{
  "dataset_name": "CoTEgo-Recursion-MCQ",
  "version": "v8_multiturn_with_move_constraint",
  "notes": "State strings use L=..._C=..._R=... with rings listed bottom-to-top. 'E' indicates empty pole. Images are named <state>.jpg. Reason questions are free-form text and are not auto-graded.",
  "poles": [
    "L",
    "C",
    "R"
  ],
  "size_order_largest_to_smallest": "G > L > S > R > O > Y",
  "problems": [
    {
      "problem_id": "L=OY_C=E_R=E__TO__L=E_C=OY_R=E",
      "n_rings": 2,
      "source_pole": "L",
      "target_pole": "C",
      "aux_pole": "R",
      "initial_state": "L=OY_C=E_R=E",
      "goal_state": "L=E_C=OY_R=E",
      "initial_image": "L=OY_C=E_R=E.jpg",
      "goal_image": "L=E_C=OY_R=E.jpg",
      "questions": [
        {
          "qid": "L=OY_C=E_R=E__TO__L=E_C=OY_R=E__Q1",
          "depth": 1,
          "substack_size": 1,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=Y_C=E_R=E -> L=E_C=Y_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=Y_C=E_R=E -> L=E_C=E_R=Y",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=Y_R=E -> L=Y_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=E_C=Y_R=E -> L=E_C=E_R=Y",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=E_R=Y -> L=Y_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=E_C=E_R=Y -> L=E_C=Y_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 2,
          "answer_text": "L=Y_C=E_R=E -> L=E_C=E_R=Y",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=OY_C=E_R=E.jpg\nGoal state image: L=E_C=OY_R=E.jpg\nInitial state: L=OY_C=E_R=E\nGoal state: L=E_C=OY_R=E\nAnswer the following questions.\n\nQuestion (depth 1): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=OY_C=E_R=E.jpg\nGoal state image: L=E_C=OY_R=E.jpg\nInitial state: L=OY_C=E_R=E\nGoal state: L=E_C=OY_R=E\nAnswer the following questions.\n\nReasoning (depth 1): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        }
      ],
      "full_problem_prompt": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=OY_C=E_R=E.jpg\nGoal state image: L=E_C=OY_R=E.jpg\nInitial state: L=OY_C=E_R=E\nGoal state: L=E_C=OY_R=E\nAnswer the following questions."
    },
    {
      "problem_id": "L=OY_C=E_R=E__TO__L=E_C=E_R=OY",
      "n_rings": 2,
      "source_pole": "L",
      "target_pole": "R",
      "aux_pole": "C",
      "initial_state": "L=OY_C=E_R=E",
      "goal_state": "L=E_C=E_R=OY",
      "initial_image": "L=OY_C=E_R=E.jpg",
      "goal_image": "L=E_C=E_R=OY.jpg",
      "questions": [
        {
          "qid": "L=OY_C=E_R=E__TO__L=E_C=E_R=OY__Q1",
          "depth": 1,
          "substack_size": 1,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=Y_C=E_R=E -> L=E_C=Y_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=Y_C=E_R=E -> L=E_C=E_R=Y",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=Y_R=E -> L=Y_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=E_C=Y_R=E -> L=E_C=E_R=Y",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=E_R=Y -> L=Y_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=E_C=E_R=Y -> L=E_C=Y_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 1,
          "answer_text": "L=Y_C=E_R=E -> L=E_C=Y_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=OY_C=E_R=E.jpg\nGoal state image: L=E_C=E_R=OY.jpg\nInitial state: L=OY_C=E_R=E\nGoal state: L=E_C=E_R=OY\nAnswer the following questions.\n\nQuestion (depth 1): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=OY_C=E_R=E.jpg\nGoal state image: L=E_C=E_R=OY.jpg\nInitial state: L=OY_C=E_R=E\nGoal state: L=E_C=E_R=OY\nAnswer the following questions.\n\nReasoning (depth 1): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        }
      ],
      "full_problem_prompt": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=OY_C=E_R=E.jpg\nGoal state image: L=E_C=E_R=OY.jpg\nInitial state: L=OY_C=E_R=E\nGoal state: L=E_C=E_R=OY\nAnswer the following questions."
    },
    {
      "problem_id": "L=E_C=OY_R=E__TO__L=OY_C=E_R=E",
      "n_rings": 2,
      "source_pole": "C",
      "target_pole": "L",
      "aux_pole": "R",
      "initial_state": "L=E_C=OY_R=E",
      "goal_state": "L=OY_C=E_R=E",
      "initial_image": "L=E_C=OY_R=E.jpg",
      "goal_image": "L=OY_C=E_R=E.jpg",
      "questions": [
        {
          "qid": "L=E_C=OY_R=E__TO__L=OY_C=E_R=E__Q1",
          "depth": 1,
          "substack_size": 1,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=Y_C=E_R=E -> L=E_C=Y_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=Y_C=E_R=E -> L=E_C=E_R=Y",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=Y_R=E -> L=Y_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=E_C=Y_R=E -> L=E_C=E_R=Y",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=E_R=Y -> L=Y_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=E_C=E_R=Y -> L=E_C=Y_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 4,
          "answer_text": "L=E_C=Y_R=E -> L=E_C=E_R=Y",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=OY_R=E.jpg\nGoal state image: L=OY_C=E_R=E.jpg\nInitial state: L=E_C=OY_R=E\nGoal state: L=OY_C=E_R=E\nAnswer the following questions.\n\nQuestion (depth 1): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=OY_R=E.jpg\nGoal state image: L=OY_C=E_R=E.jpg\nInitial state: L=E_C=OY_R=E\nGoal state: L=OY_C=E_R=E\nAnswer the following questions.\n\nReasoning (depth 1): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        }
      ],
      "full_problem_prompt": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=OY_R=E.jpg\nGoal state image: L=OY_C=E_R=E.jpg\nInitial state: L=E_C=OY_R=E\nGoal state: L=OY_C=E_R=E\nAnswer the following questions."
    },
    {
      "problem_id": "L=E_C=OY_R=E__TO__L=E_C=E_R=OY",
      "n_rings": 2,
      "source_pole": "C",
      "target_pole": "R",
      "aux_pole": "L",
      "initial_state": "L=E_C=OY_R=E",
      "goal_state": "L=E_C=E_R=OY",
      "initial_image": "L=E_C=OY_R=E.jpg",
      "goal_image": "L=E_C=E_R=OY.jpg",
      "questions": [
        {
          "qid": "L=E_C=OY_R=E__TO__L=E_C=E_R=OY__Q1",
          "depth": 1,
          "substack_size": 1,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=Y_C=E_R=E -> L=E_C=Y_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=Y_C=E_R=E -> L=E_C=E_R=Y",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=Y_R=E -> L=Y_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=E_C=Y_R=E -> L=E_C=E_R=Y",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=E_R=Y -> L=Y_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=E_C=E_R=Y -> L=E_C=Y_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 3,
          "answer_text": "L=E_C=Y_R=E -> L=Y_C=E_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=OY_R=E.jpg\nGoal state image: L=E_C=E_R=OY.jpg\nInitial state: L=E_C=OY_R=E\nGoal state: L=E_C=E_R=OY\nAnswer the following questions.\n\nQuestion (depth 1): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=OY_R=E.jpg\nGoal state image: L=E_C=E_R=OY.jpg\nInitial state: L=E_C=OY_R=E\nGoal state: L=E_C=E_R=OY\nAnswer the following questions.\n\nReasoning (depth 1): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        }
      ],
      "full_problem_prompt": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=OY_R=E.jpg\nGoal state image: L=E_C=E_R=OY.jpg\nInitial state: L=E_C=OY_R=E\nGoal state: L=E_C=E_R=OY\nAnswer the following questions."
