 1303 1305 1307 1309 1311 1313 1315 1317 1319 1321 1323 1325 1327 1329 1331 1333 1335 1337 1339 1341 1343 1345 1347 1349 1351 1353 1355 1357 1359 1361 1363 1365 1367 1369 1371 1373 1375 1377 1379 1381 1383 1385 1387 1389 1391 1393 1395 1397 1399 1401 1403 1405 1407 1409 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1396 1394 1392 1390 1388 1386 1384 1382 1380 1378 1376 1374 1372 1370 1368 1366 1364 1362 1360 1358 1356 1354 1352 1350 1348 1346 1344 1342 1340 1338 1336 1334 1332 1330 1328 1326 1324 1322 1320 1318 1316 1314 1312 1310 1308 1306 1304 1302 1300 1298 1296 1294 1292 1290 1288 1286 1284 1282 1280 1278 1276 1274 1272 1270 1268 1266 1264 1262 1260 1258 1256 1254 1252 1250 1248 1246 1244 1242 1240 1238 1236 1234 1232 1230 1228 1226 1224 1222 1220 1218 1216 1214 1212 1210 1208 1206 1204 1202 1200 1198 1196 1194 1192 1190 1188 1186 1184 1182 1180 1178 1176 1174 1172 1170 1168 1166 1164 1162 1160 1158 1156 1154 1152 1150 1148 1146 1144 1142 1140 1138 1136 1134 1132 1130 1128 1126 1124 1122 1120 1118 1116 1114 1112 1110 1108 1106 1104 1102 1100 1098 1096 1094 1092 1090 1088 1086 1084 1082 1080 1078 1076 1074 1072 1070 1068 1066 1064 1062 1060 1058 1056 1054 1052 1050 1048 1046 1044 1042 1040 1038 1036 1034 1032 1030 1028 1026 1024 1022 1020 1018 1016 1014 1012 1010 1008 1006 1004 1002 1000 998 996 994 992 990 988 986 984 982 980 978 976 974 972 970 968 966 964 962 960 958 956 954 952 950 948 946 944 942 940 938 936 934 932 930 928 926 924 922 920 918 916 914 912 910 908 906 904 902 900 898 896 894 892 890 888 886 884 882 880 878 876 874 872 870 868 866 864 862 860 858 856 854 852 850 848 846 844 842 840 838 836 834 832 830 828 826 824 822 820 818 816 814 812 810 808 806 804 802 800 798 796 794 792 790 788 786 784 782 780 778 776 774 772 770 768 766 764 762 760 758 756 754 752 750 748 746 744 742 740 738 736 734 732 730 728 726 724 722 720 718 716 714 712 710 708 706 704 702 700 698 696 694 692 690 688 686 684 682 680 678 676 674 672 670 668 666 664 662 660 658 656 654 652 650 648 646 644 642 640 638 636 634 632 630 628 626 624 622 620 618 616 614 612 610 608 606 604 602 600 598 596 594 592 590 588 586 584 582 580 578 576 574 572 570 568 566 564 562 560 558 556 554 552 550 548 546 544 542 540 538 536 534 532 530 528 526 524 522 520 518 516 514 512 510 508 506 504 502 500 498 496 494 492 490 488 486 484 482 480 478 476 474 472 470 468 466 464 462 460 458 456 454 452 450 448 446 444 442 440 438 436 434 432 430 428 426 424 422 420 418 416 414 412 410 408 406 404 402 400 398 396 394 392 390 388 386 384 382 380 378 376 374 372 370 368 366 364 362 360 358 356 354 352 350 348 346 344 342 340 338 336 334 332 330 328 326 324 322 320 318 316 314 312 310 308 306 304 302 300 298 296 294 292 290 288 286 284 282 280 278 276 274 272 270 268 266 264 262 260 258 256 254 252 250 248 246 244 242 240 238 236 234 232 230 228 226 224 222 220 218 216 214 212 210 208 206 204 202 200 198 196 194 192 190 188 186 184 182 180 178 176 174 172 170 168 166 164 162 160 158 156 154 152 150 148 146 144 142 140 138 136 134 132 130 128 126 124 122 120 118 116 114 112 110 108 106 104 102 100 98 96 94 92 90 88 86 84 82 80 78 76 74 72 70 68 66 64 62 60 58 56 54 52 50 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 \n-1\n-1\n-1\n-1\n"
+            },
+            {
+                "input": "10\r\n5833\r\n7290\r\n12015\r\n5617\r\n9952\r\n19390\r\n18619\r\n10268\r\n10416\r\n9251\r\n",
+                "output": "1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 101 103 105 107 109 111 113 115 117 119 121 123 125 127 129 131 133 135 137 139 141 143 145 147 149 151 153 155 157 159 161 163 165 167 169 171 173 175 177 179 181 183 185 187 189 191 193 195 197 199 201 203 205 207 209 211 213 215 217 219 221 223 225 227 229 231 233 235 237 239 241 243 245 247 249 251 253 255 257 259 261 263 265 267 269 271 273 275 277 279 281 283 285 287 289 291 293 295 297 299 301 303 305 307 309 311 313 315 317 319 321 323 325 327 329 331 333 335 337 339 341 343 345 347 349 351 353 355 357 359 361 363 365 367 369 371 373 375 377 379 381 383 385 387 389 391 393 395 397 399 401 403 405 407 409 411 413 415 417 419 421 423 425 427 429 431 433 435 437 439 441 443 445 447 449 451 453 455 457 459 461 463 465 467 469 471 473 475 477 479 481 483 485 487 489 491 493 495 497 499 501 503 505 507 509 511 513 515 517 519 521 523 525 527 529 531 533 535 537 539 541 543 545 547 549 551 553 555 557 559 561 563 565 567 569 571 573 575 577 579 581 583 585 587 589 591 593 595 597 599 601 603 605 607 609 611 613 615 617 619 621 623 625 627 629 631 633 635 637 639 641 643 645 647 649 651 653 655 657 659 661 663 665 667 669 671 673 675 677 679 681 683 685 687 689 691 693 695 697 699 701 703 705 707 709 711 713 715 717 719 721 723 725 727 729 731 733 735 737 739 741 743 745 747 749 751 753 755 757 759 761 763 765 767 769 771 773 775 777 779 781 783 785 787 789 791 793 795 797 799 801 803 805 807 809 811 813 815 817 819 821 823 825 827 829 831 833 835 837 839 841 843 845 847 849 851 853 855 857 859 861 863 865 867 869 871 873 875 877 879 881 883 885 887 889 891 893 895 897 899 901 903 905 907 909 911 913 915 917 919 921 923 925 927 929 931 933 935 937 939 941 943 945 947 949 951 953 955 957 959 961 963 965 967 969 971 973 975 977 979 981 983 985 987 989 991 993 995 997 999 1001 1003 1005 1007 1009 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3854 3852 3850 3848 3846 3844 3842 3840 3838 3836 3834 3832 3830 3828 3826 3824 3822 3820 3818 3816 3814 3812 3810 3808 3806 3804 3802 3800 3798 3796 3794 3792 3790 3788 3786 3784 3782 3780 3778 3776 3774 3772 3770 3768 3766 3764 3762 3760 3758 3756 3754 3752 3750 3748 3746 3744 3742 3740 3738 3736 3734 3732 3730 3728 3726 3724 3722 3720 3718 3716 3714 3712 3710 3708 3706 3704 3702 3700 3698 3696 3694 3692 3690 3688 3686 3684 3682 3680 3678 3676 3674 3672 3670 3668 3666 3664 3662 3660 3658 3656 3654 3652 3650 3648 3646 3644 3642 3640 3638 3636 3634 3632 3630 3628 3626 3624 3622 3620 3618 3616 3614 3612 3610 3608 3606 3604 3602 3600 3598 3596 3594 3592 3590 3588 3586 3584 3582 3580 3578 3576 3574 3572 3570 3568 3566 3564 3562 3560 3558 3556 3554 3552 3550 3548 3546 3544 3542 3540 3538 3536 3534 3532 3530 3528 3526 3524 3522 3520 3518 3516 3514 3512 3510 3508 3506 3504 3502 3500 3498 3496 3494 3492 3490 3488 3486 3484 3482 3480 3478 3476 3474 3472 3470 3468 3466 3464 3462 3460 3458 3456 3454 3452 3450 3448 3446 3444 3442 3440 3438 3436 3434 3432 3430 3428 3426 3424 3422 3420 3418 3416 3414 3412 3410 3408 3406 3404 3402 3400 3398 3396 3394 3392 3390 3388 3386 3384 3382 3380 3378 3376 3374 3372 3370 3368 3366 3364 3362 3360 3358 3356 3354 3352 3350 3348 3346 3344 3342 3340 3338 3336 3334 3332 3330 3328 3326 3324 3322 3320 3318 3316 3314 3312 3310 3308 3306 3304 3302 3300 3298 3296 3294 3292 3290 3288 3286 3284 3282 3280 3278 3276 3274 3272 3270 3268 3266 3264 3262 3260 3258 3256 3254 3252 3250 3248 3246 3244 3242 3240 3238 3236 3234 3232 3230 3228 3226 3224 3222 3220 3218 3216 3214 3212 3210 3208 3206 3204 3202 3200 3198 3196 3194 3192 3190 3188 3186 3184 3182 3180 3178 3176 3174 3172 3170 3168 3166 3164 3162 3160 3158 3156 3154 3152 3150 3148 3146 3144 3142 3140 3138 3136 3134 3132 3130 3128 3126 3124 3122 3120 3118 3116 3114 3112 3110 3108 3106 3104 3102 3100 3098 3096 3094 3092 3090 3088 3086 3084 3082 3080 3078 3076 3074 3072 3070 3068 3066 3064 3062 3060 3058 3056 3054 3052 3050 3048 3046 3044 3042 3040 3038 3036 3034 3032 3030 3028 3026 3024 3022 3020 3018 3016 3014 3012 3010 3008 3006 3004 3002 3000 2998 2996 2994 2992 2990 2988 2986 2984 2982 2980 2978 2976 2974 2972 2970 2968 2966 2964 2962 2960 2958 2956 2954 2952 2950 2948 2946 2944 2942 2940 2938 2936 2934 2932 2930 2928 2926 2924 2922 2920 2918 2916 2914 2912 2910 2908 2906 2904 2902 2900 2898 2896 2894 2892 2890 2888 2886 2884 2882 2880 2878 2876 2874 2872 2870 2868 2866 2864 2862 2860 2858 2856 2854 2852 2850 2848 2846 2844 2842 2840 2838 2836 2834 2832 2830 2828 2826 2824 2822 2820 2818 2816 2814 2812 2810 2808 2806 2804 2802 2800 2798 2796 2794 2792 2790 2788 2786 2784 2782 2780 2778 2776 2774 2772 2770 2768 2766 2764 2762 2760 2758 2756 2754 2752 2750 2748 2746 2744 2742 2740 2738 2736 2734 2732 2730 2728 2726 2724 2722 2720 2718 2716 2714 2712 2710 2708 2706 2704 2702 2700 2698 2696 2694 2692 2690 2688 2686 2684 2682 2680 2678 2676 2674 2672 2670 2668 2666 2664 2662 2660 2658 2656 2654 2652 2650 2648 2646 2644 2642 2640 2638 2636 2634 2632 2630 2628 2626 2624 2622 2620 2618 2616 2614 2612 2610 2608 2606 2604 2602 2600 2598 2596 2594 2592 2590 2588 2586 2584 2582 2580 2578 2576 2574 2572 2570 2568 2566 2564 2562 2560 2558 2556 2554 2552 2550 2548 2546 2544 2542 2540 2538 2536 2534 2532 2530 2528 2526 2524 2522 2520 2518 2516 2514 2512 2510 2508 2506 2504 2502 2500 2498 2496 2494 2492 2490 2488 2486 2484 2482 2480 2478 2476 2474 2472 2470 2468 2466 2464 2462 2460 2458 2456 2454 2452 2450 2448 2446 2444 2442 2440 2438 2436 2434 2432 2430 2428 2426 2424 2422 2420 2418 2416 2414 2412 2410 2408 2406 2404 2402 2400 2398 2396 2394 2392 2390 2388 2386 2384 2382 2380 2378 2376 2374 2372 2370 2368 2366 2364 2362 2360 2358 2356 2354 2352 2350 2348 2346 2344 2342 2340 2338 2336 2334 2332 2330 2328 2326 2324 2322 2320 2318 2316 2314 2312 2310 2308 2306 2304 2302 2300 2298 2296 2294 2292 2290 2288 2286 2284 2282 2280 2278 2276 2274 2272 2270 2268 2266 2264 2262 2260 2258 2256 2254 2252 2250 2248 2246 2244 2242 2240 2238 2236 2234 2232 2230 2228 2226 2224 2222 2220 2218 2216 2214 2212 2210 2208 2206 2204 2202 2200 2198 2196 2194 2192 2190 2188 2186 2184 2182 2180 2178 2176 2174 2172 2170 2168 2166 2164 2162 2160 2158 2156 2154 2152 2150 2148 2146 2144 2142 2140 2138 2136 2134 2132 2130 2128 2126 2124 2122 2120 2118 2116 2114 2112 2110 2108 2106 2104 2102 2100 2098 2096 2094 2092 2090 2088 2086 2084 2082 2080 2078 2076 2074 2072 2070 2068 2066 2064 2062 2060 2058 2056 2054 2052 2050 2048 2046 2044 2042 2040 2038 2036 2034 2032 2030 2028 2026 2024 2022 2020 2018 2016 2014 2012 2010 2008 2006 2004 2002 2000 1998 1996 1994 1992 1990 1988 1986 1984 1982 1980 1978 1976 1974 1972 1970 1968 1966 1964 1962 1960 1958 1956 1954 1952 1950 1948 1946 1944 1942 1940 1938 1936 1934 1932 1930 1928 1926 1924 1922 1920 1918 1916 1914 1912 1910 1908 1906 1904 1902 1900 1898 1896 1894 1892 1890 1888 1886 1884 1882 1880 1878 1876 1874 1872 1870 1868 1866 1864 1862 1860 1858 1856 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1054 1052 1050 1048 1046 1044 1042 1040 1038 1036 1034 1032 1030 1028 1026 1024 1022 1020 1018 1016 1014 1012 1010 1008 1006 1004 1002 1000 998 996 994 992 990 988 986 984 982 980 978 976 974 972 970 968 966 964 962 960 958 956 954 952 950 948 946 944 942 940 938 936 934 932 930 928 926 924 922 920 918 916 914 912 910 908 906 904 902 900 898 896 894 892 890 888 886 884 882 880 878 876 874 872 870 868 866 864 862 860 858 856 854 852 850 848 846 844 842 840 838 836 834 832 830 828 826 824 822 820 818 816 814 812 810 808 806 804 802 800 798 796 794 792 790 788 786 784 782 780 778 776 774 772 770 768 766 764 762 760 758 756 754 752 750 748 746 744 742 740 738 736 734 732 730 728 726 724 722 720 718 716 714 712 710 708 706 704 702 700 698 696 694 692 690 688 686 684 682 680 678 676 674 672 670 668 666 664 662 660 658 656 654 652 650 648 646 644 642 640 638 636 634 632 630 628 626 624 622 620 618 616 614 612 610 608 606 604 602 600 598 596 594 592 590 588 586 584 582 580 578 576 574 572 570 568 566 564 562 560 558 556 554 552 550 548 546 544 542 540 538 536 534 532 530 528 526 524 522 520 518 516 514 512 510 508 506 504 502 500 498 496 494 492 490 488 486 484 482 480 478 476 474 472 470 468 466 464 462 460 458 456 454 452 450 448 446 444 442 440 438 436 434 432 430 428 426 424 422 420 418 416 414 412 410 408 406 404 402 400 398 396 394 392 390 388 386 384 382 380 378 376 374 372 370 368 366 364 362 360 358 356 354 352 350 348 346 344 342 340 338 336 334 332 330 328 326 324 322 320 318 316 314 312 310 308 306 304 302 300 298 296 294 292 290 288 286 284 282 280 278 276 274 272 270 268 266 264 262 260 258 256 254 252 250 248 246 244 242 240 238 236 234 232 230 228 226 224 222 220 218 216 214 212 210 208 206 204 202 200 198 196 194 192 190 188 186 184 182 180 178 176 174 172 170 168 166 164 162 160 158 156 154 152 150 148 146 144 142 140 138 136 134 132 130 128 126 124 122 120 118 116 114 112 110 108 106 104 102 100 98 96 94 92 90 88 86 84 82 80 78 76 74 72 70 68 66 64 62 60 58 56 54 52 50 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 \n"
+            },
+            {
+                "input": "9\r\n21753\r\n21606\r\n20665\r\n22171\r\n20649\r\n21395\r\n21154\r\n21912\r\n20102\r\n",
+                "output": "1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 101 103 105 107 109 111 113 115 117 119 121 123 125 127 129 131 133 135 137 139 141 143 145 147 149 151 153 155 157 159 161 163 165 167 169 171 173 175 177 179 181 183 185 187 189 191 193 195 197 199 201 203 205 207 209 211 213 215 217 219 221 223 225 227 229 231 233 235 237 239 241 243 245 247 249 251 253 255 257 259 261 263 265 267 269 271 273 275 277 279 281 283 285 287 289 291 293 295 297 299 301 303 305 307 309 311 313 315 317 319 321 323 325 327 329 331 333 335 337 339 341 343 345 347 349 351 353 355 357 359 361 363 365 367 369 371 373 375 377 379 381 383 385 387 389 391 393 395 397 399 401 403 405 407 409 411 413 415 417 419 421 423 425 427 429 431 433 435 437 439 441 443 445 447 449 451 453 455 457 459 461 463 465 467 469 471 473 475 477 479 481 483 485 487 489 491 493 495 497 499 501 503 505 507 509 511 513 515 517 519 521 523 525 527 529 531 533 535 537 539 541 543 545 547 549 551 553 555 557 559 561 563 565 567 569 571 573 575 577 579 581 583 585 587 589 591 593 595 597 599 601 603 605 607 609 611 613 615 617 619 621 623 625 627 629 631 633 635 637 639 641 643 645 647 649 651 653 655 657 659 661 663 665 667 669 671 673 675 677 679 681 683 685 687 689 691 693 695 697 699 701 703 705 707 709 711 713 715 717 719 721 723 725 727 729 731 733 735 737 739 741 743 745 747 749 751 753 755 757 759 761 763 765 767 769 771 773 775 777 779 781 783 785 787 789 791 793 795 797 799 801 803 805 807 809 811 813 815 817 819 821 823 825 827 829 831 833 835 837 839 841 843 845 847 849 851 853 855 857 859 861 863 865 867 869 871 873 875 877 879 881 883 885 887 889 891 893 895 897 899 901 903 905 907 909 911 913 915 917 919 921 923 925 927 929 931 933 935 937 939 941 943 945 947 949 951 953 955 957 959 961 963 965 967 969 971 973 975 977 979 981 983 985 987 989 991 993 995 997 999 1001 1003 1005 1007 1009 1011 1013 1015 1017 1019 1021 1023 1025 1027 1029 1031 1033 1035 1037 1039 1041 1043 1045 1047 1049 1051 1053 1055 1057 1059 1061 1063 1065 1067 1069 1071 1073 1075 1077 1079 1081 1083 1085 1087 1089 1091 1093 1095 1097 1099 1101 1103 1105 1107 1109 1111 1113 1115 1117 1119 1121 1123 1125 1127 1129 1131 1133 1135 1137 1139 1141 1143 1145 1147 1149 1151 1153 1155 1157 1159 1161 1163 1165 1167 1169 1171 1173 1175 1177 1179 1181 1183 1185 1187 1189 1191 1193 1195 1197 1199 1201 1203 1205 1207 1209 1211 1213 1215 1217 1219 1221 1223 1225 1227 1229 1231 1233 1235 1237 1239 1241 1243 1245 1247 1249 1251 1253 1255 1257 1259 1261 1263 1265 1267 1269 1271 1273 1275 1277 1279 1281 1283 1285 1287 1289 1291 1293 1295 1297 1299 1301 1303 1305 1307 1309 1311 1313 1315 1317 1319 1321 1323 1325 1327 1329 1331 1333 1335 1337 1339 1341 1343 1345 1347 1349 1351 1353 1355 1357 1359 1361 1363 1365 1367 1369 1371 1373 1375 1377 1379 1381 1383 1385 1387 1389 1391 1393 1395 1397 1399 1401 1403 1405 1407 1409 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1294 1292 1290 1288 1286 1284 1282 1280 1278 1276 1274 1272 1270 1268 1266 1264 1262 1260 1258 1256 1254 1252 1250 1248 1246 1244 1242 1240 1238 1236 1234 1232 1230 1228 1226 1224 1222 1220 1218 1216 1214 1212 1210 1208 1206 1204 1202 1200 1198 1196 1194 1192 1190 1188 1186 1184 1182 1180 1178 1176 1174 1172 1170 1168 1166 1164 1162 1160 1158 1156 1154 1152 1150 1148 1146 1144 1142 1140 1138 1136 1134 1132 1130 1128 1126 1124 1122 1120 1118 1116 1114 1112 1110 1108 1106 1104 1102 1100 1098 1096 1094 1092 1090 1088 1086 1084 1082 1080 1078 1076 1074 1072 1070 1068 1066 1064 1062 1060 1058 1056 1054 1052 1050 1048 1046 1044 1042 1040 1038 1036 1034 1032 1030 1028 1026 1024 1022 1020 1018 1016 1014 1012 1010 1008 1006 1004 1002 1000 998 996 994 992 990 988 986 984 982 980 978 976 974 972 970 968 966 964 962 960 958 956 954 952 950 948 946 944 942 940 938 936 934 932 930 928 926 924 922 920 918 916 914 912 910 908 906 904 902 900 898 896 894 892 890 888 886 884 882 880 878 876 874 872 870 868 866 864 862 860 858 856 854 852 850 848 846 844 842 840 838 836 834 832 830 828 826 824 822 820 818 816 814 812 810 808 806 804 802 800 798 796 794 792 790 788 786 784 782 780 778 776 774 772 770 768 766 764 762 760 758 756 754 752 750 748 746 744 742 740 738 736 734 732 730 728 726 724 722 720 718 716 714 712 710 708 706 704 702 700 698 696 694 692 690 688 686 684 682 680 678 676 674 672 670 668 666 664 662 660 658 656 654 652 650 648 646 644 642 640 638 636 634 632 630 628 626 624 622 620 618 616 614 612 610 608 606 604 602 600 598 596 594 592 590 588 586 584 582 580 578 576 574 572 570 568 566 564 562 560 558 556 554 552 550 548 546 544 542 540 538 536 534 532 530 528 526 524 522 520 518 516 514 512 510 508 506 504 502 500 498 496 494 492 490 488 486 484 482 480 478 476 474 472 470 468 466 464 462 460 458 456 454 452 450 448 446 444 442 440 438 436 434 432 430 428 426 424 422 420 418 416 414 412 410 408 406 404 402 400 398 396 394 392 390 388 386 384 382 380 378 376 374 372 370 368 366 364 362 360 358 356 354 352 350 348 346 344 342 340 338 336 334 332 330 328 326 324 322 320 318 316 314 312 310 308 306 304 302 300 298 296 294 292 290 288 286 284 282 280 278 276 274 272 270 268 266 264 262 260 258 256 254 252 250 248 246 244 242 240 238 236 234 232 230 228 226 224 222 220 218 216 214 212 210 208 206 204 202 200 198 196 194 192 190 188 186 184 182 180 178 176 174 172 170 168 166 164 162 160 158 156 154 152 150 148 146 144 142 140 138 136 134 132 130 128 126 124 122 120 118 116 114 112 110 108 106 104 102 100 98 96 94 92 90 88 86 84 82 80 78 76 74 72 70 68 66 64 62 60 58 56 54 52 50 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 \n-1\n-1\n-1\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": false
+    },
+    {
+        "id": "2056/A",
+        "aliases": null,
+        "contest_id": "2056",
+        "contest_name": "Codeforces Round 997 (Div. 2)",
+        "contest_type": "CF",
+        "contest_start": 1737124500,
+        "contest_start_year": 2025,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Shape Perimeter",
+        "description": "There is an $$$m$$$ by $$$m$$$ square stamp on an infinite piece of paper. Initially, the bottom-left corner of the square stamp is aligned with the bottom-left corner of the paper. You are given two integer sequences $$$x$$$ and $$$y$$$, each of length $$$n$$$. For each step $$$i$$$ from $$$1$$$ to $$$n$$$, the following happens:\n\n- Move the stamp $$$x_i$$$ units to the right and $$$y_i$$$ units upwards.\n- Press the stamp onto the paper, leaving an $$$m$$$ by $$$m$$$ colored square at its current position.\n\nNote that the elements of sequences $$$x$$$ and $$$y$$$ have a special constraint: $$$1\\le x_i, y_i\\le m - 1$$$.\n\nNote that you do not press the stamp at the bottom-left corner of the paper. Refer to the notes section for better understanding.\n\nIt can be proven that after all the operations, the colored shape on the paper formed by the stamp is a single connected region. Find the perimeter of this colored shape.",
+        "input_format": "Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \\le t \\le 1000$$$). The description of the test cases follows.\n\nThe first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \\le n \\le 100$$$, $$$2 \\le m \\le 100$$$) — the number of operations performed and the side length of the square stamp.\n\nThe $$$i$$$-th of the next $$$n$$$ lines contains two integers $$$x_i$$$ and $$$y_i$$$ ($$$1 \\le x_i, y_i \\le m - 1$$$) — the distance that the stamp will be moved right and up during the $$$i$$$-th operation, respectively.\n\nNote that there are no constraints on the sum of $$$n$$$ over all test cases.",
+        "output_format": "For each test case, output a single integer representing the perimeter of the colored shape on the paper.",
+        "interaction_format": null,
+        "note": "In the first example, the stamp has a side length of $$$3$$$ and is pressed $$$4$$$ times at coordinates $$$(1, 1)$$$, $$$(3, 3)$$$, $$$(5, 4)$$$, and $$$(6, 6)$$$. The piece of paper looks like that afterwards:\n\nHere, the square formed by the first press is colored blue, the second red, the third green, and the fourth purple. The combined shape, whose perimeter we need to calculate, looks like that:\n\nFrom the diagram, it can be seen that this shape has a perimeter of $$$32$$$.",
+        "examples": [
+            {
+                "input": "3\n4 3\n1 1\n2 2\n2 1\n1 2\n1 2\n1 1\n6 7\n3 6\n1 1\n3 1\n6 6\n5 4\n6 1",
+                "output": "32\n8\n96"
+            }
+        ],
+        "editorial": null,
+        "rating": 800,
+        "tags": [
+            "constructive algorithms",
+            "math"
+        ],
+        "testset_size": 4,
+        "official_tests": [
+            {
+                "input": "3\r\n4 3\r\n1 1\r\n2 2\r\n2 1\r\n1 2\r\n1 2\r\n1 1\r\n6 7\r\n3 6\r\n1 1\r\n3 1\r\n6 6\r\n5 4\r\n6 1\r\n",
+                "output": "32\r\n8\r\n96\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2067/A",
+        "aliases": null,
+        "contest_id": "2067",
+        "contest_name": "Codeforces Round 1004 (Div. 2)",
+        "contest_type": "CF",
+        "contest_start": 1739284500,
+        "contest_start_year": 2025,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Adjacent Digit Sums",
+        "description": "Given two numbers $$$x, y$$$. You need to determine if there exists an integer $$$n$$$ such that $$$S(n) = x$$$, $$$S(n + 1) = y$$$.\n\nHere, $$$S(a)$$$ denotes the sum of the digits of the number $$$a$$$ in the decimal numeral system.",
+        "input_format": "Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \\le t \\le 500$$$). The description of the test cases follows.\n\nThe first line of each test case contains two integers $$$x, y$$$ ($$$1 \\le x \\le 1000, 1 \\le y \\le 1000$$$).",
+        "output_format": "For each test case, print \"NO\" if a suitable $$$n$$$ does not exist. Otherwise, output \"YES\".\n\nYou can output each letter in any case (for example, \"YES\", \"Yes\", \"yes\", \"yEs\", \"yEs\" will be recognized as a positive answer).",
+        "interaction_format": null,
+        "note": "In the first test case, for example, $$$n = 100$$$ works. $$$S(100) = 1$$$, $$$S(101) = 2$$$.\n\nIn the second test case, it can be shown that $$$S(n) \\neq S(n+1)$$$ for all $$$n$$$; therefore, the answer is No.\n\nIn the fourth test case, $$$n = 10^{111}-1$$$ works, which is a number consisting of $$$111$$$ digits of $$$9$$$.",
+        "examples": [
+            {
+                "input": "7\n1 2\n77 77\n997 999\n999 1\n1000 1\n1 11\n18 1",
+                "output": "Yes\nNo\nNo\nYes\nNo\nNo\nYes"
+            }
+        ],
+        "editorial": "Let's look at the last digit of the number $$$n$$$. If it is not equal to $$$9$$$, then in the number $$$n+1$$$, this last digit will be increased by $$$1$$$, while the rest of the number will remain unchanged. Thus, for such $$$n$$$, we have $$$S(n+1) = S(n) + 1$$$.\nIn general, if the number $$$n$$$ ends with $$$k$$$ consecutive digits of $$$9$$$, then it turns out that $$$S(n+1) = S(n) + 1 - 9k$$$, since all these $$$9$$$s will be replaced by $$$0$$$s after adding one to the number $$$n$$$.\nTherefore, if there does not exist an integer $$$k \\geq 0$$$ such that $$$y = x + 1 - 9k$$$, then the answer is definitely No. Otherwise, it is easy to see that the answer is definitely Yes.\nOne of the suitable numbers is: $$$n = \\underbrace{11 \\ldots 1}_{x-9k}\\underbrace{99 \\ldots 9}_{k}$$$. Then, $$$S(n) = x - 9k + 9k = x$$$.\nAnd $$$n + 1 = \\underbrace{11 \\ldots 1}_{x-9k-1}2\\underbrace{00 \\ldots 0}_{k}$$$. Thus, $$$S(n+1) = x - 9k - 1 + 2 = x + 1 - 9k$$$.\nTherefore, to solve the problem, we need to check whether the number $$$\\frac{x + 1 - y}{9}$$$ is an integer and non-negative.",
+        "rating": 800,
+        "tags": [
+            "brute force",
+            "constructive algorithms",
+            "math"
+        ],
+        "testset_size": 5,
+        "official_tests": [
+            {
+                "input": "7\r\n1 2\r\n77 77\r\n997 999\r\n999 1\r\n1000 1\r\n1 11\r\n18 1\r\n",
+                "output": "Yes\r\nNo\r\nNo\r\nYes\r\nNo\r\nNo\r\nYes\r\n"
+            },
+            {
+                "input": "1\r\n999 1000\r\n",
+                "output": "Yes\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2034/A",
+        "aliases": null,
+        "contest_id": "2034",
+        "contest_name": "Rayan Programming Contest 2024 - Selection (Codeforces Round 989, Div. 1 + Div. 2)",
+        "contest_type": "CF",
+        "contest_start": 1732977300,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "King Keykhosrow's Mystery",
+        "description": "There is a tale about the wise King Keykhosrow who owned a grand treasury filled with treasures from across the Persian Empire. However, to prevent theft and ensure the safety of his wealth, King Keykhosrow's vault was sealed with a magical lock that could only be opened by solving a riddle.\n\nThe riddle involves two sacred numbers $$$a$$$ and $$$b$$$. To unlock the vault, the challenger must determine the smallest key number $$$m$$$ that satisfies two conditions:\n\n- $$$m$$$ must be greater than or equal to at least one of $$$a$$$ and $$$b$$$.\n- The remainder when $$$m$$$ is divided by $$$a$$$ must be equal to the remainder when $$$m$$$ is divided by $$$b$$$.\n\nOnly by finding the smallest correct value of $$$m$$$ can one unlock the vault and access the legendary treasures!",
+        "input_format": "The first line of the input contains an integer $$$t$$$ ($$$1 \\leq t \\leq 100$$$), the number of test cases.\n\nEach test case consists of a single line containing two integers $$$a$$$ and $$$b$$$ ($$$1 \\leq a, b \\leq 1000$$$).",
+        "output_format": "For each test case, print the smallest integer $$$m$$$ that satisfies the conditions above.",
+        "interaction_format": null,
+        "note": "In the first test case, you can see that:\n\n- $$$4 \\bmod 4 = 0$$$ but $$$4 \\bmod 6 = 4$$$\n- $$$5 \\bmod 4 = 1$$$ but $$$5 \\bmod 6 = 5$$$\n- $$$6 \\bmod 4 = 2$$$ but $$$6 \\bmod 6 = 0$$$\n- $$$7 \\bmod 4 = 3$$$ but $$$7 \\bmod 6 = 1$$$\n- $$$8 \\bmod 4 = 0$$$ but $$$8 \\bmod 6 = 2$$$\n- $$$9 \\bmod 4 = 1$$$ but $$$9 \\bmod 6 = 3$$$\n- $$$10 \\bmod 4 = 2$$$ but $$$10 \\bmod 6 = 4$$$\n- $$$11 \\bmod 4 = 3$$$ but $$$11 \\bmod 6 = 5$$$",
+        "examples": [
+            {
+                "input": "2\n4 6\n472 896",
+                "output": "12\n52864"
+            }
+        ],
+        "editorial": null,
+        "rating": 800,
+        "tags": [
+            "brute force",
+            "chinese remainder theorem",
+            "math",
+            "number theory"
+        ],
+        "testset_size": 11,
+        "official_tests": [
+            {
+                "input": "2\r\n4 6\r\n472 896\r\n",
+                "output": "12\r\n52864\r\n"
+            },
+            {
+                "input": "1\r\n1000 999\r\n",
+                "output": "999000\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2038/J",
+        "aliases": null,
+        "contest_id": "2038",
+        "contest_name": "2024-2025 ICPC, NERC, Southern and Volga Russian Regional Contest (Unrated, Online Mirror, ICPC Rules, Preferably Teams)",
+        "contest_type": "ICPC",
+        "contest_start": 1731926100,
+        "contest_start_year": 2024,
+        "index": "J",
+        "time_limit": 2.0,
+        "memory_limit": 512.0,
+        "title": "Waiting for...",
+        "description": "Monocarp is waiting for a bus at the bus stop. Unfortunately, there are many people who want to ride a bus too.\n\nYou are given a list of events of two types:\n\n- B $$$b_i$$$ — a bus with $$$b_i$$$ free seats arrives at the stop;\n- P $$$p_i$$$ — $$$p_i$$$ people arrive at the stop.\n\nThese events are listed in a chronological order.\n\nWhen a bus arrives, the following happens. All people at the bus stop (except for Monocarp) try to enter the bus. If there are enough free seats for all of them, then they all enter the bus. Otherwise, some people remain at the bus stop (the number of people who enter the bus is equal to the number of free seats).\n\nIf there is still at least one free seat after all people (except for Monocarp) enter the bus, then Monocarp can decide to enter this bus as well (but he might choose to wait for another bus). For each bus, you have to determine if it is possible for Monocarp to take that bus.",
+        "input_format": "The first line contains one integer $$$n$$$ $$$(1 \\le n \\le 10^3)$$$ — the number of events.\n\nThen, $$$n$$$ lines follow. The $$$i$$$-th of them contains the description of the $$$i$$$-th event in one of the two following formats:\n\n- B $$$b_i$$$ ($$$1 \\le b_i \\le 10^6$$$) — a bus with $$$b_i$$$ free seats arrives at the stop;\n- P $$$p_i$$$ ($$$1 \\le p_i \\le 10^6$$$) — $$$p_i$$$ people arrive at the stop.\n\nAdditional constraint on the input: there is at least one event of type B.",
+        "output_format": "For each event of type B, print YES if it is possible for Monocarp to take the corresponding bus, or NO otherwise (case-insensitive).",
+        "interaction_format": null,
+        "note": null,
+        "examples": [
+            {
+                "input": "10\nP 2\nP 5\nB 8\nP 14\nB 5\nB 9\nB 3\nP 2\nB 1\nB 2",
