[
  {
    "id": "Geometric_Reasoning/Maze06/Maze06001",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06001-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 8, 9",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06002",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06002-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 6, 9",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06003",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06003-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 5, 7",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06004",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06004-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 5, 6, 7",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06005",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06005-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 2, 6",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06006",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06006-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 5, 6",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06007",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06007-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 2, 3, 6",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06008",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06008-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 5, 7, 8",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06009",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06009-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 2, 4",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06010",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06010-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 6, 7",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06011",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06011-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 2, 3, 6",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06012",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06012-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 5, 6, 7",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06013",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06013-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 8, 9",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06014",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06014-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 5, 6, 7",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06015",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06015-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 2, 5",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06016",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06016-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 2, 3, 4, 7",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06017",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06017-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 7, 8",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06018",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06018-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 6, 7",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06019",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06019-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 3, 7",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06020",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06020-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 2, 3, 5, 6, 8, 9",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06021",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06021-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 2, 3, 7",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06022",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06022-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 2, 6, 7",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06023",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06023-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 3, 4, 5, 6, 7, 9",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06024",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06024-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 6, 7",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06025",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06025-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 2, 3, 4, 5",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06026",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06026-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 2, 6, 7",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06027",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06027-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 5, 6, 9",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06028",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06028-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 9",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06029",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06029-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 3, 4, 5, 7",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06030",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06030-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 2, 3, 7",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06031",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06031-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 2, 3, 4",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06032",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06032-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 3, 6, 7, 8",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06033",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06033-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 5, 6",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06034",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06034-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 3, 5, 7",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06035",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06035-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 7, 8, 9",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06036",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06036-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 5, 9",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06037",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06037-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 8, 9",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06038",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06038-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 5, 7, 8, 9",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06039",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06039-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 2, 3, 5, 7, 8",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06040",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06040-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 6, 9",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06041",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06041-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 6, 7, 9",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06042",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06042-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 6, 7",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06043",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06043-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 5, 6, 7",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06044",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06044-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 5, 6, 7",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06045",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06045-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 2, 3, 5, 7",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06046",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06046-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 5, 8, 9",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06047",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06047-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 5, 6",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06048",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06048-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 8, 9",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06049",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06049-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 6, 7, 8",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06050",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06050-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 5",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06051",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 7). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06051-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 1, 2, 9",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06052",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06052-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 3, 4",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06053",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06053-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 3, 5",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06054",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06054-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 8, 9",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06055",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06055-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 6, 9",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06056",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06056-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 3, 4, 7",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06057",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06057-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 2, 6, 7, 8, 9",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06058",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06058-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 6, 7",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06061",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06061-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 5, 6",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06062",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06062-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 2, 4, 7",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06063",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06063-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 5, 6, 7",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06064",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06064-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 7, 8",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06065",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06065-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 6",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06066",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06066-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 5, 6, 9",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06067",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06067-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 7",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06068",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06068-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 5, 6, 9",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06069",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06069-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 7, 9",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06070",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06070-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 2, 4, 5, 6, 7",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06071",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06071-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 5, 6, 7",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06072",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06072-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 2, 5, 6, 7",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06073",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06073-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 3, 5, 7",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06074",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06074-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 6, 7, 8",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06075",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06075-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 6",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06076",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06076-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 8, 9",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06078",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06078-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 5, 7",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06079",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06079-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 5, 6, 7",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06080",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06080-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 6",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06081",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06081-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 5, 7",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06082",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06082-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 2, 3, 7",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06083",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06083-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 5",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06084",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06084-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 6",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06085",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06085-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 2, 5, 8",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06086",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06086-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 6, 8",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06087",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06087-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 2, 3, 7",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06088",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06088-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 1, 2, 3, 4, 7",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06089",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 9). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06089-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 5, 7, 8",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06090",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06090-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 2, 5, 6, 7",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06091",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06091-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 2, 6",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06092",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06092-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 3, 4, 6, 9",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06093",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06093-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 5, 6",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06094",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06094-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 5, 6",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06095",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06095-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 5, 9",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06096",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06096-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 2, 4, 7",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06097",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06097-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 2, 3, 4",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06098",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06098-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 5, 6, 7, 8",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06099",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06099-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 2, 8",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Maze06/Maze06100",
    "system_prompt": "",
    "prompt": "As shown in the figure, this is a maze created by extracting some edge lines from a 6*6 chessboard grid. The drawn lines represent impassable areas. The entrance is at the bottom-left opening, and the exit is at the top-right opening. There exists a unique shortest path from the entrance to the exit. The numbers 1-9 represent their respective grid cells, and some cells may not be traversed by this shortest path (such as 1). Which points among 1-9 must this shortest path pass through? Please provide your thought process, and finally list all the points that must be passed through in ascending order of their numbers. The format is as follows: The shortest path from the maze entrance to the exit must pass through: x, x...",
    "image_list": [
      "Maze06/Maze06100-1-1.png"
    ],
    "ground_truth": {
      "answer": "The shortest path from the maze entrance to the exit must pass through: 4, 6",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Maze06",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  }
]