    },
    {
      "problem_id": "L=E_C=E_R=OY__TO__L=OY_C=E_R=E",
      "n_rings": 2,
      "source_pole": "R",
      "target_pole": "L",
      "aux_pole": "C",
      "initial_state": "L=E_C=E_R=OY",
      "goal_state": "L=OY_C=E_R=E",
      "initial_image": "L=E_C=E_R=OY.jpg",
      "goal_image": "L=OY_C=E_R=E.jpg",
      "questions": [
        {
          "qid": "L=E_C=E_R=OY__TO__L=OY_C=E_R=E__Q1",
          "depth": 1,
          "substack_size": 1,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=Y_C=E_R=E -> L=E_C=Y_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=Y_C=E_R=E -> L=E_C=E_R=Y",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=Y_R=E -> L=Y_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=E_C=Y_R=E -> L=E_C=E_R=Y",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=E_R=Y -> L=Y_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=E_C=E_R=Y -> L=E_C=Y_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 6,
          "answer_text": "L=E_C=E_R=Y -> L=E_C=Y_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=OY.jpg\nGoal state image: L=OY_C=E_R=E.jpg\nInitial state: L=E_C=E_R=OY\nGoal state: L=OY_C=E_R=E\nAnswer the following questions.\n\nQuestion (depth 1): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=OY.jpg\nGoal state image: L=OY_C=E_R=E.jpg\nInitial state: L=E_C=E_R=OY\nGoal state: L=OY_C=E_R=E\nAnswer the following questions.\n\nReasoning (depth 1): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        }
      ],
      "full_problem_prompt": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=OY.jpg\nGoal state image: L=OY_C=E_R=E.jpg\nInitial state: L=E_C=E_R=OY\nGoal state: L=OY_C=E_R=E\nAnswer the following questions."
    },
    {
      "problem_id": "L=E_C=E_R=OY__TO__L=E_C=OY_R=E",
      "n_rings": 2,
      "source_pole": "R",
      "target_pole": "C",
      "aux_pole": "L",
      "initial_state": "L=E_C=E_R=OY",
      "goal_state": "L=E_C=OY_R=E",
      "initial_image": "L=E_C=E_R=OY.jpg",
      "goal_image": "L=E_C=OY_R=E.jpg",
      "questions": [
        {
          "qid": "L=E_C=E_R=OY__TO__L=E_C=OY_R=E__Q1",
          "depth": 1,
          "substack_size": 1,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=Y_C=E_R=E -> L=E_C=Y_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=Y_C=E_R=E -> L=E_C=E_R=Y",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=Y_R=E -> L=Y_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=E_C=Y_R=E -> L=E_C=E_R=Y",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=E_R=Y -> L=Y_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=E_C=E_R=Y -> L=E_C=Y_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 5,
          "answer_text": "L=E_C=E_R=Y -> L=Y_C=E_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=OY.jpg\nGoal state image: L=E_C=OY_R=E.jpg\nInitial state: L=E_C=E_R=OY\nGoal state: L=E_C=OY_R=E\nAnswer the following questions.\n\nQuestion (depth 1): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=OY.jpg\nGoal state image: L=E_C=OY_R=E.jpg\nInitial state: L=E_C=E_R=OY\nGoal state: L=E_C=OY_R=E\nAnswer the following questions.\n\nReasoning (depth 1): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        }
      ],
      "full_problem_prompt": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=OY.jpg\nGoal state image: L=E_C=OY_R=E.jpg\nInitial state: L=E_C=E_R=OY\nGoal state: L=E_C=OY_R=E\nAnswer the following questions."
    },
    {
      "problem_id": "L=ROY_C=E_R=E__TO__L=E_C=ROY_R=E",
      "n_rings": 3,
      "source_pole": "L",
      "target_pole": "C",
      "aux_pole": "R",
      "initial_state": "L=ROY_C=E_R=E",
      "goal_state": "L=E_C=ROY_R=E",
      "initial_image": "L=ROY_C=E_R=E.jpg",
      "goal_image": "L=E_C=ROY_R=E.jpg",
      "questions": [
        {
          "qid": "L=ROY_C=E_R=E__TO__L=E_C=ROY_R=E__Q1",
          "depth": 1,
          "substack_size": 2,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=OY_C=E_R=E -> L=E_C=OY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=OY_C=E_R=E -> L=E_C=E_R=OY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=OY_R=E -> L=OY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=E_C=OY_R=E -> L=E_C=E_R=OY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=E_R=OY -> L=OY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=E_C=E_R=OY -> L=E_C=OY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 2,
          "answer_text": "L=OY_C=E_R=E -> L=E_C=E_R=OY",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=ROY_C=E_R=E.jpg\nGoal state image: L=E_C=ROY_R=E.jpg\nInitial state: L=ROY_C=E_R=E\nGoal state: L=E_C=ROY_R=E\nAnswer the following questions.\n\nQuestion (depth 1): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=ROY_C=E_R=E.jpg\nGoal state image: L=E_C=ROY_R=E.jpg\nInitial state: L=ROY_C=E_R=E\nGoal state: L=E_C=ROY_R=E\nAnswer the following questions.\n\nReasoning (depth 1): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=ROY_C=E_R=E__TO__L=E_C=ROY_R=E__Q2",
          "depth": 2,
          "substack_size": 1,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=Y_C=E_R=E -> L=E_C=Y_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=Y_C=E_R=E -> L=E_C=E_R=Y",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=Y_R=E -> L=Y_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=E_C=Y_R=E -> L=E_C=E_R=Y",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=E_R=Y -> L=Y_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=E_C=E_R=Y -> L=E_C=Y_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 1,
          "answer_text": "L=Y_C=E_R=E -> L=E_C=Y_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=ROY_C=E_R=E.jpg\nGoal state image: L=E_C=ROY_R=E.jpg\nInitial state: L=ROY_C=E_R=E\nGoal state: L=E_C=ROY_R=E\nAnswer the following questions.\n\nQuestion (depth 2): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=ROY_C=E_R=E.jpg\nGoal state image: L=E_C=ROY_R=E.jpg\nInitial state: L=ROY_C=E_R=E\nGoal state: L=E_C=ROY_R=E\nAnswer the following questions.\n\nReasoning (depth 2): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        }
      ],
      "full_problem_prompt": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=ROY_C=E_R=E.jpg\nGoal state image: L=E_C=ROY_R=E.jpg\nInitial state: L=ROY_C=E_R=E\nGoal state: L=E_C=ROY_R=E\nAnswer the following questions."