+                "output": "YES\nNO\nNO\nYES\nNO\nYES"
+            }
+        ],
+        "editorial": null,
+        "rating": 800,
+        "tags": [
+            "greedy",
+            "implementation"
+        ],
+        "testset_size": 40,
+        "official_tests": [
+            {
+                "input": "10\r\nP 2\r\nP 5\r\nB 8\r\nP 14\r\nB 5\r\nB 9\r\nB 3\r\nP 2\r\nB 1\r\nB 2\r\n",
+                "output": "YES\r\nNO\r\nNO\r\nYES\r\nNO\r\nYES\r\n"
+            },
+            {
+                "input": "12\r\nP 12\r\nP 6\r\nB 3\r\nB 8\r\nP 1\r\nP 3\r\nP 4\r\nP 1\r\nB 25\r\nP 12\r\nB 20\r\nB 4\r\n",
+                "output": "NO\r\nNO\r\nYES\r\nYES\r\nYES\r\n"
+            },
+            {
+                "input": "10\r\nP 2\r\nP 5\r\nB 8\r\nP 14\r\nB 5\r\nB 9\r\nB 1\r\nP 2\r\nB 1\r\nB 2\r\n",
+                "output": "YES\r\nNO\r\nNO\r\nYES\r\nNO\r\nYES\r\n"
+            },
+            {
+                "input": "1\r\nB 1\r\n",
+                "output": "YES\r\n"
+            },
+            {
+                "input": "1\r\nB 10\r\n",
+                "output": "YES\r\n"
+            },
+            {
+                "input": "2\r\nP 1\r\nB 1\r\n",
+                "output": "NO\r\n"
+            },
+            {
+                "input": "2\r\nP 10\r\nB 11\r\n",
+                "output": "YES\r\n"
+            },
+            {
+                "input": "25\r\nP 67\r\nP 55\r\nP 48\r\nB 28\r\nP 19\r\nB 105\r\nB 66\r\nP 52\r\nP 86\r\nP 83\r\nB 168\r\nP 43\r\nP 57\r\nP 52\r\nB 121\r\nP 69\r\nP 31\r\nB 150\r\nB 172\r\nP 66\r\nP 83\r\nB 100\r\nP 13\r\nB 124\r\nP 88\r\n",
+                "output": "NO\r\nNO\r\nYES\r\nNO\r\nNO\r\nNO\r\nYES\r\nNO\r\nYES\r\n"
+            },
+            {
+                "input": "50\r\nP 28\r\nP 14\r\nB 85\r\nP 21\r\nP 13\r\nP 20\r\nP 31\r\nP 33\r\nP 23\r\nP 7\r\nP 37\r\nP 32\r\nB 172\r\nP 7\r\nB 112\r\nP 33\r\nP 29\r\nP 35\r\nP 32\r\nB 72\r\nB 71\r\nB 28\r\nB 90\r\nP 2\r\nP 43\r\nP 7\r\nP 49\r\nP 6\r\nB 124\r\nB 140\r\nP 45\r\nP 16\r\nP 29\r\nP 48\r\nB 12\r\nB 3\r\nP 17\r\nB 27\r\nB 176\r\nP 47\r\nB 25\r\nB 3\r\nP 10\r\nP 27\r\nP 8\r\nP 24\r\nP 3\r\nP 6\r\nP 23\r\nB 113\r\n",
+                "output": "YES\r\nNO\r\nYES\r\nNO\r\nYES\r\nYES\r\nYES\r\nYES\r\nYES\r\nNO\r\nNO\r\nNO\r\nYES\r\nNO\r\nNO\r\nNO\r\n"
+            },
+            {
+                "input": "75\r\nB 40\r\nP 44\r\nP 137\r\nP 15\r\nP 123\r\nP 5\r\nP 105\r\nP 175\r\nB 605\r\nP 50\r\nP 126\r\nP 121\r\nP 151\r\nP 103\r\nB 551\r\nP 193\r\nB 194\r\nP 93\r\nP 165\r\nB 257\r\nP 145\r\nP 55\r\nP 196\r\nP 12\r\nP 168\r\nP 171\r\nB 748\r\nP 96\r\nP 62\r\nP 139\r\nP 12\r\nP 168\r\nP 154\r\nP 148\r\nP 33\r\nP 137\r\nB 949\r\nP 82\r\nP 71\r\nP 162\r\nP 155\r\nP 169\r\nB 638\r\nP 20\r\nP 76\r\nP 135\r\nP 75\r\nP 69\r\nP 119\r\nB 496\r\nP 16\r\nP 134\r\nP 32\r\nP 52\r\nP 118\r\nP 176\r\nB 529\r\nB 155\r\nB 229\r\nP 25\r\nB 26\r\nP 152\r\nB 152\r\nP 139\r\nP 126\r\nP 12\r\nP 1\r\nP 155\r\nP 23\r\nP 2\r\nP 8\r\nB 466\r\nB 157\r\nP 55\r\nB 185\r\n",
+                "output": "YES\r\nYES\r\nNO\r\nYES\r\nNO\r\nNO\r\nNO\r\nNO\r\nYES\r\nYES\r\nYES\r\nYES\r\nYES\r\nNO\r\nNO\r\nYES\r\nYES\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2038/N",
+        "aliases": null,
+        "contest_id": "2038",
+        "contest_name": "2024-2025 ICPC, NERC, Southern and Volga Russian Regional Contest (Unrated, Online Mirror, ICPC Rules, Preferably Teams)",
+        "contest_type": "ICPC",
+        "contest_start": 1731926100,
+        "contest_start_year": 2024,
+        "index": "N",
+        "time_limit": 2.0,
+        "memory_limit": 512.0,
+        "title": "Fixing the Expression",
+        "description": "An expression is a string consisting of three characters, where the first and the last characters are digits (from $$$0$$$ to $$$9$$$), and the middle character is a comparison symbol (<, = or >).\n\nAn expression is true if the comparison symbol matches the digits (for example, if the first digit is strictly less than the last digit, the comparison symbol should be <).\n\nFor example, the expressions 1<3, 4>2, 0=0 are true, while 5>5, 7<3 are not.\n\nYou are given a string $$$s$$$, which is an expression. Change as few characters as possible so that $$$s$$$ becomes a true expression. Note that if $$$s$$$ is already true, you should leave it as it is.",
+        "input_format": "The first line contains one integer $$$t$$$ ($$$1 \\le t \\le 300$$$) — the number of test cases.\n\nEach test case consists of one line containing the string $$$s$$$ ($$$|s| = 3$$$, the first and the last characters of $$$s$$$ are digits, the second character is a comparison symbol).",
+        "output_format": "For each test case, print a string consisting of $$$3$$$ characters — a true expression which can be obtained by changing as few characters as possible in $$$s$$$. If there are multiple answers, print any of them.",
+        "interaction_format": null,
+        "note": null,
+        "examples": [
+            {
+                "input": "5\n3<7\n3>7\n8=9\n0=0\n5<3",
+                "output": "3<7\n8>7\n8<9\n0=0\n0<3"
+            }
+        ],
+        "editorial": null,
+        "rating": 800,
+        "tags": [
+            "implementation"
+        ],
+        "testset_size": 2,
+        "official_tests": [
+            {
+                "input": "5\r\n3<7\r\n3>7\r\n8=9\r\n0=0\r\n5<3\r\n",
+                "output": "3<7\r\n3<7\r\n8<9\r\n0=0\r\n5>3\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2042/A",
+        "aliases": null,
+        "contest_id": "2042",
+        "contest_name": "Educational Codeforces Round 172 (Rated for Div. 2)",
+        "contest_type": "ICPC",
+        "contest_start": 1733150100,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 2.0,
+        "memory_limit": 512.0,
+        "title": "Greedy Monocarp",
+        "description": "There are $$$n$$$ chests; the $$$i$$$-th chest initially contains $$$a_i$$$ coins. For each chest, you can choose any non-negative ($$$0$$$ or greater) number of coins to add to that chest, with one constraint: the total number of coins in all chests must become at least $$$k$$$.\n\nAfter you've finished adding coins to the chests, greedy Monocarp comes, who wants the coins. He will take the chests one by one, and since he is greedy, he will always choose the chest with the maximum number of coins. Monocarp will stop as soon as the total number of coins in chests he takes is at least $$$k$$$.\n\nYou want Monocarp to take as few coins as possible, so you have to add coins to the chests in such a way that, when Monocarp stops taking chests, he will have exactly $$$k$$$ coins. Calculate the minimum number of coins you have to add.",
+        "input_format": "The first line contains one integer $$$t$$$ ($$$1 \\le t \\le 1000$$$) — the number of test cases.\n\nEach test case consists of two lines:\n\n- the first line contains two integers $$$n$$$ and $$$k$$$ ($$$1 \\le n \\le 50$$$; $$$1 \\le k \\le 10^7$$$);\n- the second line contains $$$n$$$ integers $$$a_1, a_2, \\dots, a_n$$$ ($$$1 \\le a_i \\le k$$$).",
+        "output_format": "For each test case, print one integer — the minimum number of coins you have to add so that, when Monocarp stops taking the chests, he has exactly $$$k$$$ coins. It can be shown that under the constraints of the problem, it is always possible.",
+        "interaction_format": null,
+        "note": "In the first test case of the example, you don't have to add any coins. When Monocarp arrives, he will take the chest with $$$4$$$ coins, so he will have exactly $$$4$$$ coins.\n\nIn the second test case of the example, you can add $$$1$$$ coin to the $$$4$$$-th chest, so, when Monocarp arrives, he will take a chest with $$$4$$$ coins, then another chest with $$$4$$$ coins, and a chest with $$$2$$$ coins.\n\nIn the third test case of the example, you can add $$$3$$$ coins to the $$$1$$$-st chest and $$$5$$$ coins to the $$$2$$$-nd chest.\n\nIn the fourth test case of the example, you can add $$$1$$$ coin to the $$$1$$$-st chest and $$$1$$$ coin to the $$$3$$$-rd chest.",
+        "examples": [
+            {
+                "input": "4\n5 4\n4 1 2 3 2\n5 10\n4 1 2 3 2\n2 10\n1 1\n3 8\n3 3 3",
+                "output": "0\n1\n8\n2"
+            }
+        ],
+        "editorial": "Consider several first chests that Monocarp will take before exceeding the limit if we don't add any coins; so, this will be the set of several largest chests such that the sum of this set is $$$s \\le k$$$, but if the next chest is taken, the sum would exceed $$$k$$$.\nFor this set, the minimum number of coins that should be added is $$$k - s$$$. We can add exactly that amount if we increase the maximum element, the set of chests will include exactly the same elements, but now their sum is equal to $$$k$$$.\nNow we have to consider the case where we add coins to some chest that is not the part of this set. Suppose Monocarp took some chest $$$i$$$, which initially had $$$a_i$$$ coins, but did not take chest $$$j$$$, such that $$$a_j > a_i$$$. In order for chest $$$i$$$ to be taken, its final value must be at least $$$a_j$$$, as Monocarp selects the maximum chest from the available ones. Let's suppose that $$$x$$$ coins were added to chest $$$i$$$, so that $$$a_i + x \\ge a_j$$$. However, instead, we could have increased chest $$$j$$$ to the value $$$a_i + x$$$, and this would require fewer coins, since $$$a_j > a_i$$$. Thus, we have shown that it is not optimal to \"change\" the order of the chests, so we can always assume that Monocarp takes several chests that were the largest in the original order.",
+        "rating": 800,
+        "tags": [
+            "greedy",
+            "sortings"
+        ],
+        "testset_size": 10,
+        "official_tests": [
+            {
+                "input": "4\r\n5 4\r\n4 1 2 3 2\r\n5 10\r\n4 1 2 3 2\r\n2 10\r\n1 1\r\n3 8\r\n3 3 3\r\n",
+                "output": "0\r\n1\r\n8\r\n2\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2009/A",
+        "aliases": null,
+        "contest_id": "2009",
+        "contest_name": "Codeforces Round 971 (Div. 4)",
+        "contest_type": "ICPC",
+        "contest_start": 1725374100,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Minimize!",
+        "description": "You are given two integers $$$a$$$ and $$$b$$$ ($$$a \\leq b$$$). Over all possible integer values of $$$c$$$ ($$$a \\leq c \\leq b$$$), find the minimum value of $$$(c - a) + (b - c)$$$.",
+        "input_format": "The first line contains $$$t$$$ ($$$1 \\leq t \\leq 55$$$) — the number of test cases.\n\nEach test case contains two integers $$$a$$$ and $$$b$$$ ($$$1 \\leq a \\leq b \\leq 10$$$).",
+        "output_format": "For each test case, output the minimum possible value of $$$(c - a) + (b - c)$$$ on a new line.",
+        "interaction_format": null,
+        "note": "In the first test case, you can choose $$$c = 1$$$ and obtain an answer of $$$(1 - 1) + (2 - 1) = 1$$$. It can be shown this is the minimum value possible.\n\nIn the second test case, you can choose $$$c = 6$$$ and obtain an answer of $$$(6 - 3) + (10 - 6) = 7$$$. It can be shown this is the minimum value possible.",
+        "examples": [
+            {
+                "input": "3\n1 2\n3 10\n5 5",
+                "output": "1\n7\n0"
+            }
+        ],
+        "editorial": null,
+        "rating": 800,
+        "tags": [
+            "brute force",
+            "math"
+        ],
+        "testset_size": 2,
+        "official_tests": [
+            {
+                "input": "3\r\n1 2\r\n3 10\r\n5 5\r\n",
+                "output": "1\r\n7\r\n0\r\n"
+            },
+            {
+                "input": "55\r\n1 1\r\n1 2\r\n1 3\r\n1 4\r\n1 5\r\n1 6\r\n1 7\r\n1 8\r\n1 9\r\n1 10\r\n2 2\r\n2 3\r\n2 4\r\n2 5\r\n2 6\r\n2 7\r\n2 8\r\n2 9\r\n2 10\r\n3 3\r\n3 4\r\n3 5\r\n3 6\r\n3 7\r\n3 8\r\n3 9\r\n3 10\r\n4 4\r\n4 5\r\n4 6\r\n4 7\r\n4 8\r\n4 9\r\n4 10\r\n5 5\r\n5 6\r\n5 7\r\n5 8\r\n5 9\r\n5 10\r\n6 6\r\n6 7\r\n6 8\r\n6 9\r\n6 10\r\n7 7\r\n7 8\r\n7 9\r\n7 10\r\n8 8\r\n8 9\r\n8 10\r\n9 9\r\n9 10\r\n10 10\r\n",
+                "output": "0\r\n1\r\n2\r\n3\r\n4\r\n5\r\n6\r\n7\r\n8\r\n9\r\n0\r\n1\r\n2\r\n3\r\n4\r\n5\r\n6\r\n7\r\n8\r\n0\r\n1\r\n2\r\n3\r\n4\r\n5\r\n6\r\n7\r\n0\r\n1\r\n2\r\n3\r\n4\r\n5\r\n6\r\n0\r\n1\r\n2\r\n3\r\n4\r\n5\r\n0\r\n1\r\n2\r\n3\r\n4\r\n0\r\n1\r\n2\r\n3\r\n0\r\n1\r\n2\r\n0\r\n1\r\n0\r\n"
+            }
+        ],
+        "official_tests_complete": true,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2009/B",
+        "aliases": null,
+        "contest_id": "2009",
+        "contest_name": "Codeforces Round 971 (Div. 4)",
+        "contest_type": "ICPC",
+        "contest_start": 1725374100,
+        "contest_start_year": 2024,
+        "index": "B",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "osu!mania",
+        "description": "You are playing your favorite rhythm game, osu!mania. The layout of your beatmap consists of $$$n$$$ rows and $$$4$$$ columns. Because notes at the bottom are closer, you will process the bottommost row first and the topmost row last. Each row will contain exactly one note, represented as a '#'.\n\nFor each note $$$1, 2, \\dots, n$$$, in the order of processing, output the column in which the note appears.",
+        "input_format": "The first line contains $$$t$$$ ($$$1 \\leq t \\leq 100$$$) — the number of test cases.\n\nFor each test case, the first line contains $$$n$$$ ($$$1 \\leq n \\leq 500$$$) — the number of rows of the beatmap.\n\nThe following $$$n$$$ lines contain $$$4$$$ characters. The $$$i$$$-th line represents the $$$i$$$-th row of the beatmap from the top. It is guaranteed that the characters are either '.' or '#', and exactly one of the characters is '#'.\n\nIt is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$500$$$.",
+        "output_format": "For each test case, output $$$n$$$ integers on a new line, the column that the $$$i$$$-th note appears in for all $$$i$$$ from $$$1$$$ to $$$n$$$.",
+        "interaction_format": null,
+        "note": null,
+        "examples": [
+            {
+                "input": "3\n4\n#...\n.#..\n..#.\n...#\n2\n.#..\n.#..\n1\n...#",
+                "output": "4 3 2 1 \n2 2 \n4"
+            }
+        ],
+        "editorial": null,
+        "rating": 800,
+        "tags": [
+            "brute force",
+            "implementation"
+        ],
+        "testset_size": 3,
+        "official_tests": [
+            {
+                "input": "3\r\n4\r\n#...\r\n.#..\r\n..#.\r\n...#\r\n2\r\n.#..\r\n.#..\r\n1\r\n...#\r\n",
+                "output": "4 3 2 1 \r\n2 2 \r\n4 \r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2020/A",
+        "aliases": null,
+        "contest_id": "2020",
+        "contest_name": "Codeforces Round 976 (Div. 2) and Divide By Zero 9.0",
+        "contest_type": "CF",
+        "contest_start": 1727624100,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Find Minimum Operations",
+        "description": "You are given two integers $$$n$$$ and $$$k$$$.\n\nIn one operation, you can subtract any power of $$$k$$$ from $$$n$$$. Formally, in one operation, you can replace $$$n$$$ by $$$(n-k^x)$$$ for any non-negative integer $$$x$$$.\n\nFind the minimum number of operations required to make $$$n$$$ equal to $$$0$$$.",
+        "input_format": "Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \\le t \\le 10^4$$$). The description of the test cases follows.\n\nThe only line of each test case contains two integers $$$n$$$ and $$$k$$$ ($$$1 \\le n, k \\le 10^9$$$).",
+        "output_format": "For each test case, output the minimum number of operations on a new line.",
+        "interaction_format": null,
+        "note": "In the first test case, $$$n = 5$$$ and $$$k = 2$$$. We can perform the following sequence of operations:\n\n1. Subtract $$$2^0 = 1$$$ from $$$5$$$. The current value of $$$n$$$ becomes $$$5 - 1 = 4$$$.\n2. Subtract $$$2^2 = 4$$$ from $$$4$$$. The current value of $$$n$$$ becomes $$$4 - 4 = 0$$$.\n\nIt can be shown that there is no way to make $$$n$$$ equal to $$$0$$$ in less than $$$2$$$ operations. Thus, $$$2$$$ is the answer.\n\nIn the second test case, $$$n = 3$$$ and $$$k = 5$$$. We can perform the following sequence of operations:\n\n1. Subtract $$$5^0 = 1$$$ from $$$3$$$. The current value of $$$n$$$ becomes $$$3 - 1 = 2$$$.\n2. Subtract $$$5^0 = 1$$$ from $$$2$$$. The current value of $$$n$$$ becomes $$$2 - 1 = 1$$$.\n3. Subtract $$$5^0 = 1$$$ from $$$1$$$. The current value of $$$n$$$ becomes $$$1 - 1 = 0$$$.\n\nIt can be shown that there is no way to make $$$n$$$ equal to $$$0$$$ in less than $$$3$$$ operations. Thus, $$$3$$$ is the answer.",
+        "examples": [
+            {
+                "input": "6\n5 2\n3 5\n16 4\n100 3\n6492 10\n10 1",
+                "output": "2\n3\n1\n4\n21\n10"
+            }
+        ],
+        "editorial": "If $$$k$$$ is 1, we can only subtract 1 in each operation, and our answer will be $$$n$$$.\nNow, first, it can be seen that we have to apply at least $$$n$$$ mod $$$k$$$ operations of subtracting $$$k^0$$$ (as all the other operations will not change the value of $$$n$$$ mod $$$k$$$). Now, once $$$n$$$ is divisible by $$$k$$$, solving the problem for $$$n$$$ is equivalent to solving the problem for $$$\\frac{n}{k}$$$ as subtracting $$$k^0$$$ will be useless because if we apply the $$$k^0$$$ subtraction operation, then $$$n$$$ mod $$$k$$$ becomes $$$k-1$$$, and we have to apply $$$k-1$$$ more operations to make $$$n$$$ divisible by $$$k$$$ (as the final result, i.e., 0 is divisible by $$$k$$$). Instead of doing these $$$k$$$ operations of subtracting $$$k^0$$$, a single operation subtracting $$$k^1$$$ will do the job.\nSo, our final answer becomes the sum of the digits of $$$n$$$ in base $$$k$$$.",
+        "rating": 800,
+        "tags": [
+            "bitmasks",
+            "brute force",
+            "greedy",
+            "math",
+            "number theory"
+        ],
+        "testset_size": 6,
+        "official_tests": [
+            {
+                "input": "6\r\n5 2\r\n3 5\r\n16 4\r\n100 3\r\n6492 10\r\n10 1\r\n",
+                "output": "2\r\n3\r\n1\r\n4\r\n21\r\n10\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2024/A",
+        "aliases": null,
+        "contest_id": "2024",
+        "contest_name": "Codeforces Round 980 (Div. 2)",
+        "contest_type": "CF",
+        "contest_start": 1729415100,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Profitable Interest Rate",
+        "description": "Alice has $$$a$$$ coins. She can open a bank deposit called \"Profitable\", but the minimum amount required to open this deposit is $$$b$$$ coins.\n\nThere is also a deposit called \"Unprofitable\", which can be opened with any amount of coins. Alice noticed that if she opens the \"Unprofitable\" deposit with $$$x$$$ coins, the minimum amount required to open the \"Profitable\" deposit decreases by $$$2x$$$ coins. However, these coins cannot later be deposited into the \"Profitable\" deposit.\n\nHelp Alice determine the maximum number of coins she can deposit into the \"Profitable\" deposit if she first deposits some amount of coins (possibly $$$0$$$) into the \"Unprofitable\" deposit. If Alice can never open the \"Profitable\" deposit, output $$$0$$$.",
+        "input_format": "Each test consists of multiple test cases. The first line contains a single integer $$$t$$$ ($$$1 \\le t \\le 10^4$$$) — the number of test cases. The description of the test cases follows.\n\nA single line of each test case contains two integers $$$a$$$ and $$$b$$$ ($$$1 \\le a, b \\le 10^9$$$) — the number of coins Alice has and the initial minimum amount required to open the \"Profitable\" deposit.",
+        "output_format": "For each test case, output a single integer — the maximum number of coins that Alice can deposit into the \"Profitable\" deposit. If Alice can never open the \"Profitable\" deposit, output $$$0$$$.",
+        "interaction_format": null,
+        "note": "In the first test case, $$$a \\ge b$$$, so Alice can immediately open the \"Profitable\" deposit with all $$$10$$$ coins.\n\nIn the second test case, Alice can open the \"Unprofitable\" deposit with $$$2$$$ coins. Then she will have $$$5$$$ coins left, but the minimum amount required to open the \"Profitable\" deposit will decrease by $$$4$$$ coins, making it equal to $$$5$$$ coins. Thus, Alice will be able to open the \"Profitable\" deposit with $$$5$$$ coins.\n\nIn the third test case, Alice will not be able to open the \"Profitable\" deposit.",
+        "examples": [
+            {
+                "input": "5\n10 5\n7 9\n5 100\n1 1\n1 2",
+                "output": "10\n5\n0\n1\n0"
+            }
+        ],
+        "editorial": "Let's say we have deposited $$$x$$$ coins into the \"Unprofitable\" deposit, then we can open a \"Profitable\" deposit if $$$a - x \\geq b - 2x$$$ is satisfied. Which is equivalent to the inequality: $$$x \\geq b - a$$$. Thus, we need to open an \"Unprofitable\" deposit for $$$\\text{max}(0, b - a)$$$ coins, and open a \"Profitable\" deposit for the rest of the coins.",
+        "rating": 800,
+        "tags": [
+            "greedy",
+            "math"
+        ],
+        "testset_size": 3,
+        "official_tests": [
+            {
+                "input": "5\r\n10 5\r\n7 9\r\n5 100\r\n1 1\r\n1 2\r\n",
+                "output": "10\r\n5\r\n0\r\n1\r\n0\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2039/A",
+        "aliases": null,
+        "contest_id": "2039",
+        "contest_name": "CodeTON Round 9 (Div. 1 + Div. 2, Rated, Prizes!)",
+        "contest_type": "CF",
+        "contest_start": 1732372500,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Shohag Loves Mod",
+        "description": "Shohag has an integer $$$n$$$. Please help him find an increasing integer sequence $$$1 \\le a_1 \\lt a_2 \\lt \\ldots \\lt a_n \\le 100$$$ such that $$$a_i \\bmod i \\neq a_j \\bmod j$$$ $$$^{\\text{∗}}$$$ is satisfied over all pairs $$$1 \\le i \\lt j \\le n$$$.\n\nIt can be shown that such a sequence always exists under the given constraints.",
+        "input_format": "The first line contains a single integer $$$t$$$ ($$$1 \\le t \\le 50$$$) — the number of test cases.\n\nThe first and only line of each test case contains an integer $$$n$$$ ($$$2 \\le n \\le 50$$$).",
+        "output_format": "For each test case, print $$$n$$$ integers — the integer sequence that satisfies the conditions mentioned in the statement. If there are multiple such sequences, output any.",
+        "interaction_format": null,
+        "note": "In the first test case, the sequence is increasing, values are from $$$1$$$ to $$$100$$$ and each pair of indices satisfies the condition mentioned in the statement:\n\n- For pair $$$(1, 2)$$$, $$$a_1 \\bmod 1 = 2 \\bmod 1 = 0$$$, and $$$a_2 \\bmod 2 = 7 \\bmod 2 = 1$$$. So they are different.\n- For pair $$$(1, 3)$$$, $$$a_1 \\bmod 1 = 2 \\bmod 1 = 0$$$, and $$$a_3 \\bmod 3 = 8 \\bmod 3 = 2$$$. So they are different.\n- For pair $$$(2, 3)$$$, $$$a_2 \\bmod 2 = 7 \\bmod 2 = 1$$$, and $$$a_3 \\bmod 3 = 8 \\bmod 3 = 2$$$. So they are different.\n\nNote that you do not necessarily have to print the exact same sequence, you can print any other sequence as long as it satisfies the necessary conditions.",
+        "examples": [
+            {
+                "input": "2\n3\n6",
+                "output": "2 7 8\n2 3 32 35 69 95"
+            }
+        ],
+        "editorial": "THOUGHT: A general approach to tackle ad-hoc problems is to play around with the conditions and see if we can add more constraints to limit the search space.\nACTION: Let's analyze the modular condition $$$a_i \\bmod i$$$.\nWe know that $$$a_i \\bmod i < i$$$, and all $$$a_i \\bmod i$$$ values are distinct. Let's explore this step by step:\n• $$$a_1 \\bmod 1$$$ is always $$$0$$$.\n• $$$a_2 \\bmod 2$$$ can be $$$0$$$ or $$$1$$$. But since all $$$a_i \\bmod i$$$ values must be distinct, $$$a_2 \\bmod 2$$$ must be $$$1$$$ (otherwise, it would equal $$$a_1 \\bmod 1$$$).\n• $$$a_3 \\bmod 3$$$ can be $$$0$$$, $$$1$$$, or $$$2$$$. Similarly, $$$a_3 \\bmod 3$$$ must be $$$2$$$ (to avoid duplication with $$$a_1 \\bmod 1$$$ or $$$a_2 \\bmod 2$$$).\n• $$$\\ldots$$$\n• $$$a_n \\bmod n$$$ must be $$$n-1$$$.\nOBSERVATION: This leads to the constraint $$$a_i \\bmod i = i - 1$$$.\nTHOUGHT: Next, let's consider the increasing sequence condition.\nOBSERVATION: Since the sequence must be increasing, we add the constraint $$$a_i \\ge i$$$.\nTHOUGHT: To further limit the search space, note that $$$n$$$ can be up to $$$50$$$, and $$$a_i$$$ must be $$$\\le 100$$$.\nOBSERVATION: This suggests that we can restrict $$$a_i$$$ to values up to $$$2 \\cdot n$$$.\nTHOUGHT: Let's compile the constraints:\n1. $$$a_i \\bmod i = i - 1$$$.\n2. $$$a_i \\ge i$$$.\n3. $$$a_i \\le 2 \\cdot n$$$.\nNow we need to build the sequence that satisfies these conditions.\nACTION: Let's build the sequence starting from the end.\n• $$$a_n \\bmod n = n-1$$$. Thus, $$$a_n$$$ can be $$$n-1$$$ or $$$2 \\cdot n - 1$$$. Since $$$a_n \\ge n$$$, $$$a_n$$$ must be $$$2 \\cdot n - 1$$$.\n• $$$a_{n-1} \\bmod (n-1) = n-2$$$. So $$$a_{n-1}$$$ can be $$$n-2$$$ or $$$(n - 2) + (n - 1) = 2 \\cdot n - 3$$$. Since $$$a_{n-1} \\ge n-1$$$, $$$a_{n-1}$$$ must be $$$2 \\cdot n - 3$$$.\n• $$$\\ldots$$$\n• $$$a_i \\bmod i = i - 1$$$. Thus, $$$a_i$$$ can be $$$i - 1$$$ or $$$2 \\cdot i - 1$$$ or $$$3 \\cdot i - 1 \\ldots$$$. Since $$$a_i \\ge i$$$ and the odd numbers greater than $$$2 \\cdot i$$$ have already been used, $$$a_i$$$ must be $$$2 \\cdot i - 1$$$.\nOBSERVATION: If we limit the elements to $$$2 \\cdot n$$$, there is exactly one sequence that satisfies all conditions.\nCONCLUSION: The sequence is $$$a_i = 2 \\cdot i - 1$$$, which is $$$1, 3, 5, \\ldots, 2 \\cdot n - 1$$$.\nVALIDATION: We can validate the solution by checking if it satisfies all the constraints. $$$a_i \\bmod i = (2 \\cdot i - 1) \\bmod i = i - 1$$$. So all $$$a_i \\bmod i$$$ are distinct and values are under $$$2 \\cdot n$$$. So the sequence is valid.\nTime Complexity: $$$\\mathcal{O}(n)$$$.",
+        "rating": 800,
+        "tags": [
+            "constructive algorithms",
+            "number theory"
+        ],
+        "testset_size": 2,
+        "official_tests": [
+            {
+                "input": "2\r\n3\r\n6\r\n",
+                "output": "1 3 5 \r\n1 3 5 7 9 11 \r\n"
+            },
+            {
+                "input": "50\r\n50\r\n2\r\n3\r\n4\r\n5\r\n6\r\n7\r\n8\r\n9\r\n10\r\n11\r\n12\r\n13\r\n14\r\n15\r\n16\r\n17\r\n18\r\n19\r\n20\r\n21\r\n22\r\n23\r\n24\r\n25\r\n26\r\n27\r\n28\r\n29\r\n30\r\n31\r\n32\r\n33\r\n34\r\n35\r\n36\r\n37\r\n38\r\n39\r\n40\r\n41\r\n42\r\n43\r\n44\r\n45\r\n46\r\n47\r\n48\r\n49\r\n50\r\n",
+                "output": "1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 \n1 3 \n1 3 5 \n1 3 5 7 \n1 3 5 7 9 \n1 3 5 7 9 11 \n1 3 5 7 9 11 13 \n1 3 5 7 9 11 13 15 \n1 3 5 7 9 11 13 15 17 \n1 3 5 7 9 11 13 15 17 19 \n1 3 5 7 9 11 13 15 17 19 21 \n1 3 5 7 9 11 13 15 17 19 21 23 \n1 3 5 7 9 11 13 15 17 19 21 23 25 \n1 3 5 7 9 11 13 15 17 19 21 23 25 27 \n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 \n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 \n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 \n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 \n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 \n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 \n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 \n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 \n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 \n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 \n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 \n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 \n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 \n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 \n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 \n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 \n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 \n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 \n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 \n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 \n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 \n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 \n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 \n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 \n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 \n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 \n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 \n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 \n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 \n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 \n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 \n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 \n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 \n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 \n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 \n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 \n"
+            }
+        ],
+        "official_tests_complete": true,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2008/A",
+        "aliases": null,
+        "contest_id": "2008",
+        "contest_name": "Codeforces Round 970 (Div. 3)",
+        "contest_type": "ICPC",
+        "contest_start": 1725201300,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Sakurako's Exam",
+        "description": "Today, Sakurako has a math exam. The teacher gave the array, consisting of $$$a$$$ ones and $$$b$$$ twos.\n\nIn an array, Sakurako must place either a '+' or a '-' in front of each element so that the sum of all elements in the array equals $$$0$$$.\n\nSakurako is not sure if it is possible to solve this problem, so determine whether there is a way to assign signs such that the sum of all elements in the array equals $$$0$$$.",