    },
    {
      "problem_id": "L=ROY_C=E_R=E__TO__L=E_C=E_R=ROY",
      "n_rings": 3,
      "source_pole": "L",
      "target_pole": "R",
      "aux_pole": "C",
      "initial_state": "L=ROY_C=E_R=E",
      "goal_state": "L=E_C=E_R=ROY",
      "initial_image": "L=ROY_C=E_R=E.jpg",
      "goal_image": "L=E_C=E_R=ROY.jpg",
      "questions": [
        {
          "qid": "L=ROY_C=E_R=E__TO__L=E_C=E_R=ROY__Q1",
          "depth": 1,
          "substack_size": 2,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=OY_C=E_R=E -> L=E_C=OY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=OY_C=E_R=E -> L=E_C=E_R=OY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=OY_R=E -> L=OY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=E_C=OY_R=E -> L=E_C=E_R=OY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=E_R=OY -> L=OY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=E_C=E_R=OY -> L=E_C=OY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 1,
          "answer_text": "L=OY_C=E_R=E -> L=E_C=OY_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=ROY_C=E_R=E.jpg\nGoal state image: L=E_C=E_R=ROY.jpg\nInitial state: L=ROY_C=E_R=E\nGoal state: L=E_C=E_R=ROY\nAnswer the following questions.\n\nQuestion (depth 1): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=ROY_C=E_R=E.jpg\nGoal state image: L=E_C=E_R=ROY.jpg\nInitial state: L=ROY_C=E_R=E\nGoal state: L=E_C=E_R=ROY\nAnswer the following questions.\n\nReasoning (depth 1): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=ROY_C=E_R=E__TO__L=E_C=E_R=ROY__Q2",
          "depth": 2,
          "substack_size": 1,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=Y_C=E_R=E -> L=E_C=Y_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=Y_C=E_R=E -> L=E_C=E_R=Y",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=Y_R=E -> L=Y_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=E_C=Y_R=E -> L=E_C=E_R=Y",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=E_R=Y -> L=Y_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=E_C=E_R=Y -> L=E_C=Y_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 2,
          "answer_text": "L=Y_C=E_R=E -> L=E_C=E_R=Y",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=ROY_C=E_R=E.jpg\nGoal state image: L=E_C=E_R=ROY.jpg\nInitial state: L=ROY_C=E_R=E\nGoal state: L=E_C=E_R=ROY\nAnswer the following questions.\n\nQuestion (depth 2): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=ROY_C=E_R=E.jpg\nGoal state image: L=E_C=E_R=ROY.jpg\nInitial state: L=ROY_C=E_R=E\nGoal state: L=E_C=E_R=ROY\nAnswer the following questions.\n\nReasoning (depth 2): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        }
      ],
      "full_problem_prompt": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=ROY_C=E_R=E.jpg\nGoal state image: L=E_C=E_R=ROY.jpg\nInitial state: L=ROY_C=E_R=E\nGoal state: L=E_C=E_R=ROY\nAnswer the following questions."
    },
    {
      "problem_id": "L=E_C=ROY_R=E__TO__L=ROY_C=E_R=E",
      "n_rings": 3,
      "source_pole": "C",
      "target_pole": "L",
      "aux_pole": "R",
      "initial_state": "L=E_C=ROY_R=E",
      "goal_state": "L=ROY_C=E_R=E",
      "initial_image": "L=E_C=ROY_R=E.jpg",
      "goal_image": "L=ROY_C=E_R=E.jpg",
      "questions": [
        {
          "qid": "L=E_C=ROY_R=E__TO__L=ROY_C=E_R=E__Q1",
          "depth": 1,
          "substack_size": 2,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=OY_C=E_R=E -> L=E_C=OY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=OY_C=E_R=E -> L=E_C=E_R=OY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=OY_R=E -> L=OY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=E_C=OY_R=E -> L=E_C=E_R=OY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=E_R=OY -> L=OY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=E_C=E_R=OY -> L=E_C=OY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 4,
          "answer_text": "L=E_C=OY_R=E -> L=E_C=E_R=OY",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=ROY_R=E.jpg\nGoal state image: L=ROY_C=E_R=E.jpg\nInitial state: L=E_C=ROY_R=E\nGoal state: L=ROY_C=E_R=E\nAnswer the following questions.\n\nQuestion (depth 1): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=ROY_R=E.jpg\nGoal state image: L=ROY_C=E_R=E.jpg\nInitial state: L=E_C=ROY_R=E\nGoal state: L=ROY_C=E_R=E\nAnswer the following questions.\n\nReasoning (depth 1): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=ROY_R=E__TO__L=ROY_C=E_R=E__Q2",
          "depth": 2,
          "substack_size": 1,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=Y_C=E_R=E -> L=E_C=Y_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=Y_C=E_R=E -> L=E_C=E_R=Y",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=Y_R=E -> L=Y_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=E_C=Y_R=E -> L=E_C=E_R=Y",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=E_R=Y -> L=Y_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=E_C=E_R=Y -> L=E_C=Y_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 3,
          "answer_text": "L=E_C=Y_R=E -> L=Y_C=E_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=ROY_R=E.jpg\nGoal state image: L=ROY_C=E_R=E.jpg\nInitial state: L=E_C=ROY_R=E\nGoal state: L=ROY_C=E_R=E\nAnswer the following questions.\n\nQuestion (depth 2): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=ROY_R=E.jpg\nGoal state image: L=ROY_C=E_R=E.jpg\nInitial state: L=E_C=ROY_R=E\nGoal state: L=ROY_C=E_R=E\nAnswer the following questions.\n\nReasoning (depth 2): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        }
      ],
      "full_problem_prompt": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=ROY_R=E.jpg\nGoal state image: L=ROY_C=E_R=E.jpg\nInitial state: L=E_C=ROY_R=E\nGoal state: L=ROY_C=E_R=E\nAnswer the following questions."
    },
    {
      "problem_id": "L=E_C=ROY_R=E__TO__L=E_C=E_R=ROY",
      "n_rings": 3,
      "source_pole": "C",
      "target_pole": "R",
      "aux_pole": "L",
      "initial_state": "L=E_C=ROY_R=E",
      "goal_state": "L=E_C=E_R=ROY",
      "initial_image": "L=E_C=ROY_R=E.jpg",
      "goal_image": "L=E_C=E_R=ROY.jpg",
      "questions": [
        {
          "qid": "L=E_C=ROY_R=E__TO__L=E_C=E_R=ROY__Q1",
          "depth": 1,
          "substack_size": 2,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=OY_C=E_R=E -> L=E_C=OY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=OY_C=E_R=E -> L=E_C=E_R=OY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=OY_R=E -> L=OY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=E_C=OY_R=E -> L=E_C=E_R=OY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=E_R=OY -> L=OY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=E_C=E_R=OY -> L=E_C=OY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 3,
          "answer_text": "L=E_C=OY_R=E -> L=OY_C=E_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=ROY_R=E.jpg\nGoal state image: L=E_C=E_R=ROY.jpg\nInitial state: L=E_C=ROY_R=E\nGoal state: L=E_C=E_R=ROY\nAnswer the following questions.\n\nQuestion (depth 1): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=ROY_R=E.jpg\nGoal state image: L=E_C=E_R=ROY.jpg\nInitial state: L=E_C=ROY_R=E\nGoal state: L=E_C=E_R=ROY\nAnswer the following questions.\n\nReasoning (depth 1): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=ROY_R=E__TO__L=E_C=E_R=ROY__Q2",
          "depth": 2,
          "substack_size": 1,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=Y_C=E_R=E -> L=E_C=Y_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=Y_C=E_R=E -> L=E_C=E_R=Y",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=Y_R=E -> L=Y_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=E_C=Y_R=E -> L=E_C=E_R=Y",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=E_R=Y -> L=Y_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=E_C=E_R=Y -> L=E_C=Y_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 4,
          "answer_text": "L=E_C=Y_R=E -> L=E_C=E_R=Y",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=ROY_R=E.jpg\nGoal state image: L=E_C=E_R=ROY.jpg\nInitial state: L=E_C=ROY_R=E\nGoal state: L=E_C=E_R=ROY\nAnswer the following questions.\n\nQuestion (depth 2): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=ROY_R=E.jpg\nGoal state image: L=E_C=E_R=ROY.jpg\nInitial state: L=E_C=ROY_R=E\nGoal state: L=E_C=E_R=ROY\nAnswer the following questions.\n\nReasoning (depth 2): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        }
      ],
      "full_problem_prompt": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=ROY_R=E.jpg\nGoal state image: L=E_C=E_R=ROY.jpg\nInitial state: L=E_C=ROY_R=E\nGoal state: L=E_C=E_R=ROY\nAnswer the following questions."
    },
    {
      "problem_id": "L=E_C=E_R=ROY__TO__L=ROY_C=E_R=E",
      "n_rings": 3,
      "source_pole": "R",
      "target_pole": "L",
      "aux_pole": "C",
      "initial_state": "L=E_C=E_R=ROY",
      "goal_state": "L=ROY_C=E_R=E",
      "initial_image": "L=E_C=E_R=ROY.jpg",
      "goal_image": "L=ROY_C=E_R=E.jpg",
      "questions": [
        {
          "qid": "L=E_C=E_R=ROY__TO__L=ROY_C=E_R=E__Q1",
          "depth": 1,
          "substack_size": 2,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=OY_C=E_R=E -> L=E_C=OY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=OY_C=E_R=E -> L=E_C=E_R=OY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=OY_R=E -> L=OY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=E_C=OY_R=E -> L=E_C=E_R=OY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=E_R=OY -> L=OY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=E_C=E_R=OY -> L=E_C=OY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 6,
          "answer_text": "L=E_C=E_R=OY -> L=E_C=OY_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=ROY.jpg\nGoal state image: L=ROY_C=E_R=E.jpg\nInitial state: L=E_C=E_R=ROY\nGoal state: L=ROY_C=E_R=E\nAnswer the following questions.\n\nQuestion (depth 1): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=ROY.jpg\nGoal state image: L=ROY_C=E_R=E.jpg\nInitial state: L=E_C=E_R=ROY\nGoal state: L=ROY_C=E_R=E\nAnswer the following questions.\n\nReasoning (depth 1): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=E_R=ROY__TO__L=ROY_C=E_R=E__Q2",
          "depth": 2,
          "substack_size": 1,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=Y_C=E_R=E -> L=E_C=Y_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=Y_C=E_R=E -> L=E_C=E_R=Y",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=Y_R=E -> L=Y_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=E_C=Y_R=E -> L=E_C=E_R=Y",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=E_R=Y -> L=Y_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=E_C=E_R=Y -> L=E_C=Y_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 5,
          "answer_text": "L=E_C=E_R=Y -> L=Y_C=E_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=ROY.jpg\nGoal state image: L=ROY_C=E_R=E.jpg\nInitial state: L=E_C=E_R=ROY\nGoal state: L=ROY_C=E_R=E\nAnswer the following questions.\n\nQuestion (depth 2): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=ROY.jpg\nGoal state image: L=ROY_C=E_R=E.jpg\nInitial state: L=E_C=E_R=ROY\nGoal state: L=ROY_C=E_R=E\nAnswer the following questions.\n\nReasoning (depth 2): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        }
      ],
      "full_problem_prompt": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=ROY.jpg\nGoal state image: L=ROY_C=E_R=E.jpg\nInitial state: L=E_C=E_R=ROY\nGoal state: L=ROY_C=E_R=E\nAnswer the following questions."
    },
    {
      "problem_id": "L=E_C=E_R=ROY__TO__L=E_C=ROY_R=E",
      "n_rings": 3,
      "source_pole": "R",
      "target_pole": "C",
      "aux_pole": "L",
      "initial_state": "L=E_C=E_R=ROY",
      "goal_state": "L=E_C=ROY_R=E",
      "initial_image": "L=E_C=E_R=ROY.jpg",
      "goal_image": "L=E_C=ROY_R=E.jpg",
      "questions": [
        {
          "qid": "L=E_C=E_R=ROY__TO__L=E_C=ROY_R=E__Q1",
          "depth": 1,
          "substack_size": 2,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=OY_C=E_R=E -> L=E_C=OY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=OY_C=E_R=E -> L=E_C=E_R=OY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=OY_R=E -> L=OY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=E_C=OY_R=E -> L=E_C=E_R=OY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=E_R=OY -> L=OY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=E_C=E_R=OY -> L=E_C=OY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 5,
          "answer_text": "L=E_C=E_R=OY -> L=OY_C=E_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=ROY.jpg\nGoal state image: L=E_C=ROY_R=E.jpg\nInitial state: L=E_C=E_R=ROY\nGoal state: L=E_C=ROY_R=E\nAnswer the following questions.\n\nQuestion (depth 1): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=ROY.jpg\nGoal state image: L=E_C=ROY_R=E.jpg\nInitial state: L=E_C=E_R=ROY\nGoal state: L=E_C=ROY_R=E\nAnswer the following questions.\n\nReasoning (depth 1): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=E_R=ROY__TO__L=E_C=ROY_R=E__Q2",
          "depth": 2,
          "substack_size": 1,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=Y_C=E_R=E -> L=E_C=Y_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=Y_C=E_R=E -> L=E_C=E_R=Y",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=Y_R=E -> L=Y_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=E_C=Y_R=E -> L=E_C=E_R=Y",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=E_R=Y -> L=Y_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=E_C=E_R=Y -> L=E_C=Y_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 6,
          "answer_text": "L=E_C=E_R=Y -> L=E_C=Y_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=ROY.jpg\nGoal state image: L=E_C=ROY_R=E.jpg\nInitial state: L=E_C=E_R=ROY\nGoal state: L=E_C=ROY_R=E\nAnswer the following questions.\n\nQuestion (depth 2): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=ROY.jpg\nGoal state image: L=E_C=ROY_R=E.jpg\nInitial state: L=E_C=E_R=ROY\nGoal state: L=E_C=ROY_R=E\nAnswer the following questions.\n\nReasoning (depth 2): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        }
      ],
      "full_problem_prompt": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=ROY.jpg\nGoal state image: L=E_C=ROY_R=E.jpg\nInitial state: L=E_C=E_R=ROY\nGoal state: L=E_C=ROY_R=E\nAnswer the following questions."