+        "input_format": "The first line contains a single integer $$$t$$$ ($$$1\\le t\\le 100$$$)  — the number of test cases.\n\nThe only line of each test case contains two integers $$$a$$$ and $$$b$$$ ($$$0\\le a,b<10$$$)  — the number of '1's and the number of '2's in the array.",
+        "output_format": "For each test case, output \"Yes\" if you can make the sum of the entire array equal to $$$0$$$, and \"No\" otherwise.\n\nYou can output each letter in any case (lowercase or uppercase). For example, the strings \"yEs\", \"yes\", \"Yes\", and \"YES\" will be accepted as a positive answer.",
+        "interaction_format": null,
+        "note": "1. $$$a=0$$$, $$$b=1$$$: This means the array is $$$[2]$$$ — it is impossible to add the signs '+' or '-' to get $$$0$$$ as a result;\n2. $$$a=0$$$, $$$b=3$$$: This means the array is $$$[2, 2, 2]$$$ — it is impossible to add the signs '+' or '-' to get $$$0$$$ as a result;\n3. $$$a=2$$$, $$$b=0$$$: This means the array is $$$[1, 1]$$$ — it is possible to add the signs '+' or '-' to get $$$0$$$ as a result ($$$+1-1=0$$$);\n4. $$$a=2$$$, $$$b=3$$$: This means the array is $$$[1, 1, 2, 2, 2]$$$ — it is possible to add the signs '+' or '-' to get $$$0$$$ as a result ($$$+1+1-2-2+2=0$$$);",
+        "examples": [
+            {
+                "input": "5\n0 1\n0 3\n2 0\n2 3\n3 1",
+                "output": "NO\nNO\nYES\nYES\nNO"
+            }
+        ],
+        "editorial": "First of all, this task is the same as: \"divide an array into two arrays with equal sum\". So, obviously, we need to check if the sum of all elements is even which implies that the number of ones is even. Then, we can out half of 2s in one array and the other half in another, but if number of 2s is odd, then one array will have a greater sum then another, so we need to put two 1s there.\nSo, if we don't have two ones while the number of 2s is odd then the answer is \"NO\". Also, if sum is odd, the answer is also \"NO\". In all other cases, the answer is \"YES\".",
+        "rating": 800,
+        "tags": [
+            "brute force",
+            "constructive algorithms",
+            "greedy",
+            "math"
+        ],
+        "testset_size": 5,
+        "official_tests": [
+            {
+                "input": "5\r\n0 1\r\n0 3\r\n2 0\r\n2 3\r\n3 1\r\n",
+                "output": "NO\r\nNO\r\nYES\r\nYES\r\nNO\r\n"
+            },
+            {
+                "input": "100\r\n0 0\r\n0 1\r\n0 2\r\n0 3\r\n0 4\r\n0 5\r\n0 6\r\n0 7\r\n0 8\r\n0 9\r\n1 0\r\n1 1\r\n1 2\r\n1 3\r\n1 4\r\n1 5\r\n1 6\r\n1 7\r\n1 8\r\n1 9\r\n2 0\r\n2 1\r\n2 2\r\n2 3\r\n2 4\r\n2 5\r\n2 6\r\n2 7\r\n2 8\r\n2 9\r\n3 0\r\n3 1\r\n3 2\r\n3 3\r\n3 4\r\n3 5\r\n3 6\r\n3 7\r\n3 8\r\n3 9\r\n4 0\r\n4 1\r\n4 2\r\n4 3\r\n4 4\r\n4 5\r\n4 6\r\n4 7\r\n4 8\r\n4 9\r\n5 0\r\n5 1\r\n5 2\r\n5 3\r\n5 4\r\n5 5\r\n5 6\r\n5 7\r\n5 8\r\n5 9\r\n6 0\r\n6 1\r\n6 2\r\n6 3\r\n6 4\r\n6 5\r\n6 6\r\n6 7\r\n6 8\r\n6 9\r\n7 0\r\n7 1\r\n7 2\r\n7 3\r\n7 4\r\n7 5\r\n7 6\r\n7 7\r\n7 8\r\n7 9\r\n8 0\r\n8 1\r\n8 2\r\n8 3\r\n8 4\r\n8 5\r\n8 6\r\n8 7\r\n8 8\r\n8 9\r\n9 0\r\n9 1\r\n9 2\r\n9 3\r\n9 4\r\n9 5\r\n9 6\r\n9 7\r\n9 8\r\n9 9\r\n",
+                "output": "YES\r\nNO\r\nYES\r\nNO\r\nYES\r\nNO\r\nYES\r\nNO\r\nYES\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nYES\r\nYES\r\nYES\r\nYES\r\nYES\r\nYES\r\nYES\r\nYES\r\nYES\r\nYES\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nYES\r\nYES\r\nYES\r\nYES\r\nYES\r\nYES\r\nYES\r\nYES\r\nYES\r\nYES\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nYES\r\nYES\r\nYES\r\nYES\r\nYES\r\nYES\r\nYES\r\nYES\r\nYES\r\nYES\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nYES\r\nYES\r\nYES\r\nYES\r\nYES\r\nYES\r\nYES\r\nYES\r\nYES\r\nYES\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\n"
+            },
+            {
+                "input": "100\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n9 9\r\n",
+                "output": "NO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\n"
+            },
+            {
+                "input": "3\r\n0 1\r\n2 0\r\n3 1\r\n",
+                "output": "NO\r\nYES\r\nNO\r\n"
+            },
+            {
+                "input": "10\r\n2 3\r\n3 2\r\n1 2\r\n2 7\r\n1 4\r\n4 7\r\n3 7\r\n5 5\r\n2 6\r\n6 8\r\n",
+                "output": "YES\r\nNO\r\nNO\r\nYES\r\nNO\r\nYES\r\nNO\r\nNO\r\nYES\r\nYES\r\n"
+            }
+        ],
+        "official_tests_complete": true,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2008/B",
+        "aliases": [
+            "2012/B",
+            "2017/B"
+        ],
+        "contest_id": "2008",
+        "contest_name": "Codeforces Round 970 (Div. 3)",
+        "contest_type": "ICPC",
+        "contest_start": 1725201300,
+        "contest_start_year": 2024,
+        "index": "B",
+        "time_limit": 2.0,
+        "memory_limit": 256.0,
+        "title": "Square or Not",
+        "description": "A beautiful binary matrix is a matrix that has ones on its edges and zeros inside.\n\nExamples of four beautiful binary matrices.\n\nToday, Sakurako was playing with a beautiful binary matrix of size $$$r \\times c$$$ and created a binary string $$$s$$$ by writing down all the rows of the matrix, starting from the first and ending with the $$$r$$$-th. More formally, the element from the matrix in the $$$i$$$-th row and $$$j$$$-th column corresponds to the $$$((i-1)*c+j)$$$-th element of the string.\n\nYou need to check whether the beautiful matrix from which the string $$$s$$$ was obtained could be squared. In other words, you need to check whether the string $$$s$$$ could have been build from a square beautiful binary matrix (i.e., one where $$$r=c$$$).",
+        "input_format": "The first line contains a single integer $$$t$$$ ($$$1 \\le t \\le 10^4$$$) — the number of test cases.\n\nThe first line of each test case contains a single integer $$$n$$$ ($$$2 \\le n \\le 2 \\cdot 10^5$$$) — the length of the string.\n\nThe second line of each test case contains the string $$$s$$$ of length $$$n$$$. The string is always the result of writing out the strings of a beautiful matrix.\n\nIt is guaranteed that the sum of $$$n$$$ across all test cases does not exceed $$$2 \\cdot 10^5$$$.",
+        "output_format": "Print \"Yes\", if the original matrix could have been square, and \"No\" otherwise.",
+        "interaction_format": null,
+        "note": "For the second test case, string 1111 can be obtained from the matrix:\n\n$$$1$$$$$$1$$$$$$1$$$$$$1$$$\n\nFor the third test case, string 111101111 can be obtained from the matrix:\n\n$$$1$$$$$$1$$$$$$1$$$$$$1$$$$$$0$$$$$$1$$$$$$1$$$$$$1$$$$$$1$$$\n\nThere is no square matrix in the fourth case, such that the string can be obtained from it.",
+        "examples": [
+            {
+                "input": "5\n2\n11\n4\n1111\n9\n111101111\n9\n111111111\n12\n111110011111",
+                "output": "No\nYes\nYes\nNo\nNo"
+            }
+        ],
+        "editorial": "Assume that string was created from the beautiful binary matrix with size $$$r\\times c$$$.\nIf $$$r\\le 2$$$ or $$$c\\le 2$$$, then the whole matrix consists of '1'. This means that the string will have only one character and this is the only case such happening. So, if the whole string is constructed out of '1', we print \"Yes\" only if the size of the string is 4, since only $$$r=c=2$$$ is a good matrix for us.\nOtherwise, we have at least one '0' in the string. Let's look at what is the index of the first '0'. If it has index $$$r+1$$$, since the whole first line and the first character of the first line equal to '1', so now, we have a fixed value of $$$r$$$ (index of the first '0' minus 1) and the answer is \"Yes\" only if $$$r$$$ is the square root of $$$n$$$.",
+        "rating": 800,
+        "tags": [
+            "brute force",
+            "math",
+            "strings"
+        ],
+        "testset_size": 13,
+        "official_tests": [
+            {
+                "input": "5\r\n2\r\n11\r\n4\r\n1111\r\n9\r\n111101111\r\n9\r\n111111111\r\n12\r\n111110011111\r\n",
+                "output": "No\r\nYes\r\nYes\r\nNo\r\nNo\r\n"
+            },
+            {
+                "input": "1\r\n64\r\n1111111111111111100000000000000110000000000000011111111111111111\r\n",
+                "output": "No\r\n"
+            },
+            {
+                "input": "1\r\n64\r\n1111100110011001100110011001100110011001100110011001100110011111\r\n",
+                "output": "No\r\n"
+            },
+            {
+                "input": "1\r\n36\r\n111101101101101101101101101101101111\r\n",
+                "output": "No\r\n"
+            },
+            {
+                "input": "3\r\n64\r\n1111100110011001100110011001100110011001100110011001100110011111\r\n64\r\n1111111111111111100000000000000110000000000000011111111111111111\r\n64\r\n1111111110000001100000011000000110000001100000011000000111111111\r\n",
+                "output": "No\r\nNo\r\nYes\r\n"
+            },
+            {
+                "input": "1\r\n36\r\n111111111100000001100000001111111111\r\n",
+                "output": "No\r\n"
+            },
+            {
+                "input": "1\r\n36\r\n111111111111100000000001111111111111\r\n",
+                "output": "No\r\n"
+            },
+            {
+                "input": "3\r\n64\r\n1111111111111111100000000000000110000000000000011111111111111111\r\n64\r\n1111111110000001100000011000000110000001100000011000000111111111\r\n64\r\n1111100110011001100110011001100110011001100110011001100110011111\r\n",
+                "output": "No\r\nYes\r\nNo\r\n"
+            },
+            {
+                "input": "1\r\n36\r\n111110011001100110011001100110011111\r\n",
+                "output": "No\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2008/C",
+        "aliases": null,
+        "contest_id": "2008",
+        "contest_name": "Codeforces Round 970 (Div. 3)",
+        "contest_type": "ICPC",
+        "contest_start": 1725201300,
+        "contest_start_year": 2024,
+        "index": "C",
+        "time_limit": 2.0,
+        "memory_limit": 256.0,
+        "title": "Longest Good Array",
+        "description": "Today, Sakurako was studying arrays. An array $$$a$$$ of length $$$n$$$ is considered good if and only if:\n\n- the array $$$a$$$ is increasing, meaning $$$a_{i - 1} < a_i$$$ for all $$$2 \\le i \\le n$$$;\n- the differences between adjacent elements are increasing, meaning $$$a_i - a_{i-1} < a_{i+1} - a_i$$$ for all $$$2 \\le i < n$$$.\n\nSakurako has come up with boundaries $$$l$$$ and $$$r$$$ and wants to construct a good array of maximum length, where $$$l \\le a_i \\le r$$$ for all $$$a_i$$$.\n\nHelp Sakurako find the maximum length of a good array for the given $$$l$$$ and $$$r$$$.",
+        "input_format": "The first line contains a single integer $$$t$$$ ($$$1\\le t\\le 10^4$$$)  — the number of test cases.\n\nThe only line of each test case contains two integers $$$l$$$ and $$$r$$$ ($$$1\\le l\\le r\\le 10^9$$$).",
+        "output_format": "For each test case, output a single integer  — the length of the longest good array Sakurako can form given $$$l$$$ and $$$r$$$.",
+        "interaction_format": null,
+        "note": "For $$$l=1$$$ and $$$r=5$$$, one possible array could be $$$(1,2,5)$$$. It can be proven that an array of length $$$4$$$ does not exist for the given $$$l$$$ and $$$r$$$.\n\nFor $$$l=2$$$ and $$$r=2$$$, the only possible array is $$$(2)$$$.\n\nFor $$$l=10$$$ and $$$r=20$$$, the only possible array is $$$(10,11,13,16,20)$$$.",
+        "examples": [
+            {
+                "input": "5\n1 2\n1 5\n2 2\n10 20\n1 1000000000",
+                "output": "2\n3\n1\n5\n44721"
+            }
+        ],
+        "editorial": "We can solve this problem greedily. Let's choose the first element equal to $$$l$$$. Then, the second element should be $$$l+1$$$. The third $$$l+3$$$, and so on. In general, the $$$i-$$$th element is equal to $$$l+\\frac{i\\cdot (i+1)}{2}$$$.\nProof of this solution:\nAssume that array $$$a$$$ is the array made by our algorithm and $$$b$$$ is the array with a better answer. This means that $$$len(b)>len(a)$$$. By the construction of $$$a$$$, there exists an integer $$$i$$$ such that for all $$$j<i$$$, $$$a_j=b_j$$$ and $$$a_i<b_i$$$, because $$$a_i$$$ we choose as the smallest possible element. WLOG assume that $$$len(b)=len(a)+1=n$$$. Then $$$b_n-b_{n-1}>b_{n-1}-b_{n-2}\\ge a_{n-1}-a_{n-2}$$$. So, we can append $$$b_n$$$ to the array $$$a$$$, which leads to a contradiction.\nNow, the task is to find the biggest $$$x$$$ such that $$$l+\\frac{x\\cdot (x+1)}{2}\\le r$$$. In fact, it can be found by binary search, the formula of discriminant, or just by brute force.",
+        "rating": 800,
+        "tags": [
+            "binary search",
+            "brute force",
+            "math"
+        ],
+        "testset_size": 10,
+        "official_tests": [
+            {
+                "input": "5\r\n1 2\r\n1 5\r\n2 2\r\n10 20\r\n1 1000000000\r\n",
+                "output": "2\r\n3\r\n1\r\n5\r\n44721\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2043/A",
+        "aliases": null,
+        "contest_id": "2043",
+        "contest_name": "Educational Codeforces Round 173 (Rated for Div. 2)",
+        "contest_type": "ICPC",
+        "contest_start": 1735050900,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 2.0,
+        "memory_limit": 512.0,
+        "title": "Coin Transformation",
+        "description": "Initially, you have a coin with value $$$n$$$. You can perform the following operation any number of times (possibly zero):\n\n- transform one coin with value $$$x$$$, where $$$x$$$ is greater than $$$3$$$ ($$$x>3$$$), into two coins with value $$$\\lfloor \\frac{x}{4} \\rfloor$$$.\n\nWhat is the maximum number of coins you can have after performing this operation any number of times?",
+        "input_format": "The first line contains one integer $$$t$$$ ($$$1 \\le t \\le 10^4$$$) — the number of test cases.\n\nEach test case consists of one line containing one integer $$$n$$$ ($$$1 \\le n \\le 10^{18}$$$).",
+        "output_format": "For each test case, print one integer — the maximum number of coins you can have after performing the operation any number of times.",
+        "interaction_format": null,
+        "note": "In the first example, you have a coin of value $$$1$$$, and you can't do anything with it. So, the answer is $$$1$$$.\n\nIn the second example, you can transform a coin of value $$$5$$$ into two coins with value $$$1$$$.\n\nIn the third example, you can transform a coin of value $$$16$$$ into two coins with value $$$4$$$. Each of the resulting coins can be transformed into two coins with value $$$1$$$.",
+        "examples": [
+            {
+                "input": "4\n1\n5\n16\n1000000000000000000",
+                "output": "1\n2\n4\n536870912"
+            }
+        ],
+        "editorial": "Let's try to solve this problem \"naively\": obviously, while we have at least one coin with value $$$>3$$$, we should transform it, since it increases the number of coins we get. We can simulate this process, but the number of transformations we get might be really large, so we need to speed this up.\nLet's make it faster the following way: instead of transforming just one coin, we will transform all coins at once. So, after one operation, we will have $$$2$$$ coins with value $$$\\lfloor \\frac{x}{4} \\rfloor$$$; after two operations, we will have $$$4$$$ coins with value $$$\\lfloor \\frac{x}{16} \\rfloor$$$ each, and so on. This can be implemented using a simple while-loop: while the value of our coins is greater than $$$3$$$, we divide it by $$$4$$$ and double the number of coins. This solution works in $$$O(\\log n)$$$.\nIt is also possible to derive a formula for the answer: the number of times we need to divide a number by $$$4$$$ so it becomes less than $$$4$$$ is $$$\\lfloor \\log_4 n \\rfloor$$$, and the number of coins we will get is $$$2$$$ to the power of that expression. However, you must be very careful with this approach, because it can have severe precision issues due to the fact that standard logarithm functions work with floating-point numbers, so they are imprecise. You should use some way to calculate $$$\\lfloor \\log_4 n \\rfloor$$$ without floating-point calculations; for example, iterating (or binary searching) on the power of $$$4$$$ you need to divide the number by so that it becomes less than $$$4$$$.",
+        "rating": 800,
+        "tags": [
+            "brute force",
+            "math"
+        ],
+        "testset_size": 4,
+        "official_tests": [
+            {
+                "input": "4\r\n1\r\n5\r\n16\r\n1000000000000000000\r\n",
+                "output": "1\r\n2\r\n4\r\n536870912\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2021/A",
+        "aliases": null,
+        "contest_id": "2021",
+        "contest_name": "Codeforces Round 977 (Div. 2, based on COMPFEST 16 - Final Round)",
+        "contest_type": "CF",
+        "contest_start": 1728194700,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Meaning Mean",
+        "description": "Pak Chanek has an array $$$a$$$ of $$$n$$$ positive integers. Since he is currently learning how to calculate the floored average of two numbers, he wants to practice it on his array $$$a$$$.\n\nWhile the array $$$a$$$ has at least two elements, Pak Chanek will perform the following three-step operation:\n\n1. Pick two different indices $$$i$$$ and $$$j$$$ ($$$1 \\leq i, j \\leq |a|$$$; $$$i \\neq j$$$), note that $$$|a|$$$ denotes the current size of the array $$$a$$$.\n2. Append $$$\\lfloor \\frac{a_i+a_j}{2} \\rfloor$$$$$$^{\\text{∗}}$$$ to the end of the array.\n3. Remove elements $$$a_i$$$ and $$$a_j$$$ from the array and concatenate the remaining parts of the array.\n\nFor example, suppose that $$$a=[5,4,3,2,1,1]$$$. If we choose $$$i=1$$$ and $$$j=5$$$, the resulting array will be $$$a=[4,3,2,1,3]$$$. If we choose $$$i=4$$$ and $$$j=3$$$, the resulting array will be $$$a=[5,4,1,1,2]$$$.\n\nAfter all operations, the array will consist of a single element $$$x$$$. Find the maximum possible value of $$$x$$$ if Pak Chanek performs the operations optimally.",
+        "input_format": "Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \\le t \\le 5000$$$). The description of the test cases follows.\n\nThe first line of each test case contains a single integer $$$n$$$ ($$$2 \\le n \\le 50$$$) — the length of the array $$$a$$$.\n\nThe second line of each test case contains $$$n$$$ integers $$$a_1, a_2, \\ldots, a_n$$$ ($$$1 \\le a_i \\le 10^9$$$) — the elements of the array $$$a$$$.\n\nDo note that the sum of $$$n$$$ over all test cases is not bounded.",
+        "output_format": "For each test case, output a single integer: the maximum possible value of $$$x$$$ after all numbers have been picked.",
+        "interaction_format": null,
+        "note": "In the first test case, the array is initially $$$a=[1,7,8,4,5]$$$. Pak Chanek will perform the following operations:\n\n1. Pick $$$i=1$$$ and $$$j=2$$$, then $$$a=[8,4,5,4]$$$.\n2. Pick $$$i=3$$$ and $$$j=2$$$, then $$$a=[8,4,4]$$$.\n3. Pick $$$i=2$$$ and $$$j=3$$$, then $$$a=[8,4]$$$.\n4. Pick $$$i=1$$$ and $$$j=2$$$, then $$$a=[6]$$$.\n\nAfter all the operations, the array consists of a single element $$$x=6$$$. It can be proven that there is no series of operations that results in $$$x$$$ greater than $$$6$$$ in the end.",
+        "examples": [
+            {
+                "input": "3\n5\n1 7 8 4 5\n3\n2 6 5\n5\n5 5 5 5 5",
+                "output": "6\n4\n5"
+            }
+        ],
+        "editorial": "For now, let's ignore the floor operation, so an operation is merging two elements $$$a_i$$$ and $$$a_j$$$ into one element $$$\\frac{a_i+a_j}{2}$$$.\nConsider the end result. Each initial element in $$$a$$$ must contribute a fractional coefficient to the final result. It turns out that the sum of the coefficients is fixed (it must be $$$1$$$). That means we can greedily give the biggest values in $$$a$$$ the biggest coefficients.\nOne way to do this is by sorting $$$a$$$ in ascending order. We merge $$$a_1$$$ and $$$a_2$$$, then merge that result with $$$a_3$$$, then merge that result with $$$a_4$$$, and so on until $$$a_n$$$. If we do this, $$$a_n$$$ contributes $$$\\frac{1}{2}$$$ times, $$$a_{n-1}$$$ contributes $$$\\frac{1}{4}$$$ times, $$$a_{n-2}$$$ contributes $$$\\frac{1}{8}$$$ times, and so on. This is the optimal way to get the maximum final result.\nIt turns out that the strategy above is also the optimal strategy for the original version of the problem. So we can simulate that process to get the answer.\nTime complexity for each test case: $$$O(n\\log n)$$$",
+        "rating": 800,
+        "tags": [
+            "data structures",
+            "greedy",
+            "math",
+            "sortings"
+        ],
+        "testset_size": 5,
+        "official_tests": [
+            {
+                "input": "3\r\n5\r\n1 7 8 4 5\r\n3\r\n2 6 5\r\n5\r\n5 5 5 5 5\r\n",
+                "output": "6\r\n4\r\n5\r\n"
+            },
+            {
+                "input": "1\r\n50\r\n999999998 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999 999999999\r\n",
+                "output": "999999998\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2010/A",
+        "aliases": null,
+        "contest_id": "2010",
+        "contest_name": "Testing Round 19 (Div. 3)",
+        "contest_type": "ICPC",
+        "contest_start": 1724877300,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 2.0,
+        "memory_limit": 256.0,
+        "title": "Alternating Sum of Numbers",
+        "description": "You are given a sequence of integers. Output the alternating sum of this sequence. In other words, output $$$a_1 - a_2 + a_3 - a_4 + a_5 - \\dots$$$. That is, the signs of plus and minus alternate, starting with a plus.",
+        "input_format": "The first line of the test contains one integer $$$t$$$ ($$$1 \\le t \\le 1000$$$) — the number of test cases. Then follow $$$t$$$ test cases.\n\nThe first line of each test case contains one integer $$$n$$$ ($$$1 \\le n \\le 50$$$) — the length of the sequence. The second line of the test case contains $$$n$$$ integers $$$a_1, a_2, \\dots, a_n$$$ ($$$1 \\le a_i \\le 100$$$).",
+        "output_format": "Output $$$t$$$ lines. For each test case, output the required alternating sum of the numbers.",
+        "interaction_format": null,
+        "note": null,
+        "examples": [
+            {
+                "input": "4\n4\n1 2 3 17\n1\n100\n2\n100 100\n5\n3 1 4 1 5",
+                "output": "-15\n100\n0\n10"
+            }
+        ],
+        "editorial": null,
+        "rating": 800,
+        "tags": [
+            "brute force",
+            "implementation",
+            "math"
+        ],
+        "testset_size": 6,
+        "official_tests": [
+            {
+                "input": "4\r\n4\r\n1 2 3 17\r\n1\r\n100\r\n2\r\n100 100\r\n5\r\n3 1 4 1 5\r\n",
+                "output": "-15\r\n100\r\n0\r\n10\r\n"
+            },
+            {
+                "input": "10\r\n4\r\n1 3 1 3\r\n3\r\n4 4 4\r\n5\r\n4 5 5 1 3\r\n5\r\n5 5 5 4 1\r\n2\r\n5 4\r\n4\r\n1 1 5 1\r\n1\r\n1\r\n5\r\n5 1 3 5 3\r\n3\r\n4 4 4\r\n1\r\n1\r\n",
+                "output": "-4\r\n4\r\n6\r\n2\r\n1\r\n4\r\n1\r\n5\r\n4\r\n1\r\n"
+            },
+            {
+                "input": "1\r\n1\r\n100\r\n",
+                "output": "100\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2010/B",
+        "aliases": null,
+        "contest_id": "2010",
+        "contest_name": "Testing Round 19 (Div. 3)",
+        "contest_type": "ICPC",
+        "contest_start": 1724877300,
+        "contest_start_year": 2024,
+        "index": "B",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Three Brothers",
+        "description": "Three brothers agreed to meet. Let's number the brothers as follows: the oldest brother is number 1, the middle brother is number 2, and the youngest brother is number 3.\n\nWhen it was time for the meeting, one of the brothers was late. Given the numbers of the two brothers who arrived on time, you need to determine the number of the brother who was late.",
+        "input_format": "The first line of input contains two different integers a and b (1 ≤ a, b ≤ 3, a ≠ b) — the numbers of the brothers who arrived on time. The numbers are given in arbitrary order.",
+        "output_format": "Output a single integer — the number of the brother who was late to the meeting.",
+        "interaction_format": null,
+        "note": null,
+        "examples": [
+            {
+                "input": "3 1",
+                "output": "2"
+            }
+        ],
+        "editorial": null,
+        "rating": 800,
+        "tags": [
+            "brute force",
+            "implementation",
+            "math"
+        ],
+        "testset_size": 6,
+        "official_tests": [
+            {
+                "input": "3 1\r\n",
+                "output": "2\r\n"
+            },
+            {
+                "input": "2 1\r\n",
+                "output": "3\r\n"
+            },
+            {
+                "input": "2 3\r\n",
+                "output": "1\r\n"
+            },
+            {
+                "input": "1 2\r\n",
+                "output": "3\r\n"
+            },
+            {
+                "input": "1 3\r\n",
+                "output": "2\r\n"
+            },
+            {
+                "input": "3 2\r\n",
+                "output": "1\r\n"
+            }
+        ],
+        "official_tests_complete": true,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2047/A",
+        "aliases": null,
+        "contest_id": "2047",
+        "contest_name": "Codeforces Round 990 (Div. 2)",
+        "contest_type": "CF",
+        "contest_start": 1733207100,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Alyona and a Square Jigsaw Puzzle",
+        "description": "Alyona assembles an unusual square Jigsaw Puzzle. She does so in $$$n$$$ days in the following manner:\n\n- On the first day, she starts by placing the central piece in the center of the table.\n- On each day after the first one, she places a certain number of pieces around the central piece in clockwise order, always finishing each square layer completely before starting a new one.\n\nFor example, she places the first $$$14$$$ pieces in the following order:\n\nThe colors denote the layers. The third layer is still unfinished.\n\nAlyona is happy if at the end of the day the assembled part of the puzzle does not have any started but unfinished layers. Given the number of pieces she assembles on each day, find the number of days Alyona is happy on.",
+        "input_format": "Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \\le t \\le 500$$$). The description of the test cases follows.\n\nThe first line contains a single integer $$$n$$$ ($$$1 \\le n \\le 100$$$), the number of days.\n\nThe second line contains $$$n$$$ integers $$$a_1, a_2, \\ldots, a_n$$$ ($$$1 \\le a_i \\le 100$$$, $$$a_1 = 1$$$), where $$$a_i$$$ is the number of pieces Alyona assembles on the $$$i$$$-th day.\n\nIt is guaranteed in each test case that at the end of the $$$n$$$ days, there are no unfinished layers.",
+        "output_format": "For each test case, print a single integer: the number of days when Alyona is happy.",
+        "interaction_format": null,
+        "note": "In the first test case, in the only day Alyona finishes the only layer.\n\nIn the second test case, on the first day, Alyona finishes the first layer, and on the second day, she finishes the second layer.\n\nIn the third test case, she finishes the second layer in a few days.\n\nIn the fourth test case, she finishes the second layer and immediately starts the next one on the same day, therefore, she is not happy on that day. She is only happy on the first and last days.\n\nIn the fifth test case, Alyona is happy on the first, fourth, and last days.",
+        "examples": [
+            {
+                "input": "5\n1\n1\n2\n1 8\n5\n1 3 2 1 2\n7\n1 2 1 10 2 7 2\n14\n1 10 10 100 1 1 10 1 10 2 10 2 10 1",
+                "output": "1\n2\n2\n2\n3"
+            }
+        ],
+        "editorial": "Alyona is happy when there are no unfinished layers — that is, in front of her is a perfect square with odd side length. Since the order of pieces is fixed, it is enough to keep track of the total current size $$$s$$$ of the puzzle, and after each day check, if the $$$s$$$ is a perfect square of an odd number. The easiest way to do that is to create an additional array containing $$$1^2, 3^2, 5^3, \\ldots, 99^2$$$, and check after each day, whether $$$s$$$ is in this array.",
+        "rating": 800,
+        "tags": [
+            "implementation",
+            "math"
+        ],
+        "testset_size": 4,
+        "official_tests": [
+            {
+                "input": "5\r\n1\r\n1\r\n2\r\n1 8\r\n5\r\n1 3 2 1 2\r\n7\r\n1 2 1 10 2 7 2\r\n14\r\n1 10 10 100 1 1 10 1 10 2 10 2 10 1\r\n",
+                "output": "1\r\n2\r\n2\r\n2\r\n3\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2014/A",
+        "aliases": null,
+        "contest_id": "2014",
+        "contest_name": "Codeforces Round 974 (Div. 3)",
+        "contest_type": "ICPC",
+        "contest_start": 1726929900,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Robin Helps",
+        "description": "The heroic outlaw Robin Hood is famous for taking from the rich and giving to the poor.\n\nRobin encounters $$$n$$$ people starting from the $$$1$$$-st and ending with the $$$n$$$-th. The $$$i$$$-th person has $$$a_i$$$ gold. If $$$a_i \\ge k$$$, Robin will take all $$$a_i$$$ gold, and if $$$a_i=0$$$, Robin will give $$$1$$$ gold if he has any. Robin starts with $$$0$$$ gold.\n\nFind out how many people Robin gives gold to.",