    },
    {
      "problem_id": "L=SROY_C=E_R=E__TO__L=E_C=SROY_R=E",
      "n_rings": 4,
      "source_pole": "L",
      "target_pole": "C",
      "aux_pole": "R",
      "initial_state": "L=SROY_C=E_R=E",
      "goal_state": "L=E_C=SROY_R=E",
      "initial_image": "L=SROY_C=E_R=E.jpg",
      "goal_image": "L=E_C=SROY_R=E.jpg",
      "questions": [
        {
          "qid": "L=SROY_C=E_R=E__TO__L=E_C=SROY_R=E__Q1",
          "depth": 1,
          "substack_size": 3,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=ROY_C=E_R=E",
              "to_state": "L=E_C=ROY_R=E",
              "text": "L=ROY_C=E_R=E -> L=E_C=ROY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=ROY_C=E_R=E",
              "to_state": "L=E_C=E_R=ROY",
              "text": "L=ROY_C=E_R=E -> L=E_C=E_R=ROY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=ROY_R=E",
              "to_state": "L=ROY_C=E_R=E",
              "text": "L=E_C=ROY_R=E -> L=ROY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=ROY_R=E",
              "to_state": "L=E_C=E_R=ROY",
              "text": "L=E_C=ROY_R=E -> L=E_C=E_R=ROY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=ROY",
              "to_state": "L=ROY_C=E_R=E",
              "text": "L=E_C=E_R=ROY -> L=ROY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=ROY",
              "to_state": "L=E_C=ROY_R=E",
              "text": "L=E_C=E_R=ROY -> L=E_C=ROY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 2,
          "answer_text": "L=ROY_C=E_R=E -> L=E_C=E_R=ROY",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=SROY_C=E_R=E.jpg\nGoal state image: L=E_C=SROY_R=E.jpg\nInitial state: L=SROY_C=E_R=E\nGoal state: L=E_C=SROY_R=E\nAnswer the following questions.\n\nQuestion (depth 1): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=SROY_C=E_R=E.jpg\nGoal state image: L=E_C=SROY_R=E.jpg\nInitial state: L=SROY_C=E_R=E\nGoal state: L=E_C=SROY_R=E\nAnswer the following questions.\n\nReasoning (depth 1): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=SROY_C=E_R=E__TO__L=E_C=SROY_R=E__Q2",
          "depth": 2,
          "substack_size": 2,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=OY_C=E_R=E -> L=E_C=OY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=OY_C=E_R=E -> L=E_C=E_R=OY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=OY_R=E -> L=OY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=E_C=OY_R=E -> L=E_C=E_R=OY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=E_R=OY -> L=OY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=E_C=E_R=OY -> L=E_C=OY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 1,
          "answer_text": "L=OY_C=E_R=E -> L=E_C=OY_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=SROY_C=E_R=E.jpg\nGoal state image: L=E_C=SROY_R=E.jpg\nInitial state: L=SROY_C=E_R=E\nGoal state: L=E_C=SROY_R=E\nAnswer the following questions.\n\nQuestion (depth 2): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=SROY_C=E_R=E.jpg\nGoal state image: L=E_C=SROY_R=E.jpg\nInitial state: L=SROY_C=E_R=E\nGoal state: L=E_C=SROY_R=E\nAnswer the following questions.\n\nReasoning (depth 2): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=SROY_C=E_R=E__TO__L=E_C=SROY_R=E__Q3",
          "depth": 3,
          "substack_size": 1,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=Y_C=E_R=E -> L=E_C=Y_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=Y_C=E_R=E -> L=E_C=E_R=Y",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=Y_R=E -> L=Y_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=E_C=Y_R=E -> L=E_C=E_R=Y",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=E_R=Y -> L=Y_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=E_C=E_R=Y -> L=E_C=Y_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 2,
          "answer_text": "L=Y_C=E_R=E -> L=E_C=E_R=Y",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=SROY_C=E_R=E.jpg\nGoal state image: L=E_C=SROY_R=E.jpg\nInitial state: L=SROY_C=E_R=E\nGoal state: L=E_C=SROY_R=E\nAnswer the following questions.\n\nQuestion (depth 3): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=SROY_C=E_R=E.jpg\nGoal state image: L=E_C=SROY_R=E.jpg\nInitial state: L=SROY_C=E_R=E\nGoal state: L=E_C=SROY_R=E\nAnswer the following questions.\n\nReasoning (depth 3): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        }
      ],
      "full_problem_prompt": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=SROY_C=E_R=E.jpg\nGoal state image: L=E_C=SROY_R=E.jpg\nInitial state: L=SROY_C=E_R=E\nGoal state: L=E_C=SROY_R=E\nAnswer the following questions."
    },
    {
      "problem_id": "L=SROY_C=E_R=E__TO__L=E_C=E_R=SROY",
      "n_rings": 4,
      "source_pole": "L",
      "target_pole": "R",
      "aux_pole": "C",
      "initial_state": "L=SROY_C=E_R=E",
      "goal_state": "L=E_C=E_R=SROY",
      "initial_image": "L=SROY_C=E_R=E.jpg",
      "goal_image": "L=E_C=E_R=SROY.jpg",
      "questions": [
        {
          "qid": "L=SROY_C=E_R=E__TO__L=E_C=E_R=SROY__Q1",
          "depth": 1,
          "substack_size": 3,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=ROY_C=E_R=E",
              "to_state": "L=E_C=ROY_R=E",
              "text": "L=ROY_C=E_R=E -> L=E_C=ROY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=ROY_C=E_R=E",
              "to_state": "L=E_C=E_R=ROY",
              "text": "L=ROY_C=E_R=E -> L=E_C=E_R=ROY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=ROY_R=E",
              "to_state": "L=ROY_C=E_R=E",
              "text": "L=E_C=ROY_R=E -> L=ROY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=ROY_R=E",
              "to_state": "L=E_C=E_R=ROY",
              "text": "L=E_C=ROY_R=E -> L=E_C=E_R=ROY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=ROY",
              "to_state": "L=ROY_C=E_R=E",
              "text": "L=E_C=E_R=ROY -> L=ROY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=ROY",
              "to_state": "L=E_C=ROY_R=E",
              "text": "L=E_C=E_R=ROY -> L=E_C=ROY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 1,
          "answer_text": "L=ROY_C=E_R=E -> L=E_C=ROY_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=SROY_C=E_R=E.jpg\nGoal state image: L=E_C=E_R=SROY.jpg\nInitial state: L=SROY_C=E_R=E\nGoal state: L=E_C=E_R=SROY\nAnswer the following questions.\n\nQuestion (depth 1): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=SROY_C=E_R=E.jpg\nGoal state image: L=E_C=E_R=SROY.jpg\nInitial state: L=SROY_C=E_R=E\nGoal state: L=E_C=E_R=SROY\nAnswer the following questions.\n\nReasoning (depth 1): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=SROY_C=E_R=E__TO__L=E_C=E_R=SROY__Q2",
          "depth": 2,
          "substack_size": 2,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=OY_C=E_R=E -> L=E_C=OY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=OY_C=E_R=E -> L=E_C=E_R=OY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=OY_R=E -> L=OY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=E_C=OY_R=E -> L=E_C=E_R=OY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=E_R=OY -> L=OY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=E_C=E_R=OY -> L=E_C=OY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 2,
          "answer_text": "L=OY_C=E_R=E -> L=E_C=E_R=OY",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=SROY_C=E_R=E.jpg\nGoal state image: L=E_C=E_R=SROY.jpg\nInitial state: L=SROY_C=E_R=E\nGoal state: L=E_C=E_R=SROY\nAnswer the following questions.\n\nQuestion (depth 2): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=SROY_C=E_R=E.jpg\nGoal state image: L=E_C=E_R=SROY.jpg\nInitial state: L=SROY_C=E_R=E\nGoal state: L=E_C=E_R=SROY\nAnswer the following questions.\n\nReasoning (depth 2): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=SROY_C=E_R=E__TO__L=E_C=E_R=SROY__Q3",
          "depth": 3,
          "substack_size": 1,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=Y_C=E_R=E -> L=E_C=Y_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=Y_C=E_R=E -> L=E_C=E_R=Y",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=Y_R=E -> L=Y_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=E_C=Y_R=E -> L=E_C=E_R=Y",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=E_R=Y -> L=Y_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=E_C=E_R=Y -> L=E_C=Y_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 1,
          "answer_text": "L=Y_C=E_R=E -> L=E_C=Y_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=SROY_C=E_R=E.jpg\nGoal state image: L=E_C=E_R=SROY.jpg\nInitial state: L=SROY_C=E_R=E\nGoal state: L=E_C=E_R=SROY\nAnswer the following questions.\n\nQuestion (depth 3): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=SROY_C=E_R=E.jpg\nGoal state image: L=E_C=E_R=SROY.jpg\nInitial state: L=SROY_C=E_R=E\nGoal state: L=E_C=E_R=SROY\nAnswer the following questions.\n\nReasoning (depth 3): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        }
      ],
      "full_problem_prompt": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=SROY_C=E_R=E.jpg\nGoal state image: L=E_C=E_R=SROY.jpg\nInitial state: L=SROY_C=E_R=E\nGoal state: L=E_C=E_R=SROY\nAnswer the following questions."
    },
    {
      "problem_id": "L=E_C=SROY_R=E__TO__L=SROY_C=E_R=E",
      "n_rings": 4,
      "source_pole": "C",
      "target_pole": "L",
      "aux_pole": "R",
      "initial_state": "L=E_C=SROY_R=E",
      "goal_state": "L=SROY_C=E_R=E",
      "initial_image": "L=E_C=SROY_R=E.jpg",
      "goal_image": "L=SROY_C=E_R=E.jpg",
      "questions": [
        {
          "qid": "L=E_C=SROY_R=E__TO__L=SROY_C=E_R=E__Q1",
          "depth": 1,
          "substack_size": 3,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=ROY_C=E_R=E",
              "to_state": "L=E_C=ROY_R=E",
              "text": "L=ROY_C=E_R=E -> L=E_C=ROY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=ROY_C=E_R=E",
              "to_state": "L=E_C=E_R=ROY",
              "text": "L=ROY_C=E_R=E -> L=E_C=E_R=ROY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=ROY_R=E",
              "to_state": "L=ROY_C=E_R=E",
              "text": "L=E_C=ROY_R=E -> L=ROY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=ROY_R=E",
              "to_state": "L=E_C=E_R=ROY",
              "text": "L=E_C=ROY_R=E -> L=E_C=E_R=ROY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=ROY",
              "to_state": "L=ROY_C=E_R=E",
              "text": "L=E_C=E_R=ROY -> L=ROY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=ROY",
              "to_state": "L=E_C=ROY_R=E",
              "text": "L=E_C=E_R=ROY -> L=E_C=ROY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 4,
          "answer_text": "L=E_C=ROY_R=E -> L=E_C=E_R=ROY",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=SROY_R=E.jpg\nGoal state image: L=SROY_C=E_R=E.jpg\nInitial state: L=E_C=SROY_R=E\nGoal state: L=SROY_C=E_R=E\nAnswer the following questions.\n\nQuestion (depth 1): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=SROY_R=E.jpg\nGoal state image: L=SROY_C=E_R=E.jpg\nInitial state: L=E_C=SROY_R=E\nGoal state: L=SROY_C=E_R=E\nAnswer the following questions.\n\nReasoning (depth 1): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=SROY_R=E__TO__L=SROY_C=E_R=E__Q2",
          "depth": 2,
          "substack_size": 2,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=OY_C=E_R=E -> L=E_C=OY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=OY_C=E_R=E -> L=E_C=E_R=OY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=OY_R=E -> L=OY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=E_C=OY_R=E -> L=E_C=E_R=OY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=E_R=OY -> L=OY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=E_C=E_R=OY -> L=E_C=OY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 3,
          "answer_text": "L=E_C=OY_R=E -> L=OY_C=E_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=SROY_R=E.jpg\nGoal state image: L=SROY_C=E_R=E.jpg\nInitial state: L=E_C=SROY_R=E\nGoal state: L=SROY_C=E_R=E\nAnswer the following questions.\n\nQuestion (depth 2): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=SROY_R=E.jpg\nGoal state image: L=SROY_C=E_R=E.jpg\nInitial state: L=E_C=SROY_R=E\nGoal state: L=SROY_C=E_R=E\nAnswer the following questions.\n\nReasoning (depth 2): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=SROY_R=E__TO__L=SROY_C=E_R=E__Q3",
          "depth": 3,
          "substack_size": 1,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=Y_C=E_R=E -> L=E_C=Y_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=Y_C=E_R=E -> L=E_C=E_R=Y",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=Y_R=E -> L=Y_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=E_C=Y_R=E -> L=E_C=E_R=Y",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=E_R=Y -> L=Y_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=E_C=E_R=Y -> L=E_C=Y_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 4,
          "answer_text": "L=E_C=Y_R=E -> L=E_C=E_R=Y",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=SROY_R=E.jpg\nGoal state image: L=SROY_C=E_R=E.jpg\nInitial state: L=E_C=SROY_R=E\nGoal state: L=SROY_C=E_R=E\nAnswer the following questions.\n\nQuestion (depth 3): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=SROY_R=E.jpg\nGoal state image: L=SROY_C=E_R=E.jpg\nInitial state: L=E_C=SROY_R=E\nGoal state: L=SROY_C=E_R=E\nAnswer the following questions.\n\nReasoning (depth 3): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        }
      ],
      "full_problem_prompt": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=SROY_R=E.jpg\nGoal state image: L=SROY_C=E_R=E.jpg\nInitial state: L=E_C=SROY_R=E\nGoal state: L=SROY_C=E_R=E\nAnswer the following questions."