+        "input_format": "The first line of the input contains a single integer $$$t$$$ ($$$1\\leq t \\leq 10^4$$$) — the number of test cases.\n\nThe first line of each test case contains two integers $$$n$$$, $$$k$$$ ($$$1 \\le n \\le 50, 1 \\le k \\le 100$$$) — the number of people and the threshold at which Robin Hood takes the gold.\n\nThe second line of each test case contains $$$n$$$ integers $$$a_1, a_2, \\ldots, a_n$$$ ($$$0 \\le a_i \\le 100$$$) — the gold of each person.",
+        "output_format": "For each test case, output a single integer, the number of people that will get gold from Robin Hood.",
+        "interaction_format": null,
+        "note": "In the first test case, Robin takes $$$2$$$ gold from the first person and gives a gold to the second person.\n\nIn the second test case, Robin takes $$$3$$$ gold and gives $$$1$$$ gold to each of the next $$$2$$$ people.\n\nIn the third test case, Robin takes $$$3$$$ gold and so only gives gold to $$$3$$$ other people.",
+        "examples": [
+            {
+                "input": "4\n2 2\n2 0\n3 2\n3 0 0\n6 2\n0 3 0 0 0 0\n2 5\n5 4",
+                "output": "1\n2\n3\n0"
+            }
+        ],
+        "editorial": null,
+        "rating": 800,
+        "tags": [
+            "greedy",
+            "implementation"
+        ],
+        "testset_size": 7,
+        "official_tests": [
+            {
+                "input": "4\r\n2 2\r\n2 0\r\n3 2\r\n3 0 0\r\n6 2\r\n0 3 0 0 0 0\r\n2 5\r\n5 4\r\n",
+                "output": "1\r\n2\r\n3\r\n0\r\n"
+            },
+            {
+                "input": "1\r\n6 1\r\n1 0 0 2 0 0\r\n",
+                "output": "3\r\n"
+            },
+            {
+                "input": "1\r\n6 1\r\n3 0 1 0 0 0\r\n",
+                "output": "4\r\n"
+            },
+            {
+                "input": "1\r\n6 1\r\n3 1 0 0 0 0\r\n",
+                "output": "4\r\n"
+            },
+            {
+                "input": "1\r\n2 2\r\n2 3\r\n",
+                "output": "0\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2014/B",
+        "aliases": null,
+        "contest_id": "2014",
+        "contest_name": "Codeforces Round 974 (Div. 3)",
+        "contest_type": "ICPC",
+        "contest_start": 1726929900,
+        "contest_start_year": 2024,
+        "index": "B",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Robin Hood and the Major Oak",
+        "description": "The Major Oak in Sherwood is known for its majestic foliage, which provided shelter to Robin Hood and his band of merry men and women.\n\nThe Major Oak grows $$$i^i$$$ new leaves in the $$$i$$$-th year. It starts with $$$1$$$ leaf in year $$$1$$$.\n\nLeaves last for $$$k$$$ years on the tree. In other words, leaves grown in year $$$i$$$ last between years $$$i$$$ and $$$i+k-1$$$ inclusive.\n\nRobin considers even numbers lucky. Help Robin determine whether the Major Oak will have an even number of leaves in year $$$n$$$.",
+        "input_format": "The first line of the input contains a single integer $$$t$$$ ($$$1 \\le t \\le 10^4$$$) — the number of test cases.\n\nEach test case consists of two integers $$$n$$$, $$$k$$$ ($$$1 \\le n \\le 10^9$$$, $$$1 \\le k \\le n$$$) — the requested year and the number of years during which the leaves remain.",
+        "output_format": "For each test case, output one line, \"YES\" if in year $$$n$$$ the Major Oak will have an even number of leaves and \"NO\" otherwise.\n\nYou can output the answer in any case (upper or lower). For example, the strings \"yEs\", \"yes\", \"Yes\", and \"YES\" will be recognized as positive responses.",
+        "interaction_format": null,
+        "note": "In the first test case, there is only $$$1$$$ leaf.\n\nIn the second test case, $$$k=1$$$, so in the $$$2$$$-nd year there will be $$$2^2=4$$$ leaves.\n\nIn the third test case, $$$k=2$$$, so in the $$$2$$$-nd year there will be $$$1+2^2=5$$$ leaves.\n\nIn the fourth test case, $$$k=2$$$, so in the $$$3$$$-rd year there will be $$$2^2 + 3^3 = 4 + 27 = 31$$$ leaves.",
+        "examples": [
+            {
+                "input": "5\n1 1\n2 1\n2 2\n3 2\n4 4",
+                "output": "NO\nYES\nNO\nNO\nYES"
+            }
+        ],
+        "editorial": null,
+        "rating": 800,
+        "tags": [
+            "math"
+        ],
+        "testset_size": 4,
+        "official_tests": [
+            {
+                "input": "5\r\n1 1\r\n2 1\r\n2 2\r\n3 2\r\n4 4\r\n",
+                "output": "NO\r\nYES\r\nNO\r\nNO\r\nYES\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2025/A",
+        "aliases": null,
+        "contest_id": "2025",
+        "contest_name": "Educational Codeforces Round 170 (Rated for Div. 2)",
+        "contest_type": "ICPC",
+        "contest_start": 1728916500,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 2.0,
+        "memory_limit": 512.0,
+        "title": "Two Screens",
+        "description": "There are two screens which can display sequences of uppercase Latin letters. Initially, both screens display nothing.\n\nIn one second, you can do one of the following two actions:\n\n- choose a screen and an uppercase Latin letter, and append that letter to the end of the sequence displayed on that screen;\n- choose a screen and copy the sequence from it to the other screen, overwriting the sequence that was displayed on the other screen.\n\nYou have to calculate the minimum number of seconds you have to spend so that the first screen displays the sequence $$$s$$$, and the second screen displays the sequence $$$t$$$.",
+        "input_format": "The first line contains one integer $$$q$$$ ($$$1 \\le q \\le 500$$$) — the number of test cases.\n\nEach test case consists of two lines. The first line contains the string $$$s$$$, and the second line contains the string $$$t$$$ ($$$1 \\le |s|, |t| \\le 100$$$). Both strings consist of uppercase Latin letters.",
+        "output_format": "For each test case, print one integer — the minimum possible number of seconds you have to spend so that the first screen displays the sequence $$$s$$$, and the second screen displays the sequence $$$t$$$.",
+        "interaction_format": null,
+        "note": "In the first test case, the following sequence of actions is possible:\n\n- spend $$$6$$$ seconds to write the sequence GARAGE on the first screen;\n- copy the sequence from the first screen to the second screen;\n- spend $$$7$$$ seconds to complete the sequence on the second screen by writing FORSALE.\n\nIn the second test case, the following sequence of actions is possible:\n\n- spend $$$1$$$ second to write the sequence A on the second screen;\n- copy the sequence from the second screen to the first screen;\n- spend $$$4$$$ seconds to complete the sequence on the first screen by writing BCDE;\n- spend $$$4$$$ seconds to complete the sequence on the second screen by writing ABCD.\n\nIn the third test case, the fastest way to display the sequences is to type both of them character by character without copying, and this requires $$$16$$$ seconds.",
+        "examples": [
+            {
+                "input": "3\nGARAGE\nGARAGEFORSALE\nABCDE\nAABCD\nTRAINING\nDRAINING",
+                "output": "14\n10\n16"
+            }
+        ],
+        "editorial": "Whenever we perform the second action (copy from one screen to the other screen), we overwrite what was written on the other screen. It means that we can consider the string on the other screen to be empty before the copying, and that we only need to copy at most once.\nSo, the optimal sequence of operations should look as follows: add some characters to one of the screens, copy them to the other screen, and then finish both strings.\nWe need to copy as many characters as possible. Since after copying, we can only append new characters to the end of the strings on the screens, the string we copy must be a prefix of both of the given strings. So, we need to find the length of the longest common prefix of the given strings. This can be done in linear time if we scan these strings until we find a pair of different characters in the same positions; however, the constraints were low enough so that you could do it slower (like, for example, iterate on the length of the prefix and check that both prefixes of this length are equal).\nOkay, now let $$$l$$$ be the length of the longest common prefix. Using $$$l+1$$$ operations, we write $$$2l$$$ characters in total; so the number of operations we need will be $$$|s|+|t|-2l+l+1 = |s|+|t|+1-l$$$. However, if the longest common prefix is empty, there's no need to copy anything, and we can write both strings in $$$|s|+|t|$$$ seconds.",
+        "rating": 800,
+        "tags": [
+            "binary search",
+            "greedy",
+            "strings",
+            "two pointers"
+        ],
+        "testset_size": 6,
+        "official_tests": [
+            {
+                "input": "3\r\nGARAGE\r\nGARAGEFORSALE\r\nABCDE\r\nAABCD\r\nTRAINING\r\nDRAINING\r\n",
+                "output": "14\r\n10\r\n16\r\n"
+            },
+            {
+                "input": "1\r\nABCD\r\nDABC\r\n",
+                "output": "8\r\n"
+            },
+            {
+                "input": "1\r\nGUFY\r\nTFIG\r\n",
+                "output": "8\r\n"
+            },
+            {
+                "input": "6\r\nA\r\nA\r\nA\r\nA\r\nA\r\nA\r\nA\r\nA\r\nA\r\nA\r\nA\r\nA\r\n",
+                "output": "2\r\n2\r\n2\r\n2\r\n2\r\n2\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2029/A",
+        "aliases": null,
+        "contest_id": "2029",
+        "contest_name": "Refact.ai Match 1 (Codeforces Round 985)",
+        "contest_type": "CF",
+        "contest_start": 1731162900,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Set",
+        "description": "You are given a positive integer $$$k$$$ and a set $$$S$$$ of all integers from $$$l$$$ to $$$r$$$ (inclusive).\n\nYou can perform the following two-step operation any number of times (possibly zero):\n\n1. First, choose a number $$$x$$$ from the set $$$S$$$, such that there are at least $$$k$$$ multiples of $$$x$$$ in $$$S$$$ (including $$$x$$$ itself);\n2. Then, remove $$$x$$$ from $$$S$$$ (note that nothing else is removed).\n\nFind the maximum possible number of operations that can be performed.",
+        "input_format": "Each test contains multiple test cases. The first line of the input contains a single integer $$$t$$$ ($$$1\\le t\\le 10^4$$$) — the number of test cases. The description of test cases follows.\n\nThe only line of each test case contains three integers $$$l$$$, $$$r$$$, and $$$k$$$ ($$$1\\le l\\le r\\leq 10^9$$$, $$$1\\leq k\\le r-l+1$$$) — the minimum integer in $$$S$$$, the maximum integer in $$$S$$$, and the parameter $$$k$$$.",
+        "output_format": "For each test case, output a single integer — the maximum possible number of operations that can be performed.",
+        "interaction_format": null,
+        "note": "In the first test case, initially, $$$S = \\{3,4,5,6,7,8,9\\}$$$. One possible optimal sequence of operations is:\n\n1. Choose $$$x = 4$$$ for the first operation, since there are two multiples of $$$4$$$ in $$$S$$$: $$$4$$$ and $$$8$$$. $$$S$$$ becomes equal to $$$\\{3,5,6,7,8,9\\}$$$;\n2. Choose $$$x = 3$$$ for the second operation, since there are three multiples of $$$3$$$ in $$$S$$$: $$$3$$$, $$$6$$$, and $$$9$$$. $$$S$$$ becomes equal to $$$\\{5,6,7,8,9\\}$$$.\n\nIn the second test case, initially, $$$S=\\{4,5,6,7,8,9\\}$$$. One possible optimal sequence of operations is:\n\n1. Choose $$$x = 5$$$, $$$S$$$ becomes equal to $$$\\{4,6,7,8,9\\}$$$;\n2. Choose $$$x = 6$$$, $$$S$$$ becomes equal to $$$\\{4,7,8,9\\}$$$;\n3. Choose $$$x = 4$$$, $$$S$$$ becomes equal to $$$\\{7,8,9\\}$$$;\n4. Choose $$$x = 8$$$, $$$S$$$ becomes equal to $$$\\{7,9\\}$$$;\n5. Choose $$$x = 7$$$, $$$S$$$ becomes equal to $$$\\{9\\}$$$;\n6. Choose $$$x = 9$$$, $$$S$$$ becomes equal to $$$\\{\\}$$$.\n\nIn the third test case, initially, $$$S=\\{7,8,9\\}$$$. For each $$$x$$$ in $$$S$$$, no multiple of $$$x$$$ other than $$$x$$$ itself can be found in $$$S$$$. Since $$$k = 2$$$, you can perform no operations.\n\nIn the fourth test case, initially, $$$S=\\{2,3,4,5,6,7,8,9,10\\}$$$. One possible optimal sequence of operations is:\n\n1. Choose $$$x = 2$$$, $$$S$$$ becomes equal to $$$\\{3,4,5,6,7,8,9,10\\}$$$;\n2. Choose $$$x = 4$$$, $$$S$$$ becomes equal to $$$\\{3,5,6,7,8,9,10\\}$$$;\n3. Choose $$$x = 3$$$, $$$S$$$ becomes equal to $$$\\{5,6,7,8,9,10\\}$$$;\n4. Choose $$$x = 5$$$, $$$S$$$ becomes equal to $$$\\{6,7,8,9,10\\}$$$.",
+        "examples": [
+            {
+                "input": "8\n3 9 2\n4 9 1\n7 9 2\n2 10 2\n154 220 2\n147 294 2\n998 24435 3\n1 1000000000 2",
+                "output": "2\n6\n0\n4\n0\n1\n7148\n500000000"
+            }
+        ],
+        "editorial": null,
+        "rating": 800,
+        "tags": [
+            "greedy",
+            "math"
+        ],
+        "testset_size": 4,
+        "official_tests": [
+            {
+                "input": "8\r\n3 9 2\r\n4 9 1\r\n7 9 2\r\n2 10 2\r\n154 220 2\r\n147 294 2\r\n998 24435 3\r\n1 1000000000 2\r\n",
+                "output": "2\r\n6\r\n0\r\n4\r\n0\r\n1\r\n7148\r\n500000000\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2062/A",
+        "aliases": null,
+        "contest_id": "2062",
+        "contest_name": "Ethflow Round 1 (Codeforces Round 1001, Div. 1 + Div. 2)",
+        "contest_type": "CF",
+        "contest_start": 1737902100,
+        "contest_start_year": 2025,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 512.0,
+        "title": "String",
+        "description": "You are given a string $$$s$$$ of length $$$n$$$ consisting of $$$\\mathtt{0}$$$ and/or $$$\\mathtt{1}$$$. In one operation, you can select a non-empty subsequence $$$t$$$ from $$$s$$$ such that any two adjacent characters in $$$t$$$ are different. Then, you flip each character of $$$t$$$ ($$$\\mathtt{0}$$$ becomes $$$\\mathtt{1}$$$ and $$$\\mathtt{1}$$$ becomes $$$\\mathtt{0}$$$). For example, if $$$s=\\mathtt{\\underline{0}0\\underline{101}}$$$ and $$$t=s_1s_3s_4s_5=\\mathtt{0101}$$$, after the operation, $$$s$$$ becomes $$$\\mathtt{\\underline{1}0\\underline{010}}$$$.\n\nCalculate the minimum number of operations required to change all characters in $$$s$$$ to $$$\\mathtt{0}$$$.\n\nRecall that for a string $$$s = s_1s_2\\ldots s_n$$$, any string $$$t=s_{i_1}s_{i_2}\\ldots s_{i_k}$$$ ($$$k\\ge 1$$$) where $$$1\\leq i_1 < i_2 < \\ldots <i_k\\leq n$$$ is a subsequence of $$$s$$$.",
+        "input_format": "The first line of input contains a single integer $$$t$$$ ($$$1 \\leq t \\leq 10^4$$$) — the number of input test cases.\n\nThe only line of each test case contains the string $$$s$$$ ($$$1\\le |s|\\le 50$$$), where $$$|s|$$$ represents the length of $$$s$$$.",
+        "output_format": "For each test case, output the minimum number of operations required to change all characters in $$$s$$$ to $$$\\mathtt{0}$$$.",
+        "interaction_format": null,
+        "note": "In the first test case, you can flip $$$s_1$$$. Then $$$s$$$ becomes $$$\\mathtt{0}$$$, so the answer is $$$1$$$.\n\nIn the fourth test case, you can perform the following three operations in order:\n\n1. Flip $$$s_1s_2s_3s_4s_5$$$. Then $$$s$$$ becomes $$$\\mathtt{\\underline{01010}}$$$.\n2. Flip $$$s_2s_3s_4$$$. Then $$$s$$$ becomes $$$\\mathtt{0\\underline{010}0}$$$.\n3. Flip $$$s_3$$$. Then $$$s$$$ becomes $$$\\mathtt{00\\underline{0}00}$$$.\n\nIt can be shown that you can not change all characters in $$$s$$$ to $$$\\mathtt{0}$$$ in less than three operations, so the answer is $$$3$$$.",
+        "examples": [
+            {
+                "input": "5\n1\n000\n1001\n10101\n01100101011101",
+                "output": "1\n0\n2\n3\n8"
+            }
+        ],
+        "editorial": "Let $$$c$$$ be the number of $$$1$$$s.\nOn one hand, each operation can decrease $$$c$$$ by at most $$$1$$$, so the answer is at least $$$c$$$. On the other hand, by only operating on $$$1$$$s each time, it takes exactly $$$c$$$ operations, so the answer is at most $$$c$$$. Therefore, the answer is $$$c$$$.",
+        "rating": 800,
+        "tags": [
+            "constructive algorithms",
+            "greedy",
+            "math",
+            "strings"
+        ],
+        "testset_size": 4,
+        "official_tests": [
+            {
+                "input": "5\r\n1\r\n000\r\n1001\r\n10101\r\n01100101011101\r\n",
+                "output": "1\r\n0\r\n2\r\n3\r\n8\r\n"
+            },
+            {
+                "input": "1\r\n00000000000000000000000000000000000000000000000000\r\n",
+                "output": "0\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2053/A",
+        "aliases": null,
+        "contest_id": "2053",
+        "contest_name": "Good Bye 2024: 2025 is NEAR",
+        "contest_type": "CF",
+        "contest_start": 1735396500,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Tender Carpenter",
+        "description": "In his dream, Cocoly would go on a long holiday with no worries around him. So he would try out for many new things, such as... being a carpenter. To learn it well, Cocoly decides to become an apprentice of Master, but in front of him lies a hard task waiting for him to solve.\n\nCocoly is given an array $$$a_1, a_2,\\ldots, a_n$$$. Master calls a set of integers $$$S$$$ stable if and only if, for any possible $$$u$$$, $$$v$$$, and $$$w$$$ from the set $$$S$$$ (note that $$$u$$$, $$$v$$$, and $$$w$$$ do not necessarily have to be pairwise distinct), sticks of length $$$u$$$, $$$v$$$, and $$$w$$$ can form a non-degenerate triangle$$$^{\\text{∗}}$$$.\n\nCocoly is asked to partition the array $$$a$$$ into several (possibly, $$$1$$$ or $$$n$$$) non-empty continuous subsegments$$$^{\\text{†}}$$$, such that: for each of the subsegments, the set containing all the elements in it is stable.\n\nMaster wants Cocoly to partition $$$a$$$ in at least two different$$$^{\\text{‡}}$$$ ways. You have to help him determine whether it is possible.",
+        "input_format": "Each test contains multiple test cases. The first line of the input contains a single integer $$$t$$$ ($$$1 \\leq t \\leq 200$$$) — the number of test cases. The description of test cases follows.\n\nThe first line of each test case contains a single integer $$$n$$$ ($$$2 \\leq n \\leq 200$$$) — the length of the array $$$a$$$.\n\nThe second line contains $$$n$$$ integers $$$a_1,a_2,\\ldots,a_n$$$ ($$$1 \\leq a_i \\leq 10^5$$$) — the elements in the array $$$a$$$.",
+        "output_format": "For each test case, print $$$\\texttt{YES}$$$ if there are at least two ways to partition $$$a$$$, and $$$\\texttt{NO}$$$ otherwise.\n\nYou can output the answer in any case (upper or lower). For example, the strings $$$\\texttt{yEs}$$$, $$$\\texttt{yes}$$$, $$$\\texttt{Yes}$$$, and $$$\\texttt{YES}$$$ will be recognized as positive responses.",
+        "interaction_format": null,
+        "note": "In the first test case, here are two possible partitions:\n\n- $$$[2, 3], [5, 7]$$$, since   $$$[2, 3]$$$ is stable because sticks of lengths $$$(2, 2, 2), (2, 2, 3), (2, 3, 3), (3, 3, 3)$$$ respectively can all form non-degenerate triangles.  $$$[5, 7]$$$ is stable because sticks of lengths $$$(5, 5, 5), (5, 5, 7), (5, 7, 7), (7, 7, 7)$$$ respectively can all form non-degenerate triangles.\n- and $$$[2], [3, 5], [7]$$$, since   $$$[2]$$$ is stable because sticks of lengths $$$(2, 2, 2)$$$ respectively can form a non-degenerate triangle.  $$$[3, 5]$$$ is stable because sticks of lengths $$$(3, 3, 3), (3, 3, 5), (3, 5, 5), (5, 5, 5)$$$ respectively can all form non-degenerate triangles.  $$$[7]$$$ is stable because sticks of lengths $$$(7, 7, 7)$$$ respectively can form a non-degenerate triangle.\n\nNote that some other partitions also satisfy the constraints, such as $$$[2], [3], [5], [7]$$$ and $$$[2], [3], [5, 7]$$$.\n\nIn the second test case, Cocoly can only partition each element as a single subsegment, resulting in $$$[115], [9], [2], [28]$$$. Since we only have one possible partition, the answer is $$$\\texttt{NO}$$$.\n\nIn the third test case, please note that the partition $$$[8, 4], [1], [6], [2]$$$ does not satisfy the constraints, because $$$\\{8, 4\\}$$$ is not a stable set: sticks of lengths $$$4$$$, $$$4$$$, and $$$8$$$ cannot form a non-degenerate triangle.",
+        "examples": [
+            {
+                "input": "5\n4\n2 3 5 7\n4\n115 9 2 28\n5\n8 4 1 6 2\n6\n1 5 4 1 4 7\n2\n100000 100000",
+                "output": "YES\nNO\nNO\nYES\nYES"
+            }
+        ],
+        "editorial": "Note that: we always have a partition like $$$[a_1], [a_2], \\dots, [a_n]$$$ since $$$(x, x, x)$$$ always form a non-degenerate (equilateral) triangle.\nWe focus on the second partition scheme in which not all continuous subsegments have a length of $$$1$$$. In fact, one can also note that if a set $$$S$$$ is stable then for all $$$T \\subsetneq S$$$ ($$$T \\neq \\varnothing$$$), $$$T$$$ is also stable.\n• Short proof: If $$$\\exists u, v, w \\in T$$$ such that $$$(u, v, w)$$$ does not form a non-degenerate triangle, therefore $$$u, v, w \\in S$$$ so $$$S$$$ is not stable. Contradiction!\nTherefore, if such a partition exists, it can always be boiled down into a continuous subsegment of length $$$2$$$ and other continuous subsegments of length $$$1$$$, and the partition remains valid. So we should output $$$\\texttt{NO}$$$ if and only if for all $$$1 \\leq i < n$$$, $$$2\\min(a_i, a_{i+1}) \\leq \\max(a_i, a_{i+1})$$$.",
+        "rating": 800,
+        "tags": [
+            "dp",
+            "geometry",
+            "greedy",
+            "math"
+        ],
+        "testset_size": 5,
+        "official_tests": [
+            {
+                "input": "5\r\n4\r\n2 3 5 7\r\n4\r\n115 9 2 28\r\n5\r\n8 4 1 6 2\r\n6\r\n1 5 4 1 4 7\r\n2\r\n100000 100000\r\n",
+                "output": "YES\r\nNO\r\nNO\r\nYES\r\nYES\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2031/A",
+        "aliases": null,
+        "contest_id": "2031",
+        "contest_name": "Codeforces Round 987 (Div. 2)",
+        "contest_type": "CF",
+        "contest_start": 1731674100,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Penchick and Modern Monument",
+        "description": "Amidst skyscrapers in the bustling metropolis of Metro Manila, the newest Noiph mall in the Philippines has just been completed! The construction manager, Penchick, ordered a state-of-the-art monument to be built with $$$n$$$ pillars.\n\nThe heights of the monument's pillars can be represented as an array $$$h$$$ of $$$n$$$ positive integers, where $$$h_i$$$ represents the height of the $$$i$$$-th pillar for all $$$i$$$ between $$$1$$$ and $$$n$$$.\n\nPenchick wants the heights of the pillars to be in non-decreasing order, i.e. $$$h_i \\le h_{i + 1}$$$ for all $$$i$$$ between $$$1$$$ and $$$n - 1$$$. However, due to confusion, the monument was built such that the heights of the pillars are in non-increasing order instead, i.e. $$$h_i \\ge h_{i + 1}$$$ for all $$$i$$$ between $$$1$$$ and $$$n - 1$$$.\n\nLuckily, Penchick can modify the monument and do the following operation on the pillars as many times as necessary:\n\n- Modify the height of a pillar to any positive integer. Formally, choose an index $$$1\\le i\\le n$$$ and a positive integer $$$x$$$. Then, assign $$$h_i := x$$$.\n\nHelp Penchick determine the minimum number of operations needed to make the heights of the monument's pillars non-decreasing.",
+        "input_format": "Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \\le t \\le 1000$$$). The description of the test cases follows.\n\nThe first line of each test case contains a single integer $$$n$$$ ($$$1 \\leq n \\leq 50$$$) — the number of pillars.\n\nThe second line of each test case contains $$$n$$$ integers $$$h_1, h_2, \\ldots, h_n$$$ ($$$1 \\le h_i \\le n$$$ and $$$h_i\\ge h_{i+1}$$$) — the height of the pillars.\n\nPlease take note that the given array $$$h$$$ is non-increasing.\n\nNote that there are no constraints on the sum of $$$n$$$ over all test cases.",
+        "output_format": "For each test case, output a single integer representing the minimum number of operations needed to make the heights of the pillars non-decreasing.",
+        "interaction_format": null,
+        "note": "In the first test case, the initial heights of pillars are $$$h = [5, 4, 3, 2, 1]$$$.\n\n- In the first operation, Penchick changes the height of pillar $$$1$$$ to $$$h_1 := 2$$$.\n- In the second operation, he changes the height of pillar $$$2$$$ to $$$h_2 := 2$$$.\n- In the third operation, he changes the height of pillar $$$4$$$ to $$$h_4 := 4$$$.\n- In the fourth operation, he changes the height of pillar $$$5$$$ to $$$h_5 := 4$$$.\n\nAfter the operation, the heights of the pillars are $$$h = [2, 2, 3, 4, 4]$$$, which is non-decreasing. It can be proven that it is not possible for Penchick to make the heights of the pillars non-decreasing in fewer than $$$4$$$ operations.\n\nIn the second test case, Penchick can make the heights of the pillars non-decreasing by modifying the height of pillar $$$3$$$ to $$$h_3 := 2$$$.\n\nIn the third test case, the heights of pillars are already non-decreasing, so no operations are required.",
+        "examples": [
+            {
+                "input": "3\n5\n5 4 3 2 1\n3\n2 2 1\n1\n1",
+                "output": "4\n1\n0"
+            }
+        ],
+        "editorial": null,
+        "rating": 800,
+        "tags": [
+            "constructive algorithms",
+            "dp",
+            "greedy",
+            "math"
+        ],
+        "testset_size": 3,
+        "official_tests": [
+            {
+                "input": "3\r\n5\r\n5 4 3 2 1\r\n3\r\n2 2 1\r\n1\r\n1\r\n",
+                "output": "4\r\n1\r\n0\r\n"
+            },
+            {
+                "input": "1\r\n30\r\n30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 29\r\n",
+                "output": "1\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "1999/A",
+        "aliases": null,
+        "contest_id": "1999",
+        "contest_name": "Codeforces Round 964 (Div. 4)",
+        "contest_type": "ICPC",
+        "contest_start": 1722954900,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "A+B Again?",
+        "description": "Given a two-digit positive integer $$$n$$$, find the sum of its digits.",
+        "input_format": "The first line contains an integer $$$t$$$ ($$$1 \\leq t \\leq 90$$$) — the number of test cases.\n\nThe only line of each test case contains a single two-digit positive integer $$$n$$$ ($$$10 \\leq n \\leq 99$$$).",
+        "output_format": "For each test case, output a single integer — the sum of the digits of $$$n$$$.",
+        "interaction_format": null,
+        "note": null,
+        "examples": [
+            {
+                "input": "8\n77\n21\n40\n34\n19\n84\n10\n99",
+                "output": "14\n3\n4\n7\n10\n12\n1\n18"
+            }
+        ],
+        "editorial": "You can take the input as a string and output the sum of the two characters in the string.\nThere is also a formula: $$$\\lfloor \\frac{n}{10} \\rfloor + n \\bmod 10$$$. The first term is the tens digit, the second term — the ones digit.",
+        "rating": 800,
+        "tags": [
+            "implementation",
+            "math"
+        ],
+        "testset_size": 6,
+        "official_tests": [
+            {
+                "input": "8\r\n77\r\n21\r\n40\r\n34\r\n19\r\n84\r\n10\r\n99\r\n",
+                "output": "14\r\n3\r\n4\r\n7\r\n10\r\n12\r\n1\r\n18\r\n"
+            },
+            {
+                "input": "90\r\n10\r\n11\r\n12\r\n13\r\n14\r\n15\r\n16\r\n17\r\n18\r\n19\r\n20\r\n21\r\n22\r\n23\r\n24\r\n25\r\n26\r\n27\r\n28\r\n29\r\n30\r\n31\r\n32\r\n33\r\n34\r\n35\r\n36\r\n37\r\n38\r\n39\r\n40\r\n41\r\n42\r\n43\r\n44\r\n45\r\n46\r\n47\r\n48\r\n49\r\n50\r\n51\r\n52\r\n53\r\n54\r\n55\r\n56\r\n57\r\n58\r\n59\r\n60\r\n61\r\n62\r\n63\r\n64\r\n65\r\n66\r\n67\r\n68\r\n69\r\n70\r\n71\r\n72\r\n73\r\n74\r\n75\r\n76\r\n77\r\n78\r\n79\r\n80\r\n81\r\n82\r\n83\r\n84\r\n85\r\n86\r\n87\r\n88\r\n89\r\n90\r\n91\r\n92\r\n93\r\n94\r\n95\r\n96\r\n97\r\n98\r\n99\r\n",
+                "output": "1\r\n2\r\n3\r\n4\r\n5\r\n6\r\n7\r\n8\r\n9\r\n10\r\n2\r\n3\r\n4\r\n5\r\n6\r\n7\r\n8\r\n9\r\n10\r\n11\r\n3\r\n4\r\n5\r\n6\r\n7\r\n8\r\n9\r\n10\r\n11\r\n12\r\n4\r\n5\r\n6\r\n7\r\n8\r\n9\r\n10\r\n11\r\n12\r\n13\r\n5\r\n6\r\n7\r\n8\r\n9\r\n10\r\n11\r\n12\r\n13\r\n14\r\n6\r\n7\r\n8\r\n9\r\n10\r\n11\r\n12\r\n13\r\n14\r\n15\r\n7\r\n8\r\n9\r\n10\r\n11\r\n12\r\n13\r\n14\r\n15\r\n16\r\n8\r\n9\r\n10\r\n11\r\n12\r\n13\r\n14\r\n15\r\n16\r\n17\r\n9\r\n10\r\n11\r\n12\r\n13\r\n14\r\n15\r\n16\r\n17\r\n18\r\n"
+            },
+            {
+                "input": "90\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n",
+                "output": "12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n"
+            },
+            {
+                "input": "90\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n66\r\n67\r\n",
+                "output": "12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n12\r\n13\r\n"
+            },
+            {
+                "input": "78\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n",
+                "output": "1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n"
+            },
+            {
+                "input": "82\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n10\r\n",
+                "output": "1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n"
+            }
+        ],
+        "official_tests_complete": true,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "1999/C",
+        "aliases": null,
+        "contest_id": "1999",
+        "contest_name": "Codeforces Round 964 (Div. 4)",