    },
    {
      "problem_id": "L=E_C=SROY_R=E__TO__L=E_C=E_R=SROY",
      "n_rings": 4,
      "source_pole": "C",
      "target_pole": "R",
      "aux_pole": "L",
      "initial_state": "L=E_C=SROY_R=E",
      "goal_state": "L=E_C=E_R=SROY",
      "initial_image": "L=E_C=SROY_R=E.jpg",
      "goal_image": "L=E_C=E_R=SROY.jpg",
      "questions": [
        {
          "qid": "L=E_C=SROY_R=E__TO__L=E_C=E_R=SROY__Q1",
          "depth": 1,
          "substack_size": 3,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=ROY_C=E_R=E",
              "to_state": "L=E_C=ROY_R=E",
              "text": "L=ROY_C=E_R=E -> L=E_C=ROY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=ROY_C=E_R=E",
              "to_state": "L=E_C=E_R=ROY",
              "text": "L=ROY_C=E_R=E -> L=E_C=E_R=ROY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=ROY_R=E",
              "to_state": "L=ROY_C=E_R=E",
              "text": "L=E_C=ROY_R=E -> L=ROY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=ROY_R=E",
              "to_state": "L=E_C=E_R=ROY",
              "text": "L=E_C=ROY_R=E -> L=E_C=E_R=ROY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=ROY",
              "to_state": "L=ROY_C=E_R=E",
              "text": "L=E_C=E_R=ROY -> L=ROY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=ROY",
              "to_state": "L=E_C=ROY_R=E",
              "text": "L=E_C=E_R=ROY -> L=E_C=ROY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 3,
          "answer_text": "L=E_C=ROY_R=E -> L=ROY_C=E_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=SROY_R=E.jpg\nGoal state image: L=E_C=E_R=SROY.jpg\nInitial state: L=E_C=SROY_R=E\nGoal state: L=E_C=E_R=SROY\nAnswer the following questions.\n\nQuestion (depth 1): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=SROY_R=E.jpg\nGoal state image: L=E_C=E_R=SROY.jpg\nInitial state: L=E_C=SROY_R=E\nGoal state: L=E_C=E_R=SROY\nAnswer the following questions.\n\nReasoning (depth 1): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=SROY_R=E__TO__L=E_C=E_R=SROY__Q2",
          "depth": 2,
          "substack_size": 2,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=OY_C=E_R=E -> L=E_C=OY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=OY_C=E_R=E -> L=E_C=E_R=OY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=OY_R=E -> L=OY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=E_C=OY_R=E -> L=E_C=E_R=OY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=E_R=OY -> L=OY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=E_C=E_R=OY -> L=E_C=OY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 4,
          "answer_text": "L=E_C=OY_R=E -> L=E_C=E_R=OY",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=SROY_R=E.jpg\nGoal state image: L=E_C=E_R=SROY.jpg\nInitial state: L=E_C=SROY_R=E\nGoal state: L=E_C=E_R=SROY\nAnswer the following questions.\n\nQuestion (depth 2): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=SROY_R=E.jpg\nGoal state image: L=E_C=E_R=SROY.jpg\nInitial state: L=E_C=SROY_R=E\nGoal state: L=E_C=E_R=SROY\nAnswer the following questions.\n\nReasoning (depth 2): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=SROY_R=E__TO__L=E_C=E_R=SROY__Q3",
          "depth": 3,
          "substack_size": 1,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=Y_C=E_R=E -> L=E_C=Y_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=Y_C=E_R=E -> L=E_C=E_R=Y",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=Y_R=E -> L=Y_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=E_C=Y_R=E -> L=E_C=E_R=Y",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=E_R=Y -> L=Y_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=E_C=E_R=Y -> L=E_C=Y_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 3,
          "answer_text": "L=E_C=Y_R=E -> L=Y_C=E_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=SROY_R=E.jpg\nGoal state image: L=E_C=E_R=SROY.jpg\nInitial state: L=E_C=SROY_R=E\nGoal state: L=E_C=E_R=SROY\nAnswer the following questions.\n\nQuestion (depth 3): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=SROY_R=E.jpg\nGoal state image: L=E_C=E_R=SROY.jpg\nInitial state: L=E_C=SROY_R=E\nGoal state: L=E_C=E_R=SROY\nAnswer the following questions.\n\nReasoning (depth 3): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        }
      ],
      "full_problem_prompt": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=SROY_R=E.jpg\nGoal state image: L=E_C=E_R=SROY.jpg\nInitial state: L=E_C=SROY_R=E\nGoal state: L=E_C=E_R=SROY\nAnswer the following questions."