+        "contest_type": "ICPC",
+        "contest_start": 1722954900,
+        "contest_start_year": 2024,
+        "index": "C",
+        "time_limit": 2.0,
+        "memory_limit": 256.0,
+        "title": "Showering",
+        "description": "As a computer science student, Alex faces a hard challenge — showering. He tries to shower daily, but despite his best efforts there are always challenges. He takes $$$s$$$ minutes to shower and a day only has $$$m$$$ minutes!\n\nHe already has $$$n$$$ tasks planned for the day. Task $$$i$$$ is represented as an interval $$$(l_i$$$, $$$r_i)$$$, which means that Alex is busy and can not take a shower in that time interval (at any point in time strictly between $$$l_i$$$ and $$$r_i$$$). No two tasks overlap.\n\nGiven all $$$n$$$ time intervals, will Alex be able to shower that day? In other words, will Alex have a free time interval of length at least $$$s$$$?\n\nIn the first test case, Alex can shower for the first $$$3$$$ minutes of the day and not miss any of the tasks.",
+        "input_format": "The first line contains a single integer $$$t$$$ ($$$1 \\leq t \\leq 10^4$$$) — the number of test cases.\n\nThe first line of each test case contains three integers $$$n$$$, $$$s$$$, and $$$m$$$ ($$$1 \\leq n \\leq 2 \\cdot 10^5$$$; $$$1 \\leq s, m \\leq 10^9$$$) — the number of time intervals Alex already has planned, the amount of time Alex takes to take a shower, and the amount of minutes a day has.\n\nThen $$$n$$$ lines follow, the $$$i$$$-th of which contains two integers $$$l_i$$$ and $$$r_i$$$ ($$$0 \\leq l_i < r_i \\leq m$$$) — the time interval of the $$$i$$$-th task. No two tasks overlap.\n\nAdditional constraint on the input: $$$l_i > r_{i-1}$$$ for every $$$i > 1$$$.\n\nThe sum of $$$n$$$ over all test cases does not exceed $$$2 \\cdot 10^5$$$.",
+        "output_format": "For each test case output \"YES\" (without quotes) if Alex can take a shower for that given test case, and \"NO\" (also without quotes) otherwise.\n\nYou can output \"YES\" and \"NO\" in any case (for example, strings \"yEs\", \"yes\", and \"Yes\" will be recognized as a positive response).",
+        "interaction_format": null,
+        "note": null,
+        "examples": [
+            {
+                "input": "4\n3 3 10\n3 5\n6 8\n9 10\n3 3 10\n1 2\n3 5\n6 7\n3 3 10\n1 2\n3 5\n6 8\n3 4 10\n1 2\n6 7\n8 9",
+                "output": "YES\nYES\nNO\nYES"
+            }
+        ],
+        "editorial": "We just need to find the lengths of the gaps between intervals. Note that from the additional constraint on the input, the intervals are in sorted order. So you need to find if for any $$$i > 1$$$, $$$l_i - r_{i - 1} \\geq s$$$, and also if $$$l_1 \\geq s$$$ (the initial interval), or if $$$m - r_{n} \\geq s$$$ (the final interval).",
+        "rating": 800,
+        "tags": [
+            "greedy",
+            "implementation"
+        ],
+        "testset_size": 8,
+        "official_tests": [
+            {
+                "input": "4\r\n3 3 10\r\n3 5\r\n6 8\r\n9 10\r\n3 3 10\r\n1 2\r\n3 5\r\n6 7\r\n3 3 10\r\n1 2\r\n3 5\r\n6 8\r\n3 4 10\r\n1 2\r\n6 7\r\n8 9\r\n",
+                "output": "YES\r\nYES\r\nNO\r\nYES\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2000/A",
+        "aliases": null,
+        "contest_id": "2000",
+        "contest_name": "Codeforces Round 966 (Div. 3)",
+        "contest_type": "ICPC",
+        "contest_start": 1723560000,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Primary Task",
+        "description": "Dmitry wrote down $$$t$$$ integers on the board, and that is good. He is sure that he lost an important integer $$$n$$$ among them, and that is bad.\n\nThe integer $$$n$$$ had the form $$$\\text{10^x}$$$ ($$$x \\ge 2$$$), where the symbol '$$$\\text{^}$$$' denotes exponentiation.. Something went wrong, and Dmitry missed the symbol '$$$\\text{^}$$$' when writing the important integer. For example, instead of the integer $$$10^5$$$, he would have written $$$105$$$, and instead of $$$10^{19}$$$, he would have written $$$1019$$$.\n\nDmitry wants to understand which of the integers on the board could have been the important integer and which could not.",
+        "input_format": "The first line of the input contains one integer $$$t$$$ ($$$1 \\le t \\le 10^4$$$) — the number of integers on the board.\n\nThe next $$$t$$$ lines each contain an integer $$$a$$$ ($$$1 \\le a \\le 10000$$$) — the next integer from the board.",
+        "output_format": "For each integer on the board, output \"YES\" if it could have been the important integer and \"NO\" otherwise.\n\nYou may output each letter in any case (lowercase or uppercase). For example, the strings \"yEs\", \"yes\", \"Yes\", and \"YES\" will be accepted as a positive answer.",
+        "interaction_format": null,
+        "note": null,
+        "examples": [
+            {
+                "input": "7\n100\n1010\n101\n105\n2033\n1019\n1002",
+                "output": "NO\nYES\nNO\nYES\nNO\nYES\nNO"
+            }
+        ],
+        "editorial": "You need to check that the number has the form '$$$\\text{10x}$$$', where $$$x$$$ is an integer without leading zeros. This means it must start with '10'; if the length of the number is $$$3$$$, then the third digit must be at least $$$2$$$; if the length is greater, then the third digit must be at least $$$1$$$ (all such numbers $$$x$$$ are at least $$$10$$$).\nAdditionally, under these constraints, you can simply check that for the given number $$$a$$$, either $$$102 \\le a \\le 109$$$ or $$$1010 \\le a \\le 1099$$$ holds true.",
+        "rating": 800,
+        "tags": [
+            "implementation",
+            "math",
+            "strings"
+        ],
+        "testset_size": 5,
+        "official_tests": [
+            {
+                "input": "7\r\n100\r\n1010\r\n101\r\n105\r\n2033\r\n1019\r\n1002\r\n",
+                "output": "NO\r\nYES\r\nNO\r\nYES\r\nNO\r\nYES\r\nNO\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2000/B",
+        "aliases": null,
+        "contest_id": "2000",
+        "contest_name": "Codeforces Round 966 (Div. 3)",
+        "contest_type": "ICPC",
+        "contest_start": 1723560000,
+        "contest_start_year": 2024,
+        "index": "B",
+        "time_limit": 2.0,
+        "memory_limit": 256.0,
+        "title": "Seating in a Bus",
+        "description": "In Berland, a bus consists of a row of $$$n$$$ seats numbered from $$$1$$$ to $$$n$$$. Passengers are advised to always board the bus following these rules:\n\n- If there are no occupied seats in the bus, a passenger can sit in any free seat;\n- Otherwise, a passenger should sit in any free seat that has at least one occupied neighboring seat. In other words, a passenger can sit in a seat with index $$$i$$$ ($$$1 \\le i \\le n$$$) only if at least one of the seats with indices $$$i-1$$$ or $$$i+1$$$ is occupied.\n\nToday, $$$n$$$ passengers boarded the bus. The array $$$a$$$ chronologically records the seat numbers they occupied. That is, $$$a_1$$$ contains the seat number where the first passenger sat, $$$a_2$$$ — the seat number where the second passenger sat, and so on.\n\nYou know the contents of the array $$$a$$$. Determine whether all passengers followed the recommendations.\n\nFor example, if $$$n = 5$$$, and $$$a$$$ = [$$$5, 4, 2, 1, 3$$$], then the recommendations were not followed, as the $$$3$$$-rd passenger sat in seat number $$$2$$$, while the neighboring seats with numbers $$$1$$$ and $$$3$$$ were free.",
+        "input_format": "The first line of input contains a single integer $$$t$$$ ($$$1 \\le t \\le 10^4$$$) — the number of test cases.\n\nThe following describes the input test cases.\n\nThe first line of each test case contains exactly one integer $$$n$$$ ($$$1 \\le n \\le 2 \\cdot 10^5$$$) — the number of seats in the bus and the number of passengers who boarded the bus.\n\nThe second line of each test case contains $$$n$$$ distinct integers $$$a_i$$$ ($$$1 \\le a_i \\le n$$$) — the seats that the passengers occupied in chronological order.\n\nIt is guaranteed that the sum of $$$n$$$ values across all test cases does not exceed $$$2 \\cdot 10^5$$$, and that no passenger sits in an already occupied seat.",
+        "output_format": "For each test case, output on a separate line:\n\n- \"YES\", if all passengers followed the recommendations;\n- \"NO\" otherwise.\n\nYou may output the answer in any case (for example, the strings \"yEs\", \"yes\", \"Yes\", and \"YES\" will be recognized as a positive answer).",
+        "interaction_format": null,
+        "note": "The first test case is explained in the problem statement.",
+        "examples": [
+            {
+                "input": "4\n5\n5 4 2 1 3\n3\n2 3 1\n4\n2 3 1 4\n5\n1 2 3 5 4",
+                "output": "NO\nYES\nYES\nNO"
+            }
+        ],
+        "editorial": "To solve the problem, it is enough to use an array $$$used$$$ of size $$$2 \\cdot 10^5 + 1$$$ (maximal value $$$n + 1$$$) consisting of boolean variables.\nWhen a passenger gets on the bus to seat $$$a_i$$$, we will replace the value of $$$used[a_i]$$$ with true. Then, for all passengers, starting from the second one, we should check that at least one of the existing elements $$$used[a_i - 1]$$$ or $$$used[a_i + 1]$$$ has the value true.\nIf this condition is not satisfied, the answer to the problem is \"NO\". If at least $$$1$$$ occupied neighbouring seat always existed for all passengers at the moment they took their seat — the answer is \"YES\".",
+        "rating": 800,
+        "tags": [
+            "two pointers"
+        ],
+        "testset_size": 11,
+        "official_tests": [
+            {
+                "input": "4\r\n5\r\n5 4 2 1 3\r\n3\r\n2 3 1\r\n4\r\n2 3 1 4\r\n5\r\n1 2 3 5 4\r\n",
+                "output": "NO\r\nYES\r\nYES\r\nNO\r\n"
+            },
+            {
+                "input": "6\r\n5\r\n5 4 2 1 3\r\n3\r\n2 3 1\r\n4\r\n2 3 1 4\r\n5\r\n1 2 3 5 4\r\n5\r\n1 2 3 5 4\r\n5\r\n1 2 3 5 4\r\n",
+                "output": "NO\r\nYES\r\nYES\r\nNO\r\nNO\r\nNO\r\n"
+            },
+            {
+                "input": "1\r\n78\r\n1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78\r\n",
+                "output": "YES\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2057/A",
+        "aliases": null,
+        "contest_id": "2057",
+        "contest_name": "Hello 2025",
+        "contest_type": "CF",
+        "contest_start": 1736001300,
+        "contest_start_year": 2025,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "MEX Table",
+        "description": "One day, the schoolboy Mark misbehaved, so the teacher Sasha called him to the whiteboard.\n\nSasha gave Mark a table with $$$n$$$ rows and $$$m$$$ columns. His task is to arrange the numbers $$$0, 1, \\ldots, n \\cdot m - 1$$$ in the table (each number must be used exactly once) in such a way as to maximize the sum of MEX$$$^{\\text{∗}}$$$ across all rows and columns. More formally, he needs to maximize $$$$$$\\sum\\limits_{i = 1}^{n} \\operatorname{mex}(\\{a_{i,1}, a_{i,2}, \\ldots, a_{i,m}\\}) + \\sum\\limits_{j = 1}^{m} \\operatorname{mex}(\\{a_{1,j}, a_{2,j}, \\ldots, a_{n,j}\\}),$$$$$$ where $$$a_{i,j}$$$ is the number in the $$$i$$$-th row and $$$j$$$-th column.\n\nSasha is not interested in how Mark arranges the numbers, so he only asks him to state one number — the maximum sum of MEX across all rows and columns that can be achieved.",
+        "input_format": "Each test contains multiple test cases. The first line contains a single integer $$$t$$$ ($$$1 \\le t \\le 1000$$$) — the number of test cases. The description of the test cases follows.\n\nThe first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \\le n, m \\le 10^9$$$) — the number of rows and columns in the table, respectively.",
+        "output_format": "For each test case, output the maximum possible sum of $$$\\operatorname{mex}$$$ across all rows and columns.",
+        "interaction_format": null,
+        "note": "In the first test case, the only element is $$$0$$$, and the sum of the $$$\\operatorname{mex}$$$ of the numbers in the first row and the $$$\\operatorname{mex}$$$ of the numbers in the first column is $$$\\operatorname{mex}(\\{0\\}) + \\operatorname{mex}(\\{0\\}) = 1 + 1 = 2$$$.\n\nIn the second test case, the optimal table may look as follows:\n\n$$$3$$$$$$0$$$$$$2$$$$$$1$$$\n\nThen $$$\\sum\\limits_{i = 1}^{n} \\operatorname{mex}(\\{a_{i,1}, a_{i,2}, \\ldots, a_{i,m}\\}) + \\sum\\limits_{j = 1}^{m} \\operatorname{mex}(\\{a_{1,j}, a_{2,j}, \\ldots, a_{n,j}\\}) = \\operatorname{mex}(\\{3, 0\\}) + \\operatorname{mex}(\\{2, 1\\})$$$ $$$+ \\operatorname{mex}(\\{3, 2\\}) + \\operatorname{mex}(\\{0, 1\\}) = 1 + 0 + 0 + 2 = 3$$$.",
+        "examples": [
+            {
+                "input": "3\n1 1\n2 2\n3 5",
+                "output": "2\n3\n6"
+            }
+        ],
+        "editorial": "Note that $$$0$$$ appears in exactly one row and exactly one column. As for $$$1$$$, if it exists, it can only be placed in one of the rows or columns that contain $$$0$$$, so the answer does not exceed $$$\\operatorname{max}(n, m) + 1$$$, since only one column or row can have an answer greater than one, and in another one there can be an answer of $$$1$$$. It is not difficult to provide an example where the answer is achieved: if $$$n > m$$$, we place the numbers from $$$0$$$ to $$$n - 1$$$ in the first column; otherwise, we place the numbers from $$$0$$$ to $$$m - 1$$$ in the first row. The remaining elements can be placed arbitrarily.",
+        "rating": 800,
+        "tags": [
+            "constructive algorithms",
+            "math"
+        ],
+        "testset_size": 3,
+        "official_tests": [
+            {
+                "input": "3\r\n1 1\r\n2 2\r\n3 5\r\n",
+                "output": "2\r\n3\r\n6\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2004/A",
+        "aliases": null,
+        "contest_id": "2004",
+        "contest_name": "Educational Codeforces Round 169 (Rated for Div. 2)",
+        "contest_type": "ICPC",
+        "contest_start": 1723732500,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 2.0,
+        "memory_limit": 512.0,
+        "title": "Closest Point",
+        "description": "Consider a set of points on a line. The distance between two points $$$i$$$ and $$$j$$$ is $$$|i - j|$$$.\n\nThe point $$$i$$$ from the set is the closest to the point $$$j$$$ from the set, if there is no other point $$$k$$$ in the set such that the distance from $$$j$$$ to $$$k$$$ is strictly less than the distance from $$$j$$$ to $$$i$$$. In other words, all other points from the set have distance to $$$j$$$ greater or equal to $$$|i - j|$$$.\n\nFor example, consider a set of points $$$\\{1, 3, 5, 8\\}$$$:\n\n- for the point $$$1$$$, the closest point is $$$3$$$ (other points have distance greater than $$$|1-3| = 2$$$);\n- for the point $$$3$$$, there are two closest points: $$$1$$$ and $$$5$$$;\n- for the point $$$5$$$, the closest point is $$$3$$$ (but not $$$8$$$, since its distance is greater than $$$|3-5|$$$);\n- for the point $$$8$$$, the closest point is $$$5$$$.\n\nYou are given a set of points. You have to add an integer point into this set in such a way that it is different from every existing point in the set, and it becomes the closest point to every point in the set. Is it possible?",
+        "input_format": "The first line contains one integer $$$t$$$ ($$$1 \\le t \\le 1000$$$) — the number of test cases.\n\nEach test case consists of two lines:\n\n- the first line contains one integer $$$n$$$ ($$$2 \\le n \\le 40$$$) — the number of points in the set;\n- the second line contains $$$n$$$ integers $$$x_1, x_2, \\dots, x_n$$$ ($$$1 \\le x_1 < x_2 < \\dots < x_n \\le 100$$$) — the points from the set.",
+        "output_format": "For each test case, print YES if it is possible to add a new point according to the conditions from the statement. Otherwise, print NO.",
+        "interaction_format": null,
+        "note": "In the first example, the point $$$7$$$ will be the closest to both $$$3$$$ and $$$8$$$.\n\nIn the second example, it is impossible to add an integer point so that it becomes the closest to both $$$5$$$ and $$$6$$$, and is different from both of them.",
+        "examples": [
+            {
+                "input": "3\n2\n3 8\n2\n5 6\n6\n1 2 3 4 5 10",
+                "output": "YES\nNO\nNO"
+            }
+        ],
+        "editorial": "This problem can be solved \"naively\" by iterating on the point we add and checking that it becomes the closest point to all elements from the set. However, there is another solution which is faster and easier to implement.\nWhen we add a point, it can become the closest only to two points from the set: one to the left, and one to the right. If there are, for example, two points to the left of the point we add, then the condition on the leftmost point won't be met (the point we add won't be the closest to it).\nSo, if $$$n>2$$$, there is no way to add a point. If $$$n=2$$$, we want to insert a new point between the two existing points — so we need to check that they are not adjacent.",
+        "rating": 800,
+        "tags": [
+            "implementation",
+            "math"
+        ],
+        "testset_size": 15,
+        "official_tests": [
+            {
+                "input": "3\r\n2\r\n3 8\r\n2\r\n5 6\r\n6\r\n1 2 3 4 5 10\r\n",
+                "output": "YES\r\nNO\r\nNO\r\n"
+            },
+            {
+                "input": "1\r\n3\r\n1 2 3\r\n",
+                "output": "NO\r\n"
+            },
+            {
+                "input": "1\r\n3\r\n1 6 11\r\n",
+                "output": "NO\r\n"
+            },
+            {
+                "input": "1\r\n3\r\n1 4 7\r\n",
+                "output": "NO\r\n"
+            },
+            {
+                "input": "1\r\n5\r\n1 3 5 7 9\r\n",
+                "output": "NO\r\n"
+            },
+            {
+                "input": "1\r\n3\r\n2 4 6\r\n",
+                "output": "NO\r\n"
+            },
+            {
+                "input": "1\r\n6\r\n2 4 6 8 10 12\r\n",
+                "output": "NO\r\n"
+            },
+            {
+                "input": "1\r\n3\r\n1 3 5\r\n",
+                "output": "NO\r\n"
+            },
+            {
+                "input": "1\r\n7\r\n2 4 6 8 10 12 14\r\n",
+                "output": "NO\r\n"
+            },
+            {
+                "input": "1\r\n2\r\n1 2\r\n",
+                "output": "NO\r\n"
+            },
+            {
+                "input": "1\r\n2\r\n3 4\r\n",
+                "output": "NO\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2035/A",
+        "aliases": null,
+        "contest_id": "2035",
+        "contest_name": "Codeforces Global Round 27",
+        "contest_type": "CF",
+        "contest_start": 1730039700,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Sliding",
+        "description": "There are $$$n$$$ rows of $$$m$$$ people. Let the position in the $$$r$$$-th row and the $$$c$$$-th column be denoted by $$$(r, c)$$$. Number each person starting from $$$1$$$ in row-major order, i.e., the person numbered $$$(r-1)\\cdot m+c$$$ is initially at $$$(r,c)$$$.\n\nThe person at $$$(r, c)$$$ decides to leave. To fill the gap, let the person who left be numbered $$$i$$$. Each person numbered $$$j>i$$$ will move to the position where the person numbered $$$j-1$$$ is initially at. The following diagram illustrates the case where $$$n=2$$$, $$$m=3$$$, $$$r=1$$$, and $$$c=2$$$.\n\nCalculate the sum of the Manhattan distances of each person's movement. If a person was initially at $$$(r_0, c_0)$$$ and then moved to $$$(r_1, c_1)$$$, the Manhattan distance is $$$|r_0-r_1|+|c_0-c_1|$$$.",
+        "input_format": "The first line contains a single integer $$$t$$$ ($$$1\\le t\\le 10^4$$$) — the number of test cases.\n\nThe only line of each testcase contains $$$4$$$ integers $$$n$$$, $$$m$$$, $$$r$$$, and $$$c$$$ ($$$1\\le r\\le n\\le 10^6$$$, $$$1 \\le c \\le m \\le 10^6$$$), where $$$n$$$ is the number of rows, $$$m$$$ is the number of columns, and $$$(r,c)$$$ is the position where the person who left is initially at.",
+        "output_format": "For each test case, output a single integer denoting the sum of the Manhattan distances.",
+        "interaction_format": null,
+        "note": "For the first test case, the person numbered $$$2$$$ leaves, and the distances of the movements of the person numbered $$$3$$$, $$$4$$$, $$$5$$$, and $$$6$$$ are $$$1$$$, $$$3$$$, $$$1$$$, and $$$1$$$, respectively. So the answer is $$$1+3+1+1=6$$$.\n\nFor the second test case, the person numbered $$$3$$$ leaves, and the person numbered $$$4$$$ moves. The answer is $$$1$$$.",
+        "examples": [
+            {
+                "input": "4\n2 3 1 2\n2 2 2 1\n1 1 1 1\n1000000 1000000 1 1",
+                "output": "6\n1\n0\n1999998000000"
+            }
+        ],
+        "editorial": "The people with a smaller row major number won't move at all so we can ignore them. Now we can break everyone else into $$$2$$$ groups of people. The first group will be changing rows while the second group will stay in the same row. For the people changing rows their manhattan distance will change by a total of $$$m$$$ and there are $$$n - r$$$ of these people. For the second group of people, their manhattan distance will change by $$$1$$$ and there are $$$n \\cdot m - ((r - 1) \\cdot m + c) - (n - r)$$$ of these people. Thus, the final answer will be: $$$$$$m \\cdot (n - r) + n \\cdot m - ((r - 1) \\cdot m + c) - (n - r)$$$$$$",
+        "rating": 800,
+        "tags": [
+            "implementation",
+            "math"
+        ],
+        "testset_size": 4,
+        "official_tests": [
+            {
+                "input": "4\r\n2 3 1 2\r\n2 2 2 1\r\n1 1 1 1\r\n1000000 1000000 1 1\r\n",
+                "output": "6\r\n1\r\n0\r\n1999998000000\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "1993/A",
+        "aliases": null,
+        "contest_id": "1993",
+        "contest_name": "Codeforces Round 963 (Div. 2)",
+        "contest_type": "CF",
+        "contest_start": 1722782100,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Question Marks",
+        "description": "Tim is doing a test consisting of $$$4n$$$ questions; each question has $$$4$$$ options: 'A', 'B', 'C', and 'D'. For each option, there are exactly $$$n$$$ correct answers corresponding to that option — meaning there are $$$n$$$ questions with the answer 'A', $$$n$$$ questions with the answer 'B', $$$n$$$ questions with the answer 'C', and $$$n$$$ questions with the answer 'D'.\n\nFor each question, Tim wrote his answer on the answer sheet. If he could not figure out the answer, he would leave a question mark '?' for that question.\n\nYou are given his answer sheet of $$$4n$$$ characters. What is the maximum number of correct answers Tim can get?",
+        "input_format": "The first line contains a single integer $$$t$$$ ($$$1 \\le t \\le 1000$$$) — the number of test cases.\n\nThe first line of each test case contains an integer $$$n$$$ ($$$1 \\le n \\le 100$$$).\n\nThe second line of each test case contains a string $$$s$$$ of $$$4n$$$ characters ($$$s_i \\in \\{\\texttt{A}, \\texttt{B}, \\texttt{C}, \\texttt{D}, \\texttt{?}\\}$$$) — Tim's answers for the questions.",
+        "output_format": "For each test case, print a single integer — the maximum score that Tim can achieve.",
+        "interaction_format": null,
+        "note": "In the first test case, there is exactly one question with each answer 'A', 'B', 'C', and 'D'; so it's possible that Tim gets all his answers correct.\n\nIn the second test case, there are only two correct answers 'A' which makes him get exactly $$$2$$$ points in any case.\n\nIn the third test case, Tim can get at most $$$2$$$ correct answers with option 'A' and $$$2$$$ correct answers with option 'B'. For example, he would get $$$4$$$ points if the answers were 'AACCBBDD'.\n\nIn the fourth test case, he refuses to answer any question at all, which makes him get $$$0$$$ points.",
+        "examples": [
+            {
+                "input": "6\n1\nABCD\n2\nAAAAAAAA\n2\nAAAABBBB\n2\n????????\n3\nABCABCABCABC\n5\nACADC??ACAC?DCAABC?C",
+                "output": "4\n2\n4\n0\n9\n13"
+            }
+        ],
+        "editorial": null,
+        "rating": 800,
+        "tags": [
+            "greedy",
+            "implementation"
+        ],
+        "testset_size": 5,
+        "official_tests": [
+            {
+                "input": "6\r\n1\r\nABCD\r\n2\r\nAAAAAAAA\r\n2\r\nAAAABBBB\r\n2\r\n????????\r\n3\r\nABCABCABCABC\r\n5\r\nACADC??ACAC?DCAABC?C\r\n",
+                "output": "4\r\n2\r\n4\r\n0\r\n9\r\n13\r\n"
+            },
+            {
+                "input": "1\r\n5\r\nA?B?C?D??D?C?B?A????\r\n",
+                "output": "8\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2059/A",
+        "aliases": null,
+        "contest_id": "2059",
+        "contest_name": "Codeforces Round 1002 (Div. 2)",
+        "contest_type": "CF",
+        "contest_start": 1738506900,
+        "contest_start_year": 2025,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Milya and Two Arrays",
+        "description": "An array is called good if for any element $$$x$$$ that appears in this array, it holds that $$$x$$$ appears at least twice in this array. For example, the arrays $$$[1, 2, 1, 1, 2]$$$, $$$[3, 3]$$$, and $$$[1, 2, 4, 1, 2, 4]$$$ are good, while the arrays $$$[1]$$$, $$$[1, 2, 1]$$$, and $$$[2, 3, 4, 4]$$$ are not good.\n\nMilya has two good arrays $$$a$$$ and $$$b$$$ of length $$$n$$$. She can rearrange the elements in array $$$a$$$ in any way. After that, she obtains an array $$$c$$$ of length $$$n$$$, where $$$c_i = a_i + b_i$$$ ($$$1 \\le i \\le n$$$).\n\nDetermine whether Milya can rearrange the elements in array $$$a$$$ such that there are at least $$$3$$$ distinct elements in array $$$c$$$.",
+        "input_format": "Each test consists of multiple test cases. The first line contains a single integer $$$t$$$ ($$$1 \\le t \\le 1000$$$) — the number of test cases. The description of the test cases follows.\n\nThe first line of each test case contains a single integer $$$n$$$ ($$$3 \\le n \\le 50$$$) — the length of the arrays $$$a$$$ and $$$b$$$.\n\nThe second line of each test case contains $$$n$$$ integers $$$a_1, a_2, \\ldots, a_n$$$ ($$$1 \\le a_i \\le 10^9$$$) — the elements of the array $$$a$$$.\n\nThe third line of each test case contains $$$n$$$ integers $$$b_1, b_2, \\ldots, b_n$$$ ($$$1 \\le b_i \\le 10^9$$$) — the elements of the array $$$b$$$.",
+        "output_format": "For each test case, output $$$«$$$YES$$$»$$$ (without quotes) if it is possible to obtain at least $$$3$$$ distinct elements in array $$$c$$$, and $$$«$$$NO$$$»$$$ otherwise.\n\nYou can output each letter in any case (for example, $$$«$$$YES$$$»$$$, $$$«$$$Yes$$$»$$$, $$$«$$$yes$$$»$$$, $$$«$$$yEs$$$»$$$ will be recognized as a positive answer).",
+        "interaction_format": null,
+        "note": "In the first test case, you can swap the second and third elements. Then the array $$$a = [1, 1, 2, 2]$$$, $$$b = [1, 2, 1, 2]$$$, and then $$$c = [2, 3, 3, 4]$$$.\n\nIn the second test case, you can leave the elements unchanged. Then $$$c = [2, 3, 4, 4, 3, 2]$$$.\n\nIn the third test case, the array $$$a$$$ will not change from rearranging the elements in it. Then $$$c = [2, 2, 2]$$$, so the answer is $$$«$$$NO$$$»$$$.",
+        "examples": [
+            {
+                "input": "5\n4\n1 2 1 2\n1 2 1 2\n6\n1 2 3 3 2 1\n1 1 1 1 1 1\n3\n1 1 1\n1 1 1\n6\n1 52 52 3 1 3\n59 4 3 59 3 4\n4\n100 1 100 1\n2 2 2 2",
+                "output": "YES\nYES\nNO\nYES\nNO"
+            }
+        ],
+        "editorial": "If there are at least three distinct values in any of the arrays, the answer is $$$yes$$$. We can select any three elements from the other array and arrange them in increasing order, then the corresponding sums will also increase.\nIf each array contains exactly two distinct values, the answer is also $$$yes$$$. If we find elements $$$x$$$ and $$$y$$$ in the first array, where $$$x < y$$$, and elements $$$a$$$ and $$$b$$$ in the second array, such that $$$a < b$$$, we can match the elements to obtain the sums $$$x + a < x + b < y + b$$$. Since the arrays are good, $$$x$$$ appears at least twice, and all three sums can definitely be obtained.\nIn the remaining cases, one of the arrays will contain only equal elements. Since the second array contains no more than $$$2$$$ distinct values, it will not be possible to obtain $$$3$$$ distinct sums.",
+        "rating": 800,
+        "tags": [
+            "constructive algorithms",
+            "greedy",
+            "sortings"
+        ],
+        "testset_size": 6,
+        "official_tests": [
+            {
+                "input": "5\r\n4\r\n1 2 1 2\r\n1 2 1 2\r\n6\r\n1 2 3 3 2 1\r\n1 1 1 1 1 1\r\n3\r\n1 1 1\r\n1 1 1\r\n6\r\n1 52 52 3 1 3\r\n59 4 3 59 3 4\r\n4\r\n100 1 100 1\r\n2 2 2 2\r\n",
+                "output": "YES\r\nYES\r\nNO\r\nYES\r\nNO\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2027/A",
+        "aliases": null,
+        "contest_id": "2027",
+        "contest_name": "Codeforces Round 982 (Div. 2)",
+        "contest_type": "CF",
+        "contest_start": 1729953300,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Rectangle Arrangement",
+        "description": "You are coloring an infinite square grid, in which all cells are initially white. To do this, you are given $$$n$$$ stamps. Each stamp is a rectangle of width $$$w_i$$$ and height $$$h_i$$$.\n\nYou will use each stamp exactly once to color a rectangle of the same size as the stamp on the grid in black. You cannot rotate the stamp, and for each cell, the stamp must either cover it fully or not cover it at all. You can use the stamp at any position on the grid, even if some or all of the cells covered by the stamping area are already black.\n\nWhat is the minimum sum of the perimeters of the connected regions of black squares you can obtain after all the stamps have been used?",
+        "input_format": "Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \\le t \\le 500$$$). The description of the test cases follows.\n\nThe first line of each test case contains a single integer $$$n$$$ ($$$1 \\le n \\le 100$$$).\n\nThe $$$i$$$-th of the next $$$n$$$ lines contains two integers $$$w_i$$$ and $$$h_i$$$ ($$$1 \\le w_i, h_i \\le 100$$$).",
+        "output_format": "For each test case, output a single integer — the minimum sum of the perimeters of the connected regions of black squares you can obtain after all the stamps have been used.",
+        "interaction_format": null,
+        "note": "In the first test case, the stamps can be used as shown on the left. Each stamp is highlighted in its own color for clarity.\n\nAfter all these stamps are used, there is one black region (as shown on the right), and its perimeter is $$$20$$$. It can be shown that there is no way of using the stamps that yields a lower total perimeter.\n\nIn the second test case, the second and third stamps can be used entirely inside the first one, so the minimum perimeter is equal to $$$8$$$.",
+        "examples": [
+            {
+                "input": "5\n5\n1 5\n2 4\n3 3\n4 2\n5 1\n3\n2 2\n1 1\n1 2\n1\n3 2\n3\n100 100\n100 100\n100 100\n4\n1 4\n2 3\n1 5\n3 2",
+                "output": "20\n8\n10\n400\n16"
+            }
+        ],
+        "editorial": "We must minimize the perimeter, and an obvious way to attempt this is to maximize the overlap. To achieve this, we can place each stamp such that the lower left corner of each stamp is at the same position, like shown in the sample explanation.\nNow, we can observe that the perimeter of this shape is determined solely by the maximum height and width of any stamp, and these values cannot be reduced further. Therefore, the answer is $$$2 \\cdot \\bigl( \\max(a_i) + \\max(b_i) \\bigr)$$$.\nFurthermore, it's true that any arrangement of stamps which are fully enclosed in an outer rectangular area of $$$\\max(a_i)$$$ by $$$\\max(b_i)$$$ will yield the same perimeter.",
+        "rating": 800,
+        "tags": [
+            "geometry",
+            "implementation",
+            "math"
+        ],
+        "testset_size": 3,
+        "official_tests": [
+            {
+                "input": "5\r\n5\r\n1 5\r\n2 4\r\n3 3\r\n4 2\r\n5 1\r\n3\r\n2 2\r\n1 1\r\n1 2\r\n1\r\n3 2\r\n3\r\n100 100\r\n100 100\r\n100 100\r\n4\r\n1 4\r\n2 3\r\n1 5\r\n3 2\r\n",
+                "output": "20\r\n8\r\n10\r\n400\r\n16\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2060/A",
+        "aliases": null,
+        "contest_id": "2060",
+        "contest_name": "Codeforces Round 998 (Div. 3)",
+        "contest_type": "ICPC",
+        "contest_start": 1737297300,
+        "contest_start_year": 2025,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Fibonacciness",
+        "description": "There is an array of $$$5$$$ integers. Initially, you only know $$$a_1,a_2,a_4,a_5$$$. You may set $$$a_3$$$ to any positive integer, negative integer, or zero. The Fibonacciness of the array is the number of integers $$$i$$$ ($$$1 \\leq i \\leq 3$$$) such that $$$a_{i+2}=a_i+a_{i+1}$$$. Find the maximum Fibonacciness over all integer values of $$$a_3$$$.",
+        "input_format": "The first line contains an integer $$$t$$$ ($$$1 \\leq t \\leq 500$$$) — the number of test cases.\n\nThe only line of each test case contains four integers $$$a_1, a_2, a_4, a_5$$$ ($$$1 \\leq a_i \\leq 100$$$).",
+        "output_format": "For each test case, output the maximum Fibonacciness on a new line.",
+        "interaction_format": null,
+        "note": "In the first test case, we can set $$$a_3$$$ to $$$2$$$ to achieve the maximal Fibonacciness of $$$3$$$.\n\nIn the third test case, it can be shown that $$$2$$$ is the maximum Fibonacciness that can be achieved. This can be done by setting $$$a_3$$$ to $$$18$$$.",
+        "examples": [
+            {
+                "input": "6\n1 1 3 5\n1 3 2 1\n8 10 28 100\n100 1 100 1\n1 100 1 100\n100 100 100 100",
+                "output": "3\n2\n2\n1\n1\n2"
+            }
+        ],
+        "editorial": null,
+        "rating": 800,
+        "tags": [
+            "brute force"
+        ],
+        "testset_size": 3,
+        "official_tests": [
+            {
+                "input": "6\r\n1 1 3 5\r\n1 3 2 1\r\n8 10 28 100\r\n100 1 100 1\r\n1 100 1 100\r\n100 100 100 100\r\n",
+                "output": "3\r\n2\r\n2\r\n1\r\n1\r\n2\r\n"
+            },
+            {
+                "input": "1\r\n7 8 22 38\r\n",
+                "output": "1\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2051/A",
+        "aliases": null,
+        "contest_id": "2051",
+        "contest_name": "Codeforces Round 995 (Div. 3)",
+        "contest_type": "ICPC",
+        "contest_start": 1734878100,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 2.0,
+        "memory_limit": 256.0,
+        "title": "Preparing for the Olympiad",
+        "description": "Monocarp and Stereocarp are preparing for the Olympiad. There are $$$n$$$ days left until the Olympiad. On the $$$i$$$-th day, if Monocarp plans to practice, he will solve $$$a_i$$$ problems. Similarly, if Stereocarp plans to practice on the same day, he will solve $$$b_i$$$ problems.\n\nMonocarp can train on any day he wants. However, Stereocarp watches Monocarp and follows a different schedule: if Monocarp trained on day $$$i$$$ and $$$i < n$$$, then Stereocarp will train on day $$$(i+1)$$$.\n\nMonocarp wants to organize his training process in a way that the difference between the number of problems he solves and the number of problems Stereocarp solves is as large as possible. Formally, Monocarp wants to maximize the value of $$$(m-s)$$$, where $$$m$$$ is the number of problems he solves, and $$$s$$$ is the number of problems Stereocarp solves. Help Monocarp determine the maximum possible difference in the number of solved problems between them.",
+        "input_format": "The first line contains a single integer $$$t$$$ ($$$1 \\le t \\le 10^3$$$) — the number of test cases.\n\nThe first line of each test case contains a single integer $$$n$$$ ($$$1 \\le n \\le 100$$$).\n\nThe second line contains $$$n$$$ integers $$$a_1, a_2, \\dots, a_n$$$ ($$$1 \\le a_i \\le 100$$$).\n\nThe third line contains $$$n$$$ integers $$$b_1, b_2, \\dots, b_n$$$ ($$$1 \\le b_i \\le 100$$$).",
+        "output_format": "For each test case, print a single integer — the maximum possible difference between the number of problems Monocarp solves and the number of problems Stereocarp solves.",
+        "interaction_format": null,
+        "note": "Let's analyze the example from the statement:\n\n- In the first test case, it is optimal for Monocarp to train both days; then Stereocarp will train on day $$$2$$$.\n- In the second test case, it is optimal for Monocarp to train on the only day, and Stereocarp will not train at all.\n- In the third test case, it is optimal for Monocarp to train on the last day (and only on that day).\n- In the fourth test case, it is optimal for Monocarp to train on days $$$1, 3, 4, 6$$$; then Stereocarp will train on days $$$2, 4, 5$$$.",
+        "examples": [
+            {
+                "input": "4\n2\n3 2\n2 1\n1\n5\n8\n3\n1 1 1\n2 2 2\n6\n8 2 5 6 2 6\n8 2 7 4 3 4",
+                "output": "4\n5\n1\n16"
+            }
+        ],
+        "editorial": "Let's consider what contribution each day that Monokarp trains makes to the difference. For each day, except the last one, if Monokarp trains on that day, then the number of problems he has solved will increase by $$$a_i$$$, and the number of problems solved by Stereokarp will increase by $$$b_{i+1}$$$. Therefore, if $$$a_i - b_{i+1} > 0$$$, it is beneficial for Monokarp to train on the $$$i$$$-th day; otherwise, it is not beneficial.\nOn the last day, it is always beneficial to train, as Stereokarp will not solve anything on the day following it.",
+        "rating": 800,
+        "tags": [
+            "greedy"
+        ],
+        "testset_size": 5,
+        "official_tests": [
+            {
+                "input": "4\r\n2\r\n3 2\r\n2 1\r\n1\r\n5\r\n8\r\n3\r\n1 1 1\r\n2 2 2\r\n6\r\n8 2 5 6 2 6\r\n8 2 7 4 3 4\r\n",
+                "output": "4\r\n5\r\n1\r\n16\r\n"
+            },
+            {
+                "input": "2\r\n17\r\n21 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1\r\n25 1 1 1 1 1 1 1 1 1 1 1 1 1 1 17 19\r\n5\r\n1 2 3 5 1\r\n1 1 1 1 1\r\n",
+                "output": "21\r\n8\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2051/B",
+        "aliases": null,
+        "contest_id": "2051",
+        "contest_name": "Codeforces Round 995 (Div. 3)",
+        "contest_type": "ICPC",
+        "contest_start": 1734878100,
+        "contest_start_year": 2024,
+        "index": "B",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Journey",
+        "description": "Monocarp decided to embark on a long hiking journey.\n\nHe decided that on the first day he would walk $$$a$$$ kilometers, on the second day he would walk $$$b$$$ kilometers, on the third day he would walk $$$c$$$ kilometers, on the fourth day, just like on the first, he would walk $$$a$$$ kilometers, on the fifth day, just like on the second, he would walk $$$b$$$ kilometers, on the sixth day, just like on the third, he would walk $$$c$$$ kilometers, and so on.\n\nMonocarp will complete his journey on the day when he has walked at least $$$n$$$ kilometers in total. Your task is to determine the day on which Monocarp will complete his journey.",
+        "input_format": "The first line contains one integer $$$t$$$ ($$$1 \\le t \\le 10^4$$$) — the number of test cases.\n\nEach test case consists of one line containing four integers $$$n$$$, $$$a$$$, $$$b$$$, $$$c$$$ ($$$1 \\le n \\le 10^9$$$; $$$1 \\le a, b, c \\le 10^6$$$).",
+        "output_format": "For each test case, output one integer — the day on which Monocarp will have walked at least $$$n$$$ kilometers in total and will complete his journey.",
+        "interaction_format": null,
+        "note": "In the first example, over the first four days, Monocarp will cover $$$1 + 5 + 3 + 1 = 10$$$ kilometers. On the fifth day, he will cover another $$$5$$$ kilometers, meaning that in total over five days he will have covered $$$10 + 5 = 15$$$ kilometers. Since $$$n = 12$$$, Monocarp will complete his journey on the fifth day.\n\nIn the second example, Monocarp will cover $$$6$$$ kilometers on the first day. Since $$$n = 6$$$, Monocarp will complete his journey on the very first day.\n\nIn the third example, Monocarp will cover $$$3 + 4 + 1 + 3 + 4 + 1 = 16$$$ kilometers over the first six days. Since $$$n = 16$$$, Monocarp will complete his journey on the sixth day.",
+        "examples": [
+            {
+                "input": "4\n12 1 5 3\n6 6 7 4\n16 3 4 1\n1000000000 1 1 1",
+                "output": "5\n1\n6\n1000000000"
+            }
+        ],
+        "editorial": "Processing every day separately is too slow. Instead, we will use the fact that every three days, the number of kilometers Monocarp walks repeats, and process days in \"triples\".\nDuring every three days, Monocarp walks exactly $$$(a+b+c)$$$ kilometers, so we can do the following: while $$$n \\ge a + b + c$$$, subtract $$$(a+b+c)$$$ from $$$n$$$ and increase the answer by $$$3$$$; and finally, process the remaining days, since there will be at most $$$3$$$ of them.\nHowever, this works in $$$O(n)$$$ per test case, so it is still too slow. We need to improve the part when we subtract $$$a+b+c$$$ from $$$n$$$ until $$$n$$$ becomes less than this sum. Does this sound familiar?\nThe number of times we need to subtract $$$(a+b+c)$$$ from $$$n$$$ is exactly $$$\\lfloor \\frac{n}{a+b+c} \\rfloor$$$, and the number we get after that is $$$n \\bmod (a+b+c)$$$ by definition of integer division and remainder. This allows us to process all \"triples\" in $$$O(1)$$$, instead of running a loop in $$$O(n)$$$.\nThe solution we get works in $$$O(1)$$$ per test case.",
+        "rating": 800,
+        "tags": [
+            "binary search",
+            "math"
+        ],
+        "testset_size": 5,
+        "official_tests": [
+            {
+                "input": "4\r\n12 1 5 3\r\n6 6 7 4\r\n16 3 4 1\r\n1000000000 1 1 1\r\n",
+                "output": "5\r\n1\r\n6\r\n1000000000\r\n"
+            },
+            {
+                "input": "3\r\n10000 69 69 69\r\n84 17 1 1\r\n15 5 5 5\r\n",
+                "output": "145\r\n13\r\n3\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2033/A",
+        "aliases": null,
+        "contest_id": "2033",
+        "contest_name": "Codeforces Round 981 (Div. 3)",
+        "contest_type": "ICPC",
+        "contest_start": 1729780500,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Sakurako and Kosuke",
+        "description": "Sakurako and Kosuke decided to play some games with a dot on a coordinate line. The dot is currently located in position $$$x=0$$$. They will be taking turns, and Sakurako will be the one to start.\n\nOn the $$$i$$$-th move, the current player will move the dot in some direction by $$$2\\cdot i-1$$$ units. Sakurako will always be moving the dot in the negative direction, whereas Kosuke will always move it in the positive direction.\n\nIn other words, the following will happen:\n\n1. Sakurako will change the position of the dot by $$$-1$$$, $$$x = -1$$$ now\n2. Kosuke will change the position of the dot by $$$3$$$, $$$x = 2$$$ now\n3. Sakurako will change the position of the dot by $$$-5$$$, $$$x = -3$$$ now\n4. $$$\\cdots$$$\n\nThey will keep on playing while the absolute value of the coordinate of the dot does not exceed $$$n$$$. More formally, the game continues while $$$-n\\le x\\le n$$$. It can be proven that the game will always end.\n\nYour task is to determine who will be the one who makes the last turn.",
+        "input_format": "The first line contains one integer $$$t$$$ ($$$1\\le t\\le 100$$$) — the number of games that Sakurako and Kosuke played.\n\nEach game is described by one number $$$n$$$ ($$$1 \\le n\\le 100$$$) — the number that defines the condition when the game ends.",
+        "output_format": "For each of the $$$t$$$ games, output a line with the result of that game. If Sakurako makes the last turn, output \"Sakurako\" (without quotes); else output \"Kosuke\".",
+        "interaction_format": null,
+        "note": null,
+        "examples": [
+            {
+                "input": "4\n1\n6\n3\n98",
+                "output": "Kosuke\nSakurako\nKosuke\nSakurako"
+            }
+        ],
+        "editorial": "For this task we could just brute-force the answer by repeatedly adding or substracting the odd numbers from the initial position $$$0$$$. This would result in $$$O(n)$$$ time complexity. This is sufficient enough.",
+        "rating": 800,
+        "tags": [
+            "constructive algorithms",
+            "implementation",
+            "math"
+        ],
+        "testset_size": 2,
+        "official_tests": [
+            {
+                "input": "4\r\n1\r\n6\r\n3\r\n98\r\n",
+                "output": "Kosuke\r\nSakurako\r\nKosuke\r\nSakurako\r\n"
+            },
+            {
+                "input": "100\r\n100\r\n99\r\n98\r\n97\r\n96\r\n95\r\n94\r\n93\r\n92\r\n91\r\n90\r\n89\r\n88\r\n87\r\n86\r\n85\r\n84\r\n83\r\n82\r\n81\r\n80\r\n79\r\n78\r\n77\r\n76\r\n75\r\n74\r\n73\r\n72\r\n71\r\n70\r\n69\r\n68\r\n67\r\n66\r\n65\r\n64\r\n63\r\n62\r\n61\r\n60\r\n59\r\n58\r\n57\r\n56\r\n55\r\n54\r\n53\r\n52\r\n51\r\n50\r\n49\r\n48\r\n47\r\n46\r\n45\r\n44\r\n43\r\n42\r\n41\r\n40\r\n39\r\n38\r\n37\r\n36\r\n35\r\n34\r\n33\r\n32\r\n31\r\n30\r\n29\r\n28\r\n27\r\n26\r\n25\r\n24\r\n23\r\n22\r\n21\r\n20\r\n19\r\n18\r\n17\r\n16\r\n15\r\n14\r\n13\r\n12\r\n11\r\n10\r\n9\r\n8\r\n7\r\n6\r\n5\r\n4\r\n3\r\n2\r\n1\r\n",
+                "output": "Sakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\n"
+            }
+        ],
+        "official_tests_complete": true,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2049/A",
+        "aliases": null,
+        "contest_id": "2049",
+        "contest_name": "Codeforces Round 994 (Div. 2)",
+        "contest_type": "CF",
+        "contest_start": 1734705300,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "MEX Destruction",
+        "description": "Evirir the dragon snuck into a wizard's castle and found a mysterious contraption, and their playful instincts caused them to play with (destroy) it...\n\nEvirir the dragon found an array $$$a_1, a_2, \\ldots, a_n$$$ of $$$n$$$ non-negative integers.\n\nIn one operation, they can choose a non-empty subarray$$$^{\\text{∗}}$$$ $$$b$$$ of $$$a$$$ and replace it with the integer $$$\\operatorname{mex}(b)$$$$$$^{\\text{†}}$$$. They want to use this operation any number of times to make $$$a$$$ only contain zeros. It can be proven that this is always possible under the problem constraints.\n\nWhat is the minimum number of operations needed?",
+        "input_format": "Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \\le t \\le 200$$$). The description of the test cases follows.\n\nThe first line of each test case contains a single integer $$$n$$$ ($$$1 \\le n \\le 50$$$), the length of $$$a$$$.\n\nThe second line of each test case contains $$$n$$$ space-separated integers, $$$a_1, a_2, \\ldots, a_n$$$ ($$$0 \\le a_i \\le 100$$$).\n\nIt is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$500$$$.",
+        "output_format": "For each test case, output a single integer on a line, the minimum number of operations needed to make $$$a$$$ contain only zeros.",
+        "interaction_format": null,
+        "note": "In the first test case, Evirir can choose the subarray $$$b = [1, 2, 3]$$$ and replace it with $$$\\operatorname{mex}(1, 2, 3) = 0$$$, changing $$$a$$$ from $$$[0, \\underline{1, 2, 3}]$$$ to $$$[0, 0]$$$ (where the chosen subarray is underlined). Therefore, the answer is $$$1$$$.\n\nIn the second test case, $$$a$$$ already contains only $$$0$$$s, so no operation is needed.\n\nIn the third test case, Evirir can change $$$a$$$ as follows: $$$[1, \\underline{0, 1, 0, 1}] \\to [\\underline{1, 2}] \\to [0]$$$. Here, $$$\\operatorname{mex}(0, 1, 0, 1) = 2$$$ and $$$\\operatorname{mex}(1, 2) = 0$$$.\n\nIn the fourth test case, Evirir can choose $$$b$$$ to be the entire array $$$a$$$, changing $$$a$$$ from $$$[\\underline{3, 1, 4, 1, 5}]$$$ to $$$[0]$$$.",
+        "examples": [
+            {
+                "input": "10\n4\n0 1 2 3\n6\n0 0 0 0 0 0\n5\n1 0 1 0 1\n5\n3 1 4 1 5\n4\n3 2 1 0\n7\n9 100 0 89 12 2 3\n4\n0 3 9 0\n7\n0 7 0 2 0 7 0\n1\n0\n2\n0 1",
+                "output": "1\n0\n2\n1\n1\n2\n1\n2\n0\n1"
+            }
+        ],
+        "editorial": null,
+        "rating": 800,
+        "tags": [
+            "greedy",
+            "implementation"
+        ],
+        "testset_size": 6,
+        "official_tests": [
+            {
+                "input": "10\r\n4\r\n0 1 2 3\r\n6\r\n0 0 0 0 0 0\r\n5\r\n1 0 1 0 1\r\n5\r\n3 1 4 1 5\r\n4\r\n3 2 1 0\r\n7\r\n9 100 0 89 12 2 3\r\n4\r\n0 3 9 0\r\n7\r\n0 7 0 2 0 7 0\r\n1\r\n0\r\n2\r\n0 1\r\n",
+                "output": "1\r\n0\r\n2\r\n1\r\n1\r\n2\r\n1\r\n2\r\n0\r\n1\r\n"
+            },
+            {
+                "input": "1\r\n8\r\n1 0 1 0 1 0 1 0\r\n",
+                "output": "2\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2002/A",
+        "aliases": null,
+        "contest_id": "2002",
+        "contest_name": "EPIC Institute of Technology Round August 2024 (Div. 1 + Div. 2)",
+        "contest_type": "CF",
+        "contest_start": 1723386900,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Distanced Coloring",
+        "description": "You received an $$$n\\times m$$$ grid from a mysterious source. The source also gave you a magic positive integer constant $$$k$$$.\n\nThe source told you to color the grid with some colors, satisfying the following condition:\n\n- If $$$(x_1,y_1)$$$, $$$(x_2,y_2)$$$ are two distinct cells with the same color, then $$$\\max(|x_1-x_2|,|y_1-y_2|)\\ge k$$$.\n\nYou don't like using too many colors. Please find the minimum number of colors needed to color the grid.",
+        "input_format": "Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1\\le t\\le1000$$$). The description of the test cases follows.\n\nThe only line of each test case consists of three positive integers $$$n$$$, $$$m$$$, $$$k$$$ ($$$1\\le n,m,k\\le10^4$$$) — the dimensions of the grid and the magic constant.",
+        "output_format": "For each test case, print a single integer — the minimum number of colors needed to color the grid.",
+        "interaction_format": null,
+        "note": "In the first test case, one of the optimal constructions is:\n\nIn the second test case, the color of all cells must be pairwise different, so the answer is $$$5$$$.",
+        "examples": [
+            {
+                "input": "6\n3 3 2\n5 1 10000\n7 3 4\n3 2 7\n8 9 6\n2 5 4",
+                "output": "4\n5\n12\n6\n36\n8"
+            }
+        ],
+        "editorial": null,
+        "rating": 800,
+        "tags": [
+            "constructive algorithms",
+            "implementation",
+            "math"
+        ],
+        "testset_size": 7,
+        "official_tests": [
+            {
+                "input": "6\r\n3 3 2\r\n5 1 10000\r\n7 3 4\r\n3 2 7\r\n8 9 6\r\n2 5 4\r\n",
+                "output": "4\r\n5\r\n12\r\n6\r\n36\r\n8\r\n"
+            },
+            {
+                "input": "7\r\n10000 10000 10000\r\n10000 1 10000\r\n10000 1 1\r\n1 1 1\r\n1 10000 10000\r\n10 10000 1\r\n10000 1 10\r\n",
+                "output": "100000000\r\n10000\r\n1\r\n1\r\n10000\r\n1\r\n10\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2055/A",
+        "aliases": null,
+        "contest_id": "2055",
+        "contest_name": "Codeforces Round 996 (Div. 2)",
+        "contest_type": "CF",
+        "contest_start": 1736692500,
+        "contest_start_year": 2025,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Two Frogs",
+        "description": "Roaming through the alligator-infested Everglades, Florida Man encounters a most peculiar showdown.\n\nThere are $$$n$$$ lilypads arranged in a row, numbered from $$$1$$$ to $$$n$$$ from left to right. Alice and Bob are frogs initially positioned on distinct lilypads, $$$a$$$ and $$$b$$$, respectively. They take turns jumping, starting with Alice.\n\nDuring a frog's turn, it can jump either one space to the left or one space to the right, as long as the destination lilypad exists. For example, on Alice's first turn, she can jump to either lilypad $$$a-1$$$ or $$$a+1$$$, provided these lilypads are within bounds. It is important to note that each frog must jump during its turn and cannot remain on the same lilypad.\n\nHowever, there are some restrictions:\n\n- The two frogs cannot occupy the same lilypad. This means that Alice cannot jump to a lilypad that Bob is currently occupying, and vice versa.\n- If a frog cannot make a valid jump on its turn, it loses the game. As a result, the other frog wins.\n\nDetermine whether Alice can guarantee a win, assuming that both players play optimally. It can be proven that the game will end after a finite number of moves if both players play optimally.",
+        "input_format": "Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \\le t \\le 500$$$). The description of the test cases follows.\n\nThe first and only line of each test case contains three integers $$$n$$$, $$$a$$$, and $$$b$$$ ($$$2 \\leq n \\leq 100$$$, $$$1 \\leq a, b \\leq n$$$, $$$a\\neq b$$$) — the number of lilypads, and the starting positions of Alice and Bob, respectively.\n\nNote that there are no constraints on the sum of $$$n$$$ over all test cases.",
+        "output_format": "For each test case, print a single line containing either \"YES\" or \"NO\", representing whether or not Alice has a winning strategy.\n\nYou can output the answer in any case (upper or lower). For example, the strings \"yEs\", \"yes\", \"Yes\", and \"YES\" will be recognized as positive responses.",
+        "interaction_format": null,
+        "note": "In the first test case, Alice has no legal moves. Therefore, Alice loses on the first turn.\n\nIn the second test case, Alice can only move to lilypad $$$2$$$. Then, Bob has no legal moves. Therefore, Alice has a winning strategy in this case.\n\nIn the third test case, Alice can only move to lilypad $$$1$$$. Then, Bob can move to lilypad $$$2$$$. Alice is no longer able to move and loses, giving Bob the win. It can be shown that Bob can always win regardless of Alice's moves; hence, Alice does not have a winning strategy.",
+        "examples": [
+            {
+                "input": "5\n2 1 2\n3 3 1\n4 2 3\n5 2 4\n7 6 2",
+                "output": "NO\nYES\nNO\nYES\nYES"
+            }
+        ],
+        "editorial": "Notice that regardless of how the players move, the difference between the numbers of the lilypads the two players are standing on always alternates even and odd, depending on the starting configuration. Then, the key observation is that exactly one player has the following winning strategy:\n• Walk towards the other player, and do not stop until they are forced onto lilypad $$$1$$$ or $$$n$$$.\nIn the case that they start on lilypads with different parity, analogous logic shows that Bob wins. Therefore, for each case, we only need to check the parity of the lilypads for a constant time solution.",
+        "rating": 800,
+        "tags": [
+            "constructive algorithms",
+            "games",
+            "greedy",
+            "math"
+        ],
+        "testset_size": 4,
+        "official_tests": [
+            {
+                "input": "5\r\n2 1 2\r\n3 3 1\r\n4 2 3\r\n5 2 4\r\n7 6 2\r\n",
+                "output": "NO\r\nYES\r\nNO\r\nYES\r\nYES\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2037/A",
+        "aliases": null,
+        "contest_id": "2037",
+        "contest_name": "Codeforces Round 988 (Div. 3)",
+        "contest_type": "ICPC",
+        "contest_start": 1731854100,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Twice",
+        "description": "Kinich wakes up to the start of a new day. He turns on his phone, checks his mailbox, and finds a mysterious present. He decides to unbox the present.\n\nKinich unboxes an array $$$a$$$ with $$$n$$$ integers. Initially, Kinich's score is $$$0$$$. He will perform the following operation any number of times:\n\n- Select two indices $$$i$$$ and $$$j$$$ $$$(1 \\leq i < j \\leq n)$$$ such that neither $$$i$$$ nor $$$j$$$ has been chosen in any previous operation and $$$a_i = a_j$$$. Then, add $$$1$$$ to his score.\n\nOutput the maximum score Kinich can achieve after performing the aforementioned operation any number of times.",
+        "input_format": "The first line contains an integer $$$t$$$ ($$$1 \\leq t \\leq 500$$$) — the number of test cases.\n\nThe first line of each test case contains an integer $$$n$$$ ($$$1 \\leq n \\leq 20$$$) — the length of $$$a$$$.\n\nThe following line of each test case contains $$$n$$$ space-separated integers $$$a_1, a_2, \\ldots, a_n$$$ ($$$1 \\leq a_i \\leq n$$$).",
+        "output_format": "For each test case, output the maximum score achievable on a new line.",
+        "interaction_format": null,
+        "note": "In the first and third testcases, Kinich cannot perform any operations.\n\nIn the second testcase, Kinich can perform one operation with $$$i=1$$$ and $$$j=2$$$.\n\nIn the fourth testcase, Kinich can perform one operation with $$$i=1$$$ and $$$j=4$$$.",
+        "examples": [
+            {
+                "input": "5\n1\n1\n2\n2 2\n2\n1 2\n4\n1 2 3 1\n6\n1 2 3 1 2 3",
+                "output": "0\n1\n0\n1\n3"
+            }
+        ],
+        "editorial": null,
+        "rating": 800,
+        "tags": [
+            "implementation"
+        ],
+        "testset_size": 2,
+        "official_tests": [
+            {
+                "input": "5\r\n1\r\n1\r\n2\r\n2 2\r\n2\r\n1 2\r\n4\r\n1 2 3 1\r\n6\r\n1 2 3 1 2 3\r\n",
+                "output": "0\r\n1\r\n0\r\n1\r\n3\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2037/B",
+        "aliases": null,
+        "contest_id": "2037",
+        "contest_name": "Codeforces Round 988 (Div. 3)",
+        "contest_type": "ICPC",
+        "contest_start": 1731854100,
+        "contest_start_year": 2024,
+        "index": "B",
+        "time_limit": 2.0,
+        "memory_limit": 256.0,
+        "title": "Intercepted Inputs",
+        "description": "To help you prepare for your upcoming Codeforces contest, Citlali set a grid problem and is trying to give you a $$$n$$$ by $$$m$$$ grid through your input stream. Specifically, your input stream should contain the following:\n\n- The first line contains two integers $$$n$$$ and $$$m$$$ — the dimensions of the grid.\n- The following $$$n$$$ lines contain $$$m$$$ integers each — the values of the grid.\n\nHowever, someone has intercepted your input stream, shuffled all given integers, and put them all on one line! Now, there are $$$k$$$ integers all on one line, and you don't know where each integer originally belongs. Instead of asking Citlali to resend the input, you decide to determine the values of $$$n$$$ and $$$m$$$ yourself.\n\nOutput any possible value of $$$n$$$ and $$$m$$$ that Citlali could have provided.",
+        "input_format": "The first line contains an integer $$$t$$$ ($$$1 \\leq t \\leq 10^4$$$) — the number of test cases.\n\nThe first line of each test case contains an integer $$$k$$$ ($$$3 \\leq k \\leq 2 \\cdot 10^5$$$) — the total number of inputs in your input stream.\n\nThe following line of each test case contains $$$k$$$ integers $$$a_1, a_2, \\ldots, a_k$$$ ($$$1 \\leq a_i \\leq k$$$) — the shuffled inputs of your input stream. It is guaranteed that $$$n$$$ and $$$m$$$ are contained within the $$$k$$$ integers.\n\nIt is guaranteed that the sum of $$$k$$$ over all test cases does not exceed $$$2 \\cdot 10^5$$$.",
+        "output_format": "For each test case, output two integers, one possible value of $$$n$$$ and $$$m$$$. If multiple possible answers exist, output any.",
+        "interaction_format": null,
+        "note": "In the first test case, the initial input could have been the following:\n\n1 1\n\n2\n\nIn the second test case, the initial input could have been the following:\n\n3 3\n\n4 5 6\n\n7 8 9\n\n9 10 11",
+        "examples": [
+            {
+                "input": "5\n3\n1 1 2\n11\n3 3 4 5 6 7 8 9 9 10 11\n8\n8 4 8 3 8 2 8 1\n6\n2 1 4 5 3 3\n8\n1 2 6 3 8 5 5 3",
+                "output": "1 1\n3 3\n2 3\n4 1\n1 6"
+            }
+        ],
+        "editorial": null,
+        "rating": 800,
+        "tags": [
+            "brute force",
+            "implementation"
+        ],
+        "testset_size": 20,
+        "official_tests": [
+            {
+                "input": "5\r\n3\r\n1 1 2\r\n11\r\n3 3 4 5 6 7 8 9 9 10 11\r\n8\r\n8 4 8 3 8 2 8 1\r\n6\r\n2 1 4 5 3 3\r\n8\r\n1 2 6 3 8 5 5 3\r\n",
+                "output": "1 1\r\n3 3\r\n2 3\r\n1 4\r\n1 6\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2061/A",
+        "aliases": null,
+        "contest_id": "2061",
+        "contest_name": "IAEPC Preliminary Contest (Codeforces Round 999, Div. 1 + Div. 2)",
+        "contest_type": "CF",
+        "contest_start": 1737383700,
+        "contest_start_year": 2025,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Kevin and Arithmetic",
+        "description": "To train young Kevin's arithmetic skills, his mother devised the following problem.\n\nGiven $$$n$$$ integers $$$a_1, a_2, \\ldots, a_n$$$ and a sum $$$s$$$ initialized to $$$0$$$, Kevin performs the following operation for $$$i = 1, 2, \\ldots, n$$$ in order:\n\n- Add $$$a_i$$$ to $$$s$$$. If the resulting $$$s$$$ is even, Kevin earns a point and repeatedly divides $$$s$$$ by $$$2$$$ until it becomes odd.\n\nNote that Kevin can earn at most one point per operation, regardless of how many divisions he does.\n\nSince these divisions are considered more beneficial for Kevin's development, his mother wants to rearrange $$$a$$$ so that the number of Kevin's total points is maximized. Determine the maximum number of points.",
+        "input_format": "Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \\le t \\le 500$$$). The description of the test cases follows.\n\nThe first line of each test case contains a single integer $$$n$$$ ($$$1\\leq n \\leq 100$$$) — the number of integers.\n\nThe second line contains $$$n$$$ integers $$$a_1, a_2, \\ldots, a_n$$$ ($$$1\\leq a_i \\leq 10^9$$$).",
+        "output_format": "For each test case, output one integer — the maximum number of points.",
+        "interaction_format": null,
+        "note": "In the first test case, the only arrangement of $$$a$$$ is $$$[1]$$$. $$$s$$$ becomes $$$1$$$. Kevin earns no points.\n\nIn the second test case, the only possible arrangement of $$$a$$$ is $$$[2, 1]$$$. $$$s$$$ becomes $$$1$$$ and $$$1$$$ successively. Kevin earns points in both operations.\n\nIn the third test case, one possible arrangement of $$$a$$$ is $$$[2, 4, 6]$$$. $$$s$$$ becomes $$$1$$$, $$$5$$$, and $$$11$$$ successively. Kevin earns a point in the first operation.",
+        "examples": [
+            {
+                "input": "5\n1\n1\n2\n1 2\n3\n2 4 6\n4\n1000000000 999999999 999999998 999999997\n10\n3 1 4 1 5 9 2 6 5 3",
+                "output": "0\n2\n1\n3\n8"
+            }
+        ],
+        "editorial": "After the first operation, $$$s$$$ is always odd. You can only earn points when $$$a_i$$$ is odd after this operation. However, during the first operation, you can only earn a point if $$$a_i$$$ is even.\nLet $$$c_0$$$ and $$$c_1$$$ be the number of even and odd numbers, respectively.\nIf there is at least one even number, you can place it first, making the answer $$$c_1 + 1$$$. Otherwise, since no points can be earned during the first operation, the answer is $$$c_1 - 1$$$.\nTime complexity: $$$O(n)$$$.",
+        "rating": 800,
+        "tags": [
+            "math"
+        ],
+        "testset_size": 4,
+        "official_tests": [
+            {
+                "input": "5\r\n1\r\n1\r\n2\r\n1 2\r\n3\r\n2 4 6\r\n4\r\n1000000000 999999999 999999998 999999997\r\n10\r\n3 1 4 1 5 9 2 6 5 3\r\n",
+                "output": "0\r\n2\r\n1\r\n3\r\n8\r\n"
+            },
+            {
+                "input": "1\r\n6\r\n1000000000 900000000 900000000 800000000 694967296 998046875\r\n",
+                "output": "2\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2050/A",
+        "aliases": null,
+        "contest_id": "2050",
+        "contest_name": "Codeforces Round 991 (Div. 3)",
+        "contest_type": "ICPC",
+        "contest_start": 1733409300,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Line Breaks",
+        "description": "Kostya has a text $$$s$$$ consisting of $$$n$$$ words made up of Latin alphabet letters. He also has two strips on which he must write the text. The first strip can hold $$$m$$$ characters, while the second can hold as many as needed.\n\nKostya must choose a number $$$x$$$ and write the first $$$x$$$ words from $$$s$$$ on the first strip, while all the remaining words are written on the second strip. To save space, the words are written without gaps, but each word must be entirely on one strip.\n\nSince space on the second strip is very valuable, Kostya asks you to choose the maximum possible number $$$x$$$ such that all words $$$s_1, s_2, \\dots, s_x$$$ fit on the first strip of length $$$m$$$.",
+        "input_format": "The first line contains an integer $$$t$$$ ($$$1 \\le t \\le 1000$$$) — the number of test cases.\n\nThe first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \\le n \\le 50$$$; $$$1 \\le m \\le 500$$$) — the number of words in the list and the maximum number of characters that can be on the first strip.\n\nThe next $$$n$$$ lines contain one word $$$s_i$$$ of lowercase Latin letters, where the length of $$$s_i$$$ does not exceed $$$10$$$.",
+        "output_format": "For each test case, output the maximum number of words $$$x$$$ such that the first $$$x$$$ words have a total length of no more than $$$m$$$.",
+        "interaction_format": null,
+        "note": null,
+        "examples": [
+            {
+                "input": "5\n3 1\na\nb\nc\n2 9\nalpha\nbeta\n4 12\nhello\nworld\nand\ncodeforces\n3 2\nab\nc\nd\n3 2\nabc\nab\na",
+                "output": "1\n2\n2\n1\n0"
+            }
+        ],
+        "editorial": "An important condition in the problem: we can take $$$x$$$ words on the first line from the beginning and we cannot skip any word.\nThe main idea is to compute the total length of words as we keep adding them, and stop when we reach a word where adding the next word would exceed the capacity of the first strip (which equals $$$m$$$).\nIt is important to take into account the case when all words can be placed on the first line or none will fit, but our solution takes this into account.",
+        "rating": 800,
+        "tags": [
+            "implementation"
+        ],
+        "testset_size": 4,
+        "official_tests": [
+            {
+                "input": "5\r\n3 1\r\na\r\nb\r\nc\r\n2 9\r\nalpha\r\nbeta\r\n4 12\r\nhello\r\nworld\r\nand\r\ncodeforces\r\n3 2\r\nab\r\nc\r\nd\r\n3 2\r\nabc\r\nab\r\na\r\n",
+                "output": "1\r\n2\r\n2\r\n1\r\n0\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2032/A",
+        "aliases": null,
+        "contest_id": "2032",
+        "contest_name": "Codeforces Round 983 (Div. 2)",
+        "contest_type": "CF",
+        "contest_start": 1730471700,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Circuit",
+        "description": "Alice has just crafted a circuit with $$$n$$$ lights and $$$2n$$$ switches. Each component (a light or a switch) has two states: on or off. The lights and switches are arranged in a way that:\n\n- Each light is connected to exactly two switches.\n- Each switch is connected to exactly one light. It's unknown which light each switch is connected to.\n- When all switches are off, all lights are also off.\n- If a switch is toggled (from on to off, or vice versa), the state of the light connected to it will also toggle.\n\nAlice brings the circuit, which shows only the states of the $$$2n$$$ switches, to her sister Iris and gives her a riddle: what is the minimum and maximum number of lights that can be turned on?\n\nKnowing her little sister's antics too well, Iris takes no more than a second to give Alice a correct answer. Can you do the same?",
+        "input_format": "Each test consists of multiple test cases. The first line contains a single integer $$$t$$$ ($$$1 \\le t \\le 500$$$) — the number of test cases. The description of the test cases follows.\n\nThe first line of each test case contains a single integer $$$n$$$ ($$$1 \\le n \\le 50$$$) — the number of lights in the circuit.\n\nThe second line of each test case contains $$$2n$$$ integers $$$a_1, a_2, \\ldots, a_{2n}$$$ ($$$0 \\le a_i \\le 1$$$) — the states of the switches in the circuit. $$$a_i = 0$$$ means the $$$i$$$-th switch is off, and $$$a_i = 1$$$ means the $$$i$$$-th switch is on.",
+        "output_format": "For each test case, output two integers — the minimum and maximum number of lights, respectively, that can be turned on.",
+        "interaction_format": null,
+        "note": "In the first test case, there is only one light in the circuit, and no switch is on, so the light is certainly off.\n\nIn the second test case, there is only one light in the circuit, but one switch connected to it is on, so the light is on.\n\nIn the third test case, there is only one light in the circuit, and both switches are on, so the light is off as it was toggled twice.\n\nIn the fourth test case, to have no lights on, the switches can be arranged in this way:\n\n- Switch $$$1$$$ and switch $$$4$$$ are connected to light $$$1$$$. Since both switches are off, light $$$1$$$ is also off.\n- Switch $$$2$$$ and switch $$$6$$$ are connected to light $$$2$$$. Since both switches are off, light $$$2$$$ is also off.\n- Switch $$$3$$$ and switch $$$5$$$ are connected to light $$$3$$$. Both switches are on, so light $$$3$$$ is toggled twice from its initial off state, and thus also stays off.\n\nAnd to have $$$2$$$ lights on, the switches can be arranged in this way:\n\n- Switch $$$1$$$ and switch $$$2$$$ are connected to light $$$1$$$. Since both switches are off, light $$$1$$$ is also off.\n- Switch $$$3$$$ and switch $$$4$$$ are connected to light $$$2$$$. Since switch $$$3$$$ is on and switch $$$4$$$ is off, light $$$2$$$ is toggled once from its initial off state, so it is on.\n- Switch $$$5$$$ and switch $$$6$$$ are connected to light $$$3$$$. Since switch $$$5$$$ is on and switch $$$6$$$ is off, light $$$3$$$ is toggled once from its initial off state, so it is on.",
+        "examples": [
+            {
+                "input": "5\n1\n0 0\n1\n0 1\n1\n1 1\n3\n0 0 1 0 1 0\n3\n0 1 1 1 0 0",
+                "output": "0 0\n1 1\n0 0\n0 2\n1 3"
+            }
+        ],
+        "editorial": "Observe that an even number of switch toggles on the same light will not change that light's status. In other words, a light is on if and only if exactly one of the two switches connecting to it is on.\nLet's denote $$$\\text{cnt}_0$$$ and $$$\\text{cnt}_1$$$ as the number of off switches and on switches in the circuit. We see that\n• The maximum number of on lights is $$$\\min(\\text{cnt}_0, \\text{cnt}_1)$$$: we can't achieve more than this amount since any on light decreases both $$$\\text{cnt}_0$$$ and $$$\\text{cnt}_1$$$ by $$$1$$$, and we can achieve this amount by matching $$$\\min(\\text{cnt}_0, \\text{cnt}_1)$$$ pairs of one off switch and one on switch, and the rest can be matched arbitrarily.\n• The minimum number of on lights is $$$\\text{cnt}_0 \\bmod 2$$$. Since $$$\\text{cnt}_0 + \\text{cnt}_1 = 2n$$$, they have the same parity. Then, we can easily see that when both $$$\\text{cnt}_0$$$ and $$$\\text{cnt}_1$$$ are even, we can match $$$n$$$ pairs of the same type of switches, so there are no on lights in this case. When both $$$\\text{cnt}_0$$$ and $$$\\text{cnt}_1$$$ are odd, we must match one on switch with one off switch to make $$$\\text{cnt}_0$$$ and $$$\\text{cnt}_1$$$ even, so there is one on light in this case.\nThe calculation of $$$\\text{cnt}_0$$$ and $$$\\text{cnt}_1$$$ can be easily done by a simple iteration over the switches.\nTime complexity: $$$\\mathcal{O}(n)$$$.",
+        "rating": 800,
+        "tags": [
+            "greedy",
+            "implementation",
+            "math",
+            "number theory"
+        ],
+        "testset_size": 4,
+        "official_tests": [
+            {
+                "input": "5\r\n1\r\n0 0\r\n1\r\n0 1\r\n1\r\n1 1\r\n3\r\n0 0 1 0 1 0\r\n3\r\n0 1 1 1 0 0\r\n",
+                "output": "0 0\r\n1 1\r\n0 0\r\n0 2\r\n1 3\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2003/A",
+        "aliases": null,
+        "contest_id": "2003",
+        "contest_name": "Codeforces Round 968 (Div. 2)",
+        "contest_type": "CF",
+        "contest_start": 1724596500,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Turtle and Good Strings",
+        "description": "Turtle thinks a string $$$s$$$ is a good string if there exists a sequence of strings $$$t_1, t_2, \\ldots, t_k$$$ ($$$k$$$ is an arbitrary integer) such that:\n\n- $$$k \\ge 2$$$.\n- $$$s = t_1 + t_2 + \\ldots + t_k$$$, where $$$+$$$ represents the concatenation operation. For example, $$$\\texttt{abc} = \\texttt{a} + \\texttt{bc}$$$.\n- For all $$$1 \\le i < j \\le k$$$, the first character of $$$t_i$$$ isn't equal to the last character of $$$t_j$$$.\n\nTurtle is given a string $$$s$$$ consisting of lowercase Latin letters. Please tell him whether the string $$$s$$$ is a good string!",
+        "input_format": "Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \\le t \\le 500$$$). The description of the test cases follows.\n\nThe first line of each test case contains a single integer $$$n$$$ ($$$2 \\le n \\le 100$$$) — the length of the string.\n\nThe second line of each test case contains a string $$$s$$$ of length $$$n$$$, consisting of lowercase Latin letters.",
+        "output_format": "For each test case, output \"YES\" if the string $$$s$$$ is a good string, and \"NO\" otherwise.\n\nYou can output the answer in any case (upper or lower). For example, the strings \"yEs\", \"yes\", \"Yes\", and \"YES\" will be recognized as positive responses.",
+        "interaction_format": null,
+        "note": "In the first test case, the sequence of strings $$$\\texttt{a}, \\texttt{a}$$$ satisfies the condition $$$s = t_1 + t_2 + \\ldots + t_k$$$, but the first character of $$$t_1$$$ is equal to the last character of $$$t_2$$$. It can be seen that there doesn't exist any sequence of strings which satisfies all of the conditions, so the answer is \"NO\".\n\nIn the third test case, the sequence of strings $$$\\texttt{ab}, \\texttt{cb}$$$ satisfies all of the conditions.\n\nIn the fourth test case, the sequence of strings $$$\\texttt{abca}, \\texttt{bcab}, \\texttt{cabc}$$$ satisfies all of the conditions.",
+        "examples": [
+            {
+                "input": "4\n2\naa\n3\naba\n4\nabcb\n12\nabcabcabcabc",
+                "output": "No\nnO\nYes\nYES"
+            }
+        ],
+        "editorial": "A necessary condition for $$$s$$$ to be good is that $$$s_1 \\ne s_n$$$. For a string $$$s$$$ where $$$s_1 \\ne s_n$$$, let $$$t_1$$$ be a string composed of just the single character $$$s_1$$$, and let $$$t_2$$$ be a string composed of the $$$n - 1$$$ characters from $$$s_2$$$ to $$$s_n$$$. In this way, the condition is satisfied.\nTherefore, if $$$s_1 \\ne s_n$$$, the answer is \"Yes\"; otherwise, the answer is \"No\".\nTime complexity: $$$O(n)$$$ per test case.",
+        "rating": 800,
+        "tags": [
+            "greedy",
+            "strings"
+        ],
+        "testset_size": 5,
+        "official_tests": [
+            {
+                "input": "4\r\n2\r\naa\r\n3\r\naba\r\n4\r\nabcb\r\n12\r\nabcabcabcabc\r\n",
+                "output": "No\r\nNo\r\nYes\r\nYes\r\n"
+            },
+            {
+                "input": "1\r\n7\r\nhonglan\r\n",
+                "output": "Yes\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2003/B",
+        "aliases": null,
+        "contest_id": "2003",
+        "contest_name": "Codeforces Round 968 (Div. 2)",
+        "contest_type": "CF",
+        "contest_start": 1724596500,
+        "contest_start_year": 2024,
+        "index": "B",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Turtle and Piggy Are Playing a Game 2",
+        "description": "Turtle and Piggy are playing a game on a sequence. They are given a sequence $$$a_1, a_2, \\ldots, a_n$$$, and Turtle goes first. Turtle and Piggy alternate in turns (so, Turtle does the first turn, Piggy does the second, Turtle does the third, etc.).\n\nThe game goes as follows:\n\n- Let the current length of the sequence be $$$m$$$. If $$$m = 1$$$, the game ends.\n- If the game does not end and it's Turtle's turn, then Turtle must choose an integer $$$i$$$ such that $$$1 \\le i \\le m - 1$$$, set $$$a_i$$$ to $$$\\max(a_i, a_{i + 1})$$$, and remove $$$a_{i + 1}$$$.\n- If the game does not end and it's Piggy's turn, then Piggy must choose an integer $$$i$$$ such that $$$1 \\le i \\le m - 1$$$, set $$$a_i$$$ to $$$\\min(a_i, a_{i + 1})$$$, and remove $$$a_{i + 1}$$$.\n\nTurtle wants to maximize the value of $$$a_1$$$ in the end, while Piggy wants to minimize the value of $$$a_1$$$ in the end. Find the value of $$$a_1$$$ in the end if both players play optimally.\n\nYou can refer to notes for further clarification.",
+        "input_format": "Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \\le t \\le 10^4$$$). The description of the test cases follows.\n\nThe first line of each test case contains a single integer $$$n$$$ ($$$2 \\le n \\le 10^5$$$) — the length of the sequence.\n\nThe second line of each test case contains $$$n$$$ integers $$$a_1, a_2, \\ldots, a_n$$$ ($$$1 \\le a_i \\le 10^5$$$) — the elements of the sequence $$$a$$$.\n\nIt is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$10^5$$$.",
+        "output_format": "For each test case, output a single integer — the value of $$$a_1$$$ in the end if both players play optimally.",
+        "interaction_format": null,
+        "note": "In the first test case, initially $$$a = [1, 2]$$$. Turtle can only choose $$$i = 1$$$. Then he will set $$$a_1$$$ to $$$\\max(a_1, a_2) = 2$$$ and remove $$$a_2$$$. The sequence $$$a$$$ becomes $$$[2]$$$. Then the length of the sequence becomes $$$1$$$, and the game will end. The value of $$$a_1$$$ is $$$2$$$, so you should output $$$2$$$.\n\nIn the second test case, one of the possible game processes is as follows:\n\n- Initially $$$a = [1, 1, 2]$$$.\n- Turtle can choose $$$i = 2$$$. Then he will set $$$a_2$$$ to $$$\\max(a_2, a_3) = 2$$$ and remove $$$a_3$$$. The sequence $$$a$$$ will become $$$[1, 2]$$$.\n- Piggy can choose $$$i = 1$$$. Then he will set $$$a_1$$$ to $$$\\min(a_1, a_2) = 1$$$ and remove $$$a_2$$$. The sequence $$$a$$$ will become $$$[1]$$$.\n- The length of the sequence becomes $$$1$$$, so the game will end. The value of $$$a_1$$$ will be $$$1$$$ in the end.\n\nIn the fourth test case, one of the possible game processes is as follows:\n\n- Initially $$$a = [3, 1, 2, 2, 3]$$$.\n- Turtle can choose $$$i = 4$$$. Then he will set $$$a_4$$$ to $$$\\max(a_4, a_5) = 3$$$ and remove $$$a_5$$$. The sequence $$$a$$$ will become $$$[3, 1, 2, 3]$$$.\n- Piggy can choose $$$i = 3$$$. Then he will set $$$a_3$$$ to $$$\\min(a_3, a_4) = 2$$$ and remove $$$a_4$$$. The sequence $$$a$$$ will become $$$[3, 1, 2]$$$.\n- Turtle can choose $$$i = 2$$$. Then he will set $$$a_2$$$ to $$$\\max(a_2, a_3) = 2$$$ and remove $$$a_3$$$. The sequence $$$a$$$ will become $$$[3, 2]$$$.\n- Piggy can choose $$$i = 1$$$. Then he will set $$$a_1$$$ to $$$\\min(a_1, a_2) = 2$$$ and remove $$$a_2$$$. The sequence $$$a$$$ will become $$$[2]$$$.\n- The length of the sequence becomes $$$1$$$, so the game will end. The value of $$$a_1$$$ will be $$$2$$$ in the end.",
+        "examples": [
+            {
+                "input": "5\n2\n1 2\n3\n1 1 2\n3\n1 2 3\n5\n3 1 2 2 3\n10\n10 2 5 2 7 9 2 5 10 7",
+                "output": "2\n1\n2\n2\n7"
+            }
+        ],
+        "editorial": "The problem can be rephrased as follows: the first player can remove a number adjacent to a larger number, while the second player can remove a number adjacent to a smaller number.\nTo maximize the value of the last remaining number, in the $$$i$$$-th round of the first player's moves, he will remove the $$$i$$$-th smallest number in the sequence, and this is always achievable. Similarly, in the $$$i$$$-th round of the second player's moves, he will remove the $$$i$$$-th largest number.\nThus, the answer is the $$$\\left\\lfloor\\frac{n}{2}\\right\\rfloor + 1$$$-th smallest number in the original sequence.\nTime complexity: $$$O(n \\log n)$$$ per test case.",
+        "rating": 800,
+        "tags": [
+            "games",
+            "greedy",
+            "sortings"
+        ],
+        "testset_size": 10,
+        "official_tests": [
+            {
+                "input": "5\r\n2\r\n1 2\r\n3\r\n1 1 2\r\n3\r\n1 2 3\r\n5\r\n3 1 2 2 3\r\n10\r\n10 2 5 2 7 9 2 5 10 7\r\n",
+                "output": "2\r\n1\r\n2\r\n2\r\n7\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2048/A",
+        "aliases": null,
+        "contest_id": "2048",
+        "contest_name": "Codeforces Global Round 28",
+        "contest_type": "CF",
+        "contest_start": 1734618900,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Kevin and Combination Lock",
+        "description": "Kevin is trapped in Lakeside Village by Grace. At the exit of the village, there is a combination lock that can only be unlocked if Kevin solves it.\n\nThe combination lock starts with an integer $$$ x $$$. Kevin can perform one of the following two operations zero or more times:\n\n1. If $$$ x \\neq 33 $$$, he can select two consecutive digits $$$ 3 $$$ from $$$ x $$$ and remove them simultaneously. For example, if $$$ x = 13\\,323 $$$, he can remove the second and third $$$ 3 $$$, changing $$$ x $$$ to $$$ 123 $$$.\n2. If $$$ x \\geq 33 $$$, he can change $$$ x $$$ to $$$ x - 33 $$$. For example, if $$$ x = 99 $$$, he can choose this operation to change $$$ x $$$ to $$$ 99 - 33 = 66 $$$.\n\nWhen the value of $$$ x $$$ on the combination lock becomes $$$ 0 $$$, Kevin can unlock the lock and escape from Lakeside Village. Please determine whether it is possible for Kevin to unlock the combination lock and escape.",
+        "input_format": "Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \\le t \\le 10^4$$$).\n\nThe only line of each test case contains a positive integer $$$x$$$ ($$$1\\leq x\\leq 10^9$$$).",
+        "output_format": "For each test case, output \"YES\" or \"NO\" (without quotes) in one line, representing whether Kevin can unlock the combination lock and escape. You can output the answer in any case (upper or lower). For example, the strings \"yEs\", \"yes\", \"Yes\", and \"YES\" will be recognized as positive responses.",
+        "interaction_format": null,
+        "note": "For the first test case, $$$165\\xrightarrow{-33}132\\xrightarrow{-33}99\\xrightarrow{-33}66\\xrightarrow{-33}33\\xrightarrow{-33}0$$$.\n\nFor the second test case, $$$6369\\xrightarrow{-33}6{\\color{red}{33}}6\\xrightarrow{\\text{remove \"33\"}}66\\xrightarrow{-33}33\\xrightarrow{-33}0$$$.\n\nFor the third test case, it can be proven that, regardless of the operations performed, $$$666$$$ cannot be transformed into $$$0$$$.",
+        "examples": [
+            {
+                "input": "5\n165\n6369\n666\n114514\n133333332",
+                "output": "YES\nYES\nNO\nNO\nYES"
+            }
+        ],
+        "editorial": "Hint #1: How much does the number decrease when we remove two consecutive $$$3$$$s?\nWhen we transform $$$\\overline{x33y}$$$ into $$$\\overline{xy}$$$ (where $$$x$$$ and $$$y$$$ are decimal numbers), the actual value changes from $$$10^{p+2} \\cdot x + 33 \\cdot 10^p + y$$$ to $$$10^p \\cdot x + y$$$. The decrease is $$$99 \\cdot 10^p \\cdot x + 33 \\cdot 10^p$$$. It is easy to see that $$$33 \\mid (99 \\cdot 10^p \\cdot x + 33 \\cdot 10^p)$$$. Therefore, we can replace the operation of \"removing two consecutive $$$3$$$s\" with a series of \"$$$-33$$$\" operations.\nHence, we only need to determine whether $$$x$$$ can be reduced to $$$0$$$ using a series of $$$-33$$$ operations, which is equivalent to checking whether $$$x \\bmod 33$$$ equals zero.",
+        "rating": 800,
+        "tags": [
+            "brute force",
+            "greedy",
+            "implementation",
+            "math",
+            "number theory"
+        ],
+        "testset_size": 5,
+        "official_tests": [
+            {
+                "input": "5\r\n165\r\n6369\r\n666\r\n114514\r\n133333332\r\n",
+                "output": "YES\r\nYES\r\nNO\r\nNO\r\nYES\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2065/A",
+        "aliases": null,
+        "contest_id": "2065",
+        "contest_name": "Codeforces Round 1003 (Div. 4)",
+        "contest_type": "ICPC",
+        "contest_start": 1739111700,
+        "contest_start_year": 2025,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Skibidus and Amog'u",
+        "description": "Skibidus lands on a foreign planet, where the local Amog tribe speaks the Amog'u language. In Amog'u, there are two forms of nouns, which are singular and plural.\n\nGiven that the root of the noun is transcribed as $$$S$$$, the two forms are transcribed as:\n\n- Singular: $$$S$$$ $$$+$$$ \"us\"\n- Plural: $$$S$$$ $$$+$$$ \"i\"\n\nHere, $$$+$$$ denotes string concatenation. For example, abc $$$+$$$ def $$$=$$$ abcdef.\n\nFor example, when $$$S$$$ is transcribed as \"amog\", then the singular form is transcribed as \"amogus\", and the plural form is transcribed as \"amogi\". Do note that Amog'u nouns can have an empty root — in specific, \"us\" is the singular form of \"i\" (which, on an unrelated note, means \"imposter\" and \"imposters\" respectively).\n\nGiven a transcribed Amog'u noun in singular form, please convert it to the transcription of the corresponding plural noun.",
+        "input_format": "Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \\le t \\le 100$$$). The description of the test cases follows.\n\nThe only line of each test case contains a string $$$W$$$, which is a transcribed Amog'u noun in singular form. It is guaranteed that $$$W$$$ consists of only lowercase English letters, has a length of at most $$$10$$$, and ends with \"us\".",
+        "output_format": "For each test case, output the transcription of the corresponding plural noun on a separate line.",
+        "interaction_format": null,
+        "note": null,
+        "examples": [
+            {
+                "input": "9\nus\nsus\nfungus\ncactus\nsussus\namogus\nchungus\nntarsus\nskibidus",
+                "output": "i\nsi\nfungi\ncacti\nsussi\namogi\nchungi\nntarsi\nskibidi"
+            }
+        ],
+        "editorial": null,
+        "rating": 800,
+        "tags": [
+            "brute force",
+            "constructive algorithms",
+            "greedy",
+            "implementation",
+            "strings"
+        ],
+        "testset_size": 3,
+        "official_tests": [
+            {
+                "input": "9\r\nus\r\nsus\r\nfungus\r\ncactus\r\nsussus\r\namogus\r\nchungus\r\nntarsus\r\nskibidus\r\n",
+                "output": "i\r\nsi\r\nfungi\r\ncacti\r\nsussi\r\namogi\r\nchungi\r\nntarsi\r\nskibidi\r\n"
+            },
+            {
+                "input": "5\r\nabus\r\ntoorestus\r\ntoortus\r\ntrtus\r\nhmeus\r\n",
+                "output": "abi\r\ntooresti\r\ntoorti\r\ntrti\r\nhmei\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2065/B",
+        "aliases": null,
+        "contest_id": "2065",
+        "contest_name": "Codeforces Round 1003 (Div. 4)",
+        "contest_type": "ICPC",
+        "contest_start": 1739111700,
+        "contest_start_year": 2025,
+        "index": "B",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Skibidus and Ohio",
+        "description": "Skibidus is given a string $$$s$$$ that consists of lowercase Latin letters. If $$$s$$$ contains more than $$$1$$$ letter, he can:\n\n- Choose an index $$$i$$$ ($$$1 \\leq i \\leq |s| - 1$$$, $$$|s|$$$ denotes the current length of $$$s$$$) such that $$$s_i = s_{i+1}$$$. Replace $$$s_i$$$ with any lowercase Latin letter of his choice. Remove $$$s_{i+1}$$$ from the string.\n\nSkibidus must determine the minimum possible length he can achieve through any number of operations.",
+        "input_format": "The first line contains an integer $$$t$$$ ($$$1 \\leq t \\leq 100$$$) — the number of test cases.\n\nThe only line of each test case contains a string $$$s$$$ ($$$1 \\leq |s| \\leq 100$$$). It is guaranteed $$$s$$$ only contains lowercase Latin letters.",
+        "output_format": "For each test case, output an integer on the new line, the minimum achievable length of $$$s$$$.",
+        "interaction_format": null,
+        "note": "In the first test case, Skibidus can:\n\n- Perform an operation on $$$i = 2$$$. He replaces $$$s_2$$$ with b and removes $$$s_3$$$ from the string. Then, $$$s$$$ becomes bb.\n- Perform an operation on $$$i = 1$$$. He replaces $$$s_1$$$ with b and removes $$$s_2$$$ from the string. Then, $$$s$$$ becomes b.\n- Because $$$s$$$ only contains $$$1$$$ letter, Skibidus cannot perform any more operations.\n\nTherefore, the answer is $$$1$$$ for the first test case.\n\nIn the second test case, he cannot perform an operation on any index. Therefore, the answer is still the length of the initial string, $$$8$$$.",
+        "examples": [
+            {
+                "input": "4\nbaa\nskibidus\ncc\nohio",
+                "output": "1\n8\n1\n4"
+            }
+        ],
+        "editorial": null,
+        "rating": 800,
+        "tags": [
+            "strings"
+        ],
+        "testset_size": 3,
+        "official_tests": [
+            {
+                "input": "4\r\nbaa\r\nskibidus\r\ncc\r\nohio\r\n",
+                "output": "1\r\n8\r\n1\r\n4\r\n"
+            },
+            {
+                "input": "22\r\nab\r\naabcdef\r\naab\r\nbbcdkjnm\r\nbbbbbbbb\r\nfkgfkgkafss\r\nsgsgrgtyyfagth\r\ngagytbytbtsbh\r\nbbbbsvbtyhw\r\ndunfuuuu\r\nnufuimmu\r\nyiivvjm\r\nyiivjm\r\na\r\nssbjhlfj\r\na\r\nb\r\nc\r\nf\r\nd\r\nf\r\nk\r\n",
+                "output": "2\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n13\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n1\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2007/A",
+        "aliases": null,
+        "contest_id": "2007",
+        "contest_name": "Codeforces Round 969 (Div. 2)",
+        "contest_type": "CF",
+        "contest_start": 1725028500,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Dora's Set",
+        "description": "Dora has a set $$$s$$$ containing integers. In the beginning, she will put all integers in $$$[l, r]$$$ into the set $$$s$$$. That is, an integer $$$x$$$ is initially contained in the set if and only if $$$l \\leq x \\leq r$$$. Then she allows you to perform the following operations:\n\n- Select three distinct integers $$$a$$$, $$$b$$$, and $$$c$$$ from the set $$$s$$$, such that $$$\\gcd(a, b) = \\gcd(b, c) = \\gcd(a, c) = 1^\\dagger$$$.\n- Then, remove these three integers from the set $$$s$$$.\n\nWhat is the maximum number of operations you can perform?\n\n$$$^\\dagger$$$Recall that $$$\\gcd(x, y)$$$ means the greatest common divisor of integers $$$x$$$ and $$$y$$$.",
+        "input_format": "Each test consists of multiple test cases. The first line contains a single integer $$$t$$$ ($$$1 \\leq t \\leq 500$$$) — the number of test cases. The description of the test cases follows.\n\nThe only line of each test case contains two integers $$$l$$$ and $$$r$$$ ($$$1 \\leq l \\leq r \\leq 1000$$$) — the range of integers in the initial set.",
+        "output_format": "For each test case, output a single integer — the maximum number of operations you can perform.",
+        "interaction_format": null,
+        "note": "In the first test case, you can choose $$$a = 1$$$, $$$b = 2$$$, $$$c = 3$$$ in the only operation, since $$$\\gcd(1, 2) = \\gcd(2, 3) = \\gcd(1, 3) = 1$$$, and then there are no more integers in the set, so no more operations can be performed.\n\nIn the second test case, you can choose $$$a = 3$$$, $$$b = 5$$$, $$$c = 7$$$ in the only operation.\n\nIn the third test case, you can choose $$$a = 11$$$, $$$b = 19$$$, $$$c = 20$$$ in the first operation, $$$a = 13$$$, $$$b = 14$$$, $$$c = 15$$$ in the second operation, and $$$a = 10$$$, $$$b = 17$$$, $$$c = 21$$$ in the third operation. After the three operations, the set $$$s$$$ contains the following integers: $$$12$$$, $$$16$$$, $$$18$$$. It can be proven that it's impossible to perform more than $$$3$$$ operations.",
+        "examples": [
+            {
+                "input": "8\n1 3\n3 7\n10 21\n2 8\n51 60\n2 15\n10 26\n1 1000",
+                "output": "1\n1\n3\n1\n2\n3\n4\n250"
+            }
+        ],
+        "editorial": "In a pair of $$$(a, b, c)$$$ there are at least two odd integers and at most one even integer. That's because if two even integers occur at the same time, their $$$\\gcd$$$ will be at least $$$2$$$.\nIt's optimal to make full use of the odd integers, so we try to choose two odd integers in a pair. In fact, this is always possible. Note that two consecutive odd integers always have $$$\\gcd=1$$$, and consecutive integers also have $$$\\gcd=1$$$, so we can choose three consecutive integers $$$(a,b,c)$$$ in the form of $$$2k - 1, 2k, 2k + 1$$$. We can see that this is always optimal. Therefore, the answer is: the result of dividing the number of odd numbers in $$$[l, r]$$$ by $$$2$$$ and rounding down.\nThe time complexity is: $$$\\mathcal O(T)$$$ or $$$\\mathcal O(\\sum (r - l))$$$.",
+        "rating": 800,
+        "tags": [
+            "greedy",
+            "math",
+            "number theory"
+        ],
+        "testset_size": 5,
+        "official_tests": [
+            {
+                "input": "8\r\n1 3\r\n3 7\r\n10 21\r\n2 8\r\n51 60\r\n2 15\r\n10 26\r\n1 1000\r\n",
+                "output": "1\r\n1\r\n3\r\n1\r\n2\r\n3\r\n4\r\n250\r\n"
+            },
+            {
+                "input": "1\r\n503 765\r\n",
+                "output": "66\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2036/A",
+        "aliases": null,
+        "contest_id": "2036",
+        "contest_name": "Codeforces Round 984 (Div. 3)",
+        "contest_type": "ICPC",
+        "contest_start": 1730558100,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Quintomania",
+        "description": "Boris Notkin composes melodies. He represents them as a sequence of notes, where each note is encoded as an integer from $$$0$$$ to $$$127$$$ inclusive. The interval between two notes $$$a$$$ and $$$b$$$ is equal to $$$|a - b|$$$ semitones.\n\nBoris considers a melody perfect if the interval between each two adjacent notes is either $$$5$$$ semitones or $$$7$$$ semitones.\n\nAfter composing his latest melodies, he enthusiastically shows you his collection of works. Help Boris Notkin understand whether his melodies are perfect.",
+        "input_format": "The first line contains an integer $$$t$$$ ($$$1 \\leq t \\leq 1000$$$) — the number of melodies.\n\nEach melody is described by two lines.\n\nThe first line contains an integer $$$n$$$ ($$$2 \\leq n \\leq 50$$$) — the number of notes in the melody.\n\nThe second line contains $$$n$$$ integers $$$a_{1}, a_{2}, \\dots, a_{n}$$$ ($$$0 \\leq a_{i} \\leq 127$$$) — the notes of the melody.",
+        "output_format": "For each melody, output \"YES\", if it is perfect; otherwise, output \"NO\".\n\nYou can output the answer in any case (upper or lower). For example, the strings \"yEs\", \"yes\", \"Yes\", and \"YES\" will be recognized as positive responses.",
+        "interaction_format": null,
+        "note": null,
+        "examples": [
+            {
+                "input": "8\n2\n114 109\n2\n17 10\n3\n76 83 88\n8\n38 45 38 80 85 92 99 106\n5\n63 58 65 58 65\n8\n117 124 48 53 48 43 54 49\n5\n95 102 107 114 121\n10\n72 77 82 75 70 75 68 75 68 75",
+                "output": "YES\nYES\nYES\nNO\nYES\nNO\nYES\nYES"
+            }
+        ],
+        "editorial": "If for all $$$i$$$ $$$(1 \\leq i \\leq n - 1)$$$ $$$|a_i - a_{i+1}| = 5$$$ or $$$|a_i - a_{i+1}| = 7$$$, the answer to the problem is — \"YES\"; otherwise — \"NO\".",
+        "rating": 800,
+        "tags": [
+            "implementation"
+        ],
+        "testset_size": 4,
+        "official_tests": [
+            {
+                "input": "8\r\n2\r\n114 109\r\n2\r\n17 10\r\n3\r\n76 83 88\r\n8\r\n38 45 38 80 85 92 99 106\r\n5\r\n63 58 65 58 65\r\n8\r\n117 124 48 53 48 43 54 49\r\n5\r\n95 102 107 114 121\r\n10\r\n72 77 82 75 70 75 68 75 68 75\r\n",
+                "output": "YES\r\nYES\r\nYES\r\nNO\r\nYES\r\nNO\r\nYES\r\nYES\r\n"
+            },
+            {
+                "input": "19\r\n38\r\n1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126 121 126 121 126 121 126 121 126 121 126 121 126\r\n2\r\n48 1\r\n2\r\n48 53\r\n2\r\n48 59\r\n2\r\n20 48\r\n2\r\n43 48\r\n2\r\n1 2\r\n2\r\n1 2\r\n2\r\n1 2\r\n2\r\n1 2\r\n2\r\n1 2\r\n2\r\n1 2\r\n2\r\n1 2\r\n2\r\n1 2\r\n2\r\n1 2\r\n2\r\n1 2\r\n2\r\n1 2\r\n2\r\n1 2\r\n2\r\n1 2\r\n",
+                "output": "YES\r\nNO\r\nYES\r\nNO\r\nNO\r\nYES\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\nNO\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2036/B",
+        "aliases": null,
+        "contest_id": "2036",
+        "contest_name": "Codeforces Round 984 (Div. 3)",
+        "contest_type": "ICPC",
+        "contest_start": 1730558100,
+        "contest_start_year": 2024,
+        "index": "B",
+        "time_limit": 2.0,
+        "memory_limit": 256.0,
+        "title": "Startup",
+        "description": "Arseniy came up with another business plan — to sell soda from a vending machine! For this, he purchased a machine with $$$n$$$ shelves, as well as $$$k$$$ bottles, where the $$$i$$$-th bottle is characterized by the brand index $$$b_i$$$ and the cost $$$c_i$$$.\n\nYou can place any number of bottles on each shelf, but all bottles on the same shelf must be of the same brand.\n\nArseniy knows that all the bottles he puts on the shelves of the machine will be sold. Therefore, he asked you to calculate the maximum amount he can earn.",
+        "input_format": "The first line contains one integer $$$t$$$ ($$$1 \\le t \\le 10^4$$$) — the number of test cases.\n\nThe first line of each test case contains two integers $$$n$$$ and $$$k$$$ ($$$1 \\le n, k \\le 2 \\cdot 10^5$$$), where $$$n$$$ is the number of shelves in the machine, and $$$k$$$ is the number of bottles available to Arseniy.\n\nThe next $$$k$$$ lines contain two integers $$$b_i$$$ and $$$c_i$$$ ($$$1 \\le b_i \\le k, 1 \\le c_i \\le 1000$$$) — the brand and cost of the $$$i$$$-th bottle.\n\nIt is also guaranteed that the sum of $$$n$$$ across all test cases does not exceed $$$2 \\cdot 10^5$$$ and that the sum of $$$k$$$ across all test cases also does not exceed $$$2 \\cdot 10^5$$$.",
+        "output_format": "For each test case, output one integer — the maximum amount that Arseniy can earn.",
+        "interaction_format": null,
+        "note": "In the first test case, Arseniy has $$$3$$$ shelves in the vending machine. He can place, for example, two bottles of the brand $$$2$$$ on the first shelf and a bottle of the brand $$$1$$$ on the second shelf. Then the total cost of the bottles would be $$$6 + 7 + 15 = 28$$$.\n\nIn the second test case, he has only one shelf. It is not difficult to show that the optimal option is to place a bottle of the brand $$$1$$$ on it. Then the total cost will be $$$15$$$.\n\nIn the third test case, he has as many as $$$6$$$ shelves, so he can place all available bottles with a total cost of $$$7 + 5 = 12$$$.",
+        "examples": [
+            {
+                "input": "4\n3 3\n2 6\n2 7\n1 15\n1 3\n2 6\n2 7\n1 15\n6 2\n1 7\n2 5\n190000 1\n1 1000",
+                "output": "28\n15\n12\n1000"
+            }
+        ],
+        "editorial": "Let's create an array $$$brand\\_cost$$$ of length $$$k$$$ and fill it so that $$$brand\\_cost_i$$$ stores the cost of all bottles of brand $$$i+1$$$. Then we sort the array in non-increasing order and calculate the sum of its first $$$min(n, k)$$$ elements, which will be the answer to the problem.",
+        "rating": 800,
+        "tags": [
+            "greedy",
+            "sortings"
+        ],
+        "testset_size": 24,
+        "official_tests": [
+            {
+                "input": "4\r\n3 3\r\n2 6\r\n2 7\r\n1 15\r\n1 3\r\n2 6\r\n2 7\r\n1 15\r\n6 2\r\n1 7\r\n2 5\r\n190000 1\r\n1 1000\r\n",
+                "output": "28\r\n15\r\n12\r\n1000\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2040/A",
+        "aliases": null,
+        "contest_id": "2040",
+        "contest_name": "Codeforces Round 992 (Div. 2)",
+        "contest_type": "CF",
+        "contest_start": 1733668500,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Game of Division",
+        "description": "You are given an array of integers $$$a_1, a_2, \\ldots, a_n$$$ of length $$$n$$$ and an integer $$$k$$$.\n\nTwo players are playing a game. The first player chooses an index $$$1 \\le i \\le n$$$. Then the second player chooses a different index $$$1 \\le j \\le n, i \\neq j$$$. The first player wins if $$$|a_i - a_j|$$$ is not divisible by $$$k$$$. Otherwise, the second player wins.\n\nWe play as the first player. Determine whether it is possible to win, and if so, which index $$$i$$$ should be chosen.\n\nThe absolute value of a number $$$x$$$ is denoted by $$$|x|$$$ and is equal to $$$x$$$ if $$$x \\ge 0$$$, and $$$-x$$$ otherwise.",
+        "input_format": "Each test contains multiple test cases. The first line of input contains a single integer $$$t$$$ ($$$1 \\le t \\le 100$$$) — the number of test cases. The description of the test cases follows.\n\nThe first line of each test case contains two integers $$$n$$$ and $$$k$$$ ($$$1 \\le n \\le 100$$$; $$$1 \\le k \\le 100$$$) — the length of the array and the number $$$k$$$.\n\nThe second line of each test case contains $$$n$$$ integers $$$a_1, a_2, \\ldots, a_n$$$ ($$$1 \\le a_i \\le 100$$$) — the elements of the array $$$a$$$.",
+        "output_format": "For each test case, if it is impossible for the first player to win, print \"NO\" (without quotes).\n\nOtherwise, print \"YES\" (without quotes) and on the next line the appropriate index $$$1 \\le i \\le n$$$. If there are multiple solutions, print any of them.\n\nYou can output each letter in any case (lowercase or uppercase). For example, the strings \"yEs\", \"yes\", \"Yes\" and \"YES\" will be recognized as a positive answer.",
+        "interaction_format": null,
+        "note": "In the first test case, the first player can choose $$$a_2 = 2$$$. Then:\n\n- If the second player chooses $$$a_1 = 1$$$, the resulting difference is $$$|2 - 1| = 1$$$ which is not divisible by $$$k = 2$$$.\n- If the second player chooses $$$a_3 = 3$$$, the resulting difference is $$$|2 - 3| = 1$$$ which is not divisible by $$$k = 2$$$.\n\nIn the second test case:\n\n- If the first player chooses $$$a_1 = 1$$$ and then the second player chooses $$$a_4 = 5$$$, the resulting difference is $$$|1 - 5| = 4$$$ which is divisible by $$$k = 2$$$.\n- If the first player chooses $$$a_2 = 2$$$ and then the second player chooses $$$a_3 = 4$$$, the resulting difference is $$$|2 - 4| = 2$$$ which is divisible by $$$k = 2$$$.\n- If the first player chooses $$$a_3 = 4$$$ and then the second player chooses $$$a_2 = 2$$$, the resulting difference is $$$|4 - 2| = 2$$$ which is divisible by $$$k = 2$$$.\n- If the first player chooses $$$a_4 = 5$$$ and then the second player chooses $$$a_1 = 1$$$, the resulting difference is $$$|5 - 1| = 4$$$ which is divisible by $$$k = 2$$$.",
+        "examples": [
+            {
+                "input": "7\n3 2\n1 2 3\n4 2\n1 2 4 5\n5 3\n10 7 3 4 5\n5 3\n1 31 15 55 36\n2 1\n17 17\n2 2\n17 18\n1 3\n6",
+                "output": "YES\n2\nNO\nYES\n3\nNO\nNO\nYES\n2\nYES\n1"
+            }
+        ],
+        "editorial": "$$$|x - y|$$$ is divisible by $$$k$$$ if and only if $$$x \\mod k = y \\mod k$$$. Let's split all numbers into groups according to the value $$$x \\mod k$$$. The second player wins if he chooses a number from the same group. This means that the first player must choose the number that is the only one in its group.",
+        "rating": 800,
+        "tags": [
+            "games",
+            "math"
+        ],
+        "testset_size": 4,
+        "official_tests": [
+            {
+                "input": "7\r\n3 2\r\n1 2 3\r\n4 2\r\n1 2 4 5\r\n5 3\r\n10 7 3 4 5\r\n5 3\r\n1 31 15 55 36\r\n2 1\r\n17 17\r\n2 2\r\n17 18\r\n1 3\r\n6\r\n",
+                "output": "YES\r\n2\r\nNO\r\nYES\r\n3\r\nNO\r\nNO\r\nYES\r\n2\r\nYES\r\n1\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2022/A",
+        "aliases": null,
+        "contest_id": "2022",
+        "contest_name": "Codeforces Round 978 (Div. 2)",
+        "contest_type": "CF",
+        "contest_start": 1728848100,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Bus to Pénjamo",
+        "description": "There are $$$n$$$ families travelling to Pénjamo to witness Mexico's largest-ever \"walking a chicken on a leash\" marathon. The $$$i$$$-th family has $$$a_i$$$ family members. All families will travel using a single bus consisting of $$$r$$$ rows with $$$2$$$ seats each.\n\nA person is considered happy if:\n\n- Another family member is seated in the same row as them, or\n- They are sitting alone in their row (with an empty seat next to them).\n\nDetermine the maximum number of happy people in an optimal seating arrangement. Note that everyone must be seated in the bus.\n\nIt is guaranteed that all family members will fit on the bus. Formally, it is guaranteed that $$$\\displaystyle\\sum_{i=1}^{n}a_i \\le 2r$$$.",
+        "input_format": "Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \\le t \\le 1000$$$). The description of the test cases follows.\n\nThe first line of each test case contains two integers $$$n$$$ and $$$r$$$ ($$$1 \\le n \\le 100$$$; $$$1 \\le r \\le 500$$$) — the number of families and the number of rows in the bus.\n\nThe second line contains $$$n$$$ integers $$$a_1, a_2, \\ldots, a_n$$$ ($$$1 \\le a_i \\le 10$$$) — the number of family members in each family.",
+        "output_format": "For each test case, output the maximum number of happy people in an optimal seating arrangement.",
+        "interaction_format": null,
+        "note": "In the first test case, the two members of the first family can sit together in the first row, while the two members of the second family can sit together in the second row. The remaining member of the second family can sit in the third row along with a member of the third family. This seating arrangement is shown below, where the $$$4$$$ happy people are colored green.\n\n$$$\\color{green}{1}$$$$$$\\color{green}{1}$$$$$$\\color{green}{2}$$$$$$\\color{green}{2}$$$$$$2$$$$$$3$$$\n\nIn the second test case, a possible seating arrangement with $$$6$$$ happy people is shown below.\n\n$$$\\color{green}{3}$$$$$$\\color{green}{3}$$$$$$\\color{green}{1}$$$$$$\\color{green}{1}$$$$$$\\color{green}{2}$$$$$$\\color{green}{2}$$$\n\nIn the third test case, a possible seating arrangement with $$$6$$$ happy people is shown below.\n\n$$$\\color{green}{4}$$$$$$\\color{green}{4}$$$$$$\\color{green}{}$$$$$$\\color{green}{2}$$$$$$\\color{green}{3}$$$$$$\\color{green}{3}$$$$$$\\color{green}{1}$$$$$$\\color{green}{}$$$$$$\\color{green}{}$$$$$$\\color{green}{}$$$",
+        "examples": [
+            {
+                "input": "4\n3 3\n2 3 1\n3 3\n2 2 2\n4 5\n1 1 2 2\n4 5\n3 1 1 3",
+                "output": "4\n6\n6\n6"
+            }
+        ],
+        "editorial": null,
+        "rating": 800,
+        "tags": [
+            "constructive algorithms",
+            "greedy",
+            "implementation",
+            "math"
+        ],
+        "testset_size": 4,
+        "official_tests": [
+            {
+                "input": "4\r\n3 3\r\n2 3 1\r\n3 3\r\n2 2 2\r\n4 5\r\n1 1 2 2\r\n4 5\r\n3 1 1 3\r\n",
+                "output": "4\r\n6\r\n6\r\n6\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2013/A",
+        "aliases": null,
+        "contest_id": "2013",
+        "contest_name": "Codeforces Round 973 (Div. 2)",
+        "contest_type": "CF",
+        "contest_start": 1726842900,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Zhan's Blender",
+        "description": "Today, a club fair was held at \"NSPhM\". In order to advertise his pastry club, Zhan decided to demonstrate the power of his blender.\n\nTo demonstrate the power of his blender, Zhan has $$$n$$$ fruits.\n\nThe blender can mix up to $$$x$$$ fruits per second.\n\nIn each second, Zhan can put up to $$$y$$$ fruits into the blender. After that, the blender will blend $$$\\min(x, c)$$$ fruits, where $$$c$$$ is the number of fruits inside the blender. After blending, blended fruits are removed from the blender.\n\nHelp Zhan determine the minimum amount of time required for Zhan to blend all fruits.",
+        "input_format": "Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \\le t \\le 1000$$$). The description of the test cases follows.\n\nThe first line of each test case contains one integer $$$n$$$ ($$$0 \\le n \\le 10^9$$$) — the number of fruits Zhan has.\n\nThe second line of each test case contains two integers $$$x$$$ and $$$y$$$ ($$$1 \\le x, y \\le 10^9$$$) — the number of fruits the blender can blend per second and the number of fruits Zhan can put into the blender per second.",
+        "output_format": "For each testcase, output a single integer — the minimum number of seconds to blend all fruits.",
+        "interaction_format": null,
+        "note": "In the first example, you can first put $$$2$$$ fruits in the blender. After that, the blender will mix these $$$2$$$ fruits, and in the end, there will be $$$0$$$ fruits left in the blender. Then you can put $$$3$$$ fruits into the blender, after which the blender will mix these $$$3$$$ fruits.\n\nIn the second example, you can put $$$1$$$ fruit into the blender $$$3$$$ times.\n\nIn the third example, you can first put $$$3$$$ fruits into the blender, then add another $$$3$$$ fruits.",
+        "examples": [
+            {
+                "input": "5\n5\n3 4\n3\n1 2\n6\n4 3\n100\n4 3\n9\n3 3",
+                "output": "2\n3\n2\n34\n3"
+            }
+        ],
+        "editorial": "Давайте рассмотрим 2 случая.\n1) если $$$x >= y$$$, то можно заметить, что блендер каждую секунду будет смешивать $$$\\min(c, y)$$$ фруктов, следовательно, ответом будет $$$(n + y - 1) / y$$$.\n2) если $$$x < y$$$, блендер будет смешивать $$$\\min(c, x)$$$ фруктов, ответ будет $$$(n + x - 1) / x$$$ аналогично.\nОтвет будет $$$(n + \\min(x, y)) / \\min(x, y)$$$.",
+        "rating": 800,
+        "tags": [
+            "constructive algorithms",
+            "math"
+        ],
+        "testset_size": 3,
+        "official_tests": [
+            {
+                "input": "5\r\n5\r\n3 4\r\n3\r\n1 2\r\n6\r\n4 3\r\n100\r\n4 3\r\n9\r\n3 3\r\n",
+                "output": "2\r\n3\r\n2\r\n34\r\n3\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2044/A",
+        "aliases": null,
+        "contest_id": "2044",
+        "contest_name": "Codeforces Round 993 (Div. 4)",
+        "contest_type": "ICPC",
+        "contest_start": 1734273300,
+        "contest_start_year": 2024,
+        "index": "A",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Easy Problem",
+        "description": "Cube is given an integer $$$n$$$. She wants to know how many ordered pairs of positive integers $$$(a,b)$$$ there are such that $$$a=n-b$$$. Since Cube is not very good at math, please help her!",
+        "input_format": "The first line contains an integer $$$t$$$ ($$$1 \\leq t \\leq 99$$$) — the number of test cases.\n\nThe only line of each test case contains an integer $$$n$$$ ($$$2 \\leq n \\leq 100$$$).",
+        "output_format": "For each test case, output the number of ordered pairs $$$(a, b)$$$ on a new line.",
+        "interaction_format": null,
+        "note": "In the first test case, the only ordered pair that works is $$$(a,b)=(1,1)$$$.\n\nIn the second test case, the three ordered pairs of $$$(a,b)$$$ that work are $$$(3,1), (2,2), (1,3)$$$.",
+        "examples": [
+            {
+                "input": "3\n2\n4\n6",
+                "output": "1\n3\n5"
+            }
+        ],
+        "editorial": null,
+        "rating": 800,
+        "tags": [
+            "brute force",
+            "math"
+        ],
+        "testset_size": 3,
+        "official_tests": [
+            {
+                "input": "3\r\n2\r\n4\r\n6\r\n",
+                "output": "1\r\n3\r\n5\r\n"
+            },
+            {
+                "input": "99\r\n2\r\n3\r\n4\r\n5\r\n6\r\n7\r\n8\r\n9\r\n10\r\n11\r\n12\r\n13\r\n14\r\n15\r\n16\r\n17\r\n18\r\n19\r\n20\r\n21\r\n22\r\n23\r\n24\r\n25\r\n26\r\n27\r\n28\r\n29\r\n30\r\n31\r\n32\r\n33\r\n34\r\n35\r\n36\r\n37\r\n38\r\n39\r\n40\r\n41\r\n42\r\n43\r\n44\r\n45\r\n46\r\n47\r\n48\r\n49\r\n50\r\n51\r\n52\r\n53\r\n54\r\n55\r\n56\r\n57\r\n58\r\n59\r\n60\r\n61\r\n62\r\n63\r\n64\r\n65\r\n66\r\n67\r\n68\r\n69\r\n70\r\n71\r\n72\r\n73\r\n74\r\n75\r\n76\r\n77\r\n78\r\n79\r\n80\r\n81\r\n82\r\n83\r\n84\r\n85\r\n86\r\n87\r\n88\r\n89\r\n90\r\n91\r\n92\r\n93\r\n94\r\n95\r\n96\r\n97\r\n98\r\n99\r\n100\r\n",
+                "output": "1\r\n2\r\n3\r\n4\r\n5\r\n6\r\n7\r\n8\r\n9\r\n10\r\n11\r\n12\r\n13\r\n14\r\n15\r\n16\r\n17\r\n18\r\n19\r\n20\r\n21\r\n22\r\n23\r\n24\r\n25\r\n26\r\n27\r\n28\r\n29\r\n30\r\n31\r\n32\r\n33\r\n34\r\n35\r\n36\r\n37\r\n38\r\n39\r\n40\r\n41\r\n42\r\n43\r\n44\r\n45\r\n46\r\n47\r\n48\r\n49\r\n50\r\n51\r\n52\r\n53\r\n54\r\n55\r\n56\r\n57\r\n58\r\n59\r\n60\r\n61\r\n62\r\n63\r\n64\r\n65\r\n66\r\n67\r\n68\r\n69\r\n70\r\n71\r\n72\r\n73\r\n74\r\n75\r\n76\r\n77\r\n78\r\n79\r\n80\r\n81\r\n82\r\n83\r\n84\r\n85\r\n86\r\n87\r\n88\r\n89\r\n90\r\n91\r\n92\r\n93\r\n94\r\n95\r\n96\r\n97\r\n98\r\n99\r\n"
+            },
+            {
+                "input": "99\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n100\r\n",
+                "output": "99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n99\r\n"
+            }
+        ],
+        "official_tests_complete": true,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2044/B",
+        "aliases": null,
+        "contest_id": "2044",
+        "contest_name": "Codeforces Round 993 (Div. 4)",
+        "contest_type": "ICPC",
+        "contest_start": 1734273300,
+        "contest_start_year": 2024,
+        "index": "B",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Normal Problem",
+        "description": "A string consisting of only characters 'p', 'q', and 'w' is painted on a glass window of a store. Ship walks past the store, standing directly in front of the glass window, and observes string $$$a$$$. Ship then heads inside the store, looks directly at the same glass window, and observes string $$$b$$$.\n\nShip gives you string $$$a$$$. Your job is to find and output $$$b$$$.",
+        "input_format": "The first line contains an integer $$$t$$$ ($$$1 \\leq t \\leq 100$$$) — the number of test cases.\n\nThe only line of each test case contains a string $$$a$$$ ($$$1 \\leq |a| \\leq 100$$$) — the string Ship observes from outside the store. It is guaranteed that $$$a$$$ only contains characters 'p', 'q', and 'w'.",
+        "output_format": "For each test case, output string $$$b$$$, the string Ship observes from inside the store, on a new line.",
+        "interaction_format": null,
+        "note": null,
+        "examples": [
+            {
+                "input": "5\nqwq\nppppp\npppwwwqqq\nwqpqwpqwwqp\npqpqpqpq",
+                "output": "pwp\nqqqqq\npppwwwqqq\nqpwwpqwpqpw\npqpqpqpq"
+            }
+        ],
+        "editorial": null,
+        "rating": 800,
+        "tags": [
+            "implementation",
+            "strings"
+        ],
+        "testset_size": 3,
+        "official_tests": [
+            {
+                "input": "5\r\nqwq\r\nppppp\r\npppwwwqqq\r\nwqpqwpqwwqp\r\npqpqpqpq\r\n",
+                "output": "pwp\r\nqqqqq\r\npppwwwqqq\r\nqpwwpqwpqpw\r\npqpqpqpq\r\n"
+            },
+            {
+                "input": "1\r\npwwwqppppwppwwwpwwwppqpwwqppqpppwwwwpwwwwpwwwwqwqqwwww\r\n",
+                "output": "wwwwppwpwwwwqwwwwqwwwwqqqpqqpwwqpqqwwwqwwwqqwqqqqpwwwq\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    },
+    {
+        "id": "2044/C",
+        "aliases": null,
+        "contest_id": "2044",
+        "contest_name": "Codeforces Round 993 (Div. 4)",
+        "contest_type": "ICPC",
+        "contest_start": 1734273300,
+        "contest_start_year": 2024,
+        "index": "C",
+        "time_limit": 1.0,
+        "memory_limit": 256.0,
+        "title": "Hard Problem",
+        "description": "Ball is the teacher in Paperfold University. The seats of his classroom are arranged in $$$2$$$ rows with $$$m$$$ seats each.\n\nBall is teaching $$$a + b + c$$$ monkeys, and he wants to assign as many monkeys to a seat as possible. Ball knows that $$$a$$$ of them only want to sit in row $$$1$$$, $$$b$$$ of them only want to sit in row $$$2$$$, and $$$c$$$ of them have no preference. Only one monkey may sit in each seat, and each monkey's preference must be followed if it is seated.\n\nWhat is the maximum number of monkeys that Ball can seat?",
+        "input_format": "The first line contains an integer $$$t$$$ ($$$1 \\leq t \\leq 10^4$$$) — the number of test cases.\n\nEach test case contains four integers $$$m$$$, $$$a$$$, $$$b$$$, and $$$c$$$ ($$$1 \\leq m, a, b, c \\leq 10^8$$$).",
+        "output_format": "For each test case, output the maximum number of monkeys you can seat.",
+        "interaction_format": null,
+        "note": "In the second test case, $$$6$$$ monkeys want to sit in the front row, but only $$$3$$$ seats are available. The monkeys that have no preference and the monkeys who prefer sitting in the second row can sit in the second row together. Thus, the answer is $$$3+2=5$$$.",
+        "examples": [
+            {
+                "input": "5\n10 5 5 10\n3 6 1 1\n15 14 12 4\n1 1 1 1\n420 6 9 69",
+                "output": "20\n5\n30\n2\n84"
+            }
+        ],
+        "editorial": null,
+        "rating": 800,
+        "tags": [
+            "greedy",
+            "math"
+        ],
+        "testset_size": 4,
+        "official_tests": [
+            {
+                "input": "5\r\n10 5 5 10\r\n3 6 1 1\r\n15 14 12 4\r\n1 1 1 1\r\n420 6 9 69\r\n",
+                "output": "20\r\n5\r\n30\r\n2\r\n84\r\n"
+            }
+        ],
+        "official_tests_complete": false,
+        "input_mode": "stdio",
+        "generated_checker": null,
+        "executable": true
+    }
+]
\ No newline at end of file