    },
    {
      "problem_id": "L=E_C=E_R=SROY__TO__L=SROY_C=E_R=E",
      "n_rings": 4,
      "source_pole": "R",
      "target_pole": "L",
      "aux_pole": "C",
      "initial_state": "L=E_C=E_R=SROY",
      "goal_state": "L=SROY_C=E_R=E",
      "initial_image": "L=E_C=E_R=SROY.jpg",
      "goal_image": "L=SROY_C=E_R=E.jpg",
      "questions": [
        {
          "qid": "L=E_C=E_R=SROY__TO__L=SROY_C=E_R=E__Q1",
          "depth": 1,
          "substack_size": 3,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=ROY_C=E_R=E",
              "to_state": "L=E_C=ROY_R=E",
              "text": "L=ROY_C=E_R=E -> L=E_C=ROY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=ROY_C=E_R=E",
              "to_state": "L=E_C=E_R=ROY",
              "text": "L=ROY_C=E_R=E -> L=E_C=E_R=ROY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=ROY_R=E",
              "to_state": "L=ROY_C=E_R=E",
              "text": "L=E_C=ROY_R=E -> L=ROY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=ROY_R=E",
              "to_state": "L=E_C=E_R=ROY",
              "text": "L=E_C=ROY_R=E -> L=E_C=E_R=ROY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=ROY",
              "to_state": "L=ROY_C=E_R=E",
              "text": "L=E_C=E_R=ROY -> L=ROY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=ROY",
              "to_state": "L=E_C=ROY_R=E",
              "text": "L=E_C=E_R=ROY -> L=E_C=ROY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 6,
          "answer_text": "L=E_C=E_R=ROY -> L=E_C=ROY_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=SROY.jpg\nGoal state image: L=SROY_C=E_R=E.jpg\nInitial state: L=E_C=E_R=SROY\nGoal state: L=SROY_C=E_R=E\nAnswer the following questions.\n\nQuestion (depth 1): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=SROY.jpg\nGoal state image: L=SROY_C=E_R=E.jpg\nInitial state: L=E_C=E_R=SROY\nGoal state: L=SROY_C=E_R=E\nAnswer the following questions.\n\nReasoning (depth 1): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=E_R=SROY__TO__L=SROY_C=E_R=E__Q2",
          "depth": 2,
          "substack_size": 2,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=OY_C=E_R=E -> L=E_C=OY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=OY_C=E_R=E -> L=E_C=E_R=OY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=OY_R=E -> L=OY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=E_C=OY_R=E -> L=E_C=E_R=OY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=E_R=OY -> L=OY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=E_C=E_R=OY -> L=E_C=OY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 5,
          "answer_text": "L=E_C=E_R=OY -> L=OY_C=E_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=SROY.jpg\nGoal state image: L=SROY_C=E_R=E.jpg\nInitial state: L=E_C=E_R=SROY\nGoal state: L=SROY_C=E_R=E\nAnswer the following questions.\n\nQuestion (depth 2): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=SROY.jpg\nGoal state image: L=SROY_C=E_R=E.jpg\nInitial state: L=E_C=E_R=SROY\nGoal state: L=SROY_C=E_R=E\nAnswer the following questions.\n\nReasoning (depth 2): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=E_R=SROY__TO__L=SROY_C=E_R=E__Q3",
          "depth": 3,
          "substack_size": 1,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=Y_C=E_R=E -> L=E_C=Y_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=Y_C=E_R=E -> L=E_C=E_R=Y",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=Y_R=E -> L=Y_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=E_C=Y_R=E -> L=E_C=E_R=Y",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=E_R=Y -> L=Y_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=E_C=E_R=Y -> L=E_C=Y_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 6,
          "answer_text": "L=E_C=E_R=Y -> L=E_C=Y_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=SROY.jpg\nGoal state image: L=SROY_C=E_R=E.jpg\nInitial state: L=E_C=E_R=SROY\nGoal state: L=SROY_C=E_R=E\nAnswer the following questions.\n\nQuestion (depth 3): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=SROY.jpg\nGoal state image: L=SROY_C=E_R=E.jpg\nInitial state: L=E_C=E_R=SROY\nGoal state: L=SROY_C=E_R=E\nAnswer the following questions.\n\nReasoning (depth 3): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        }
      ],
      "full_problem_prompt": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=SROY.jpg\nGoal state image: L=SROY_C=E_R=E.jpg\nInitial state: L=E_C=E_R=SROY\nGoal state: L=SROY_C=E_R=E\nAnswer the following questions."
    },
    {
      "problem_id": "L=E_C=E_R=SROY__TO__L=E_C=SROY_R=E",
      "n_rings": 4,
      "source_pole": "R",
      "target_pole": "C",
      "aux_pole": "L",
      "initial_state": "L=E_C=E_R=SROY",
      "goal_state": "L=E_C=SROY_R=E",
      "initial_image": "L=E_C=E_R=SROY.jpg",
      "goal_image": "L=E_C=SROY_R=E.jpg",
      "questions": [
        {
          "qid": "L=E_C=E_R=SROY__TO__L=E_C=SROY_R=E__Q1",
          "depth": 1,
          "substack_size": 3,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=ROY_C=E_R=E",
              "to_state": "L=E_C=ROY_R=E",
              "text": "L=ROY_C=E_R=E -> L=E_C=ROY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=ROY_C=E_R=E",
              "to_state": "L=E_C=E_R=ROY",
              "text": "L=ROY_C=E_R=E -> L=E_C=E_R=ROY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=ROY_R=E",
              "to_state": "L=ROY_C=E_R=E",
              "text": "L=E_C=ROY_R=E -> L=ROY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=ROY_R=E",
              "to_state": "L=E_C=E_R=ROY",
              "text": "L=E_C=ROY_R=E -> L=E_C=E_R=ROY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=ROY",
              "to_state": "L=ROY_C=E_R=E",
              "text": "L=E_C=E_R=ROY -> L=ROY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=ROY",
              "to_state": "L=E_C=ROY_R=E",
              "text": "L=E_C=E_R=ROY -> L=E_C=ROY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 5,
          "answer_text": "L=E_C=E_R=ROY -> L=ROY_C=E_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=SROY.jpg\nGoal state image: L=E_C=SROY_R=E.jpg\nInitial state: L=E_C=E_R=SROY\nGoal state: L=E_C=SROY_R=E\nAnswer the following questions.\n\nQuestion (depth 1): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=SROY.jpg\nGoal state image: L=E_C=SROY_R=E.jpg\nInitial state: L=E_C=E_R=SROY\nGoal state: L=E_C=SROY_R=E\nAnswer the following questions.\n\nReasoning (depth 1): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=E_R=SROY__TO__L=E_C=SROY_R=E__Q2",
          "depth": 2,
          "substack_size": 2,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=OY_C=E_R=E -> L=E_C=OY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=OY_C=E_R=E -> L=E_C=E_R=OY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=OY_R=E -> L=OY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=E_C=OY_R=E -> L=E_C=E_R=OY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=E_R=OY -> L=OY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=E_C=E_R=OY -> L=E_C=OY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 6,
          "answer_text": "L=E_C=E_R=OY -> L=E_C=OY_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=SROY.jpg\nGoal state image: L=E_C=SROY_R=E.jpg\nInitial state: L=E_C=E_R=SROY\nGoal state: L=E_C=SROY_R=E\nAnswer the following questions.\n\nQuestion (depth 2): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=SROY.jpg\nGoal state image: L=E_C=SROY_R=E.jpg\nInitial state: L=E_C=E_R=SROY\nGoal state: L=E_C=SROY_R=E\nAnswer the following questions.\n\nReasoning (depth 2): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=E_R=SROY__TO__L=E_C=SROY_R=E__Q3",
          "depth": 3,
          "substack_size": 1,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=Y_C=E_R=E -> L=E_C=Y_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=Y_C=E_R=E -> L=E_C=E_R=Y",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=Y_R=E -> L=Y_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=E_C=Y_R=E -> L=E_C=E_R=Y",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=E_R=Y -> L=Y_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=E_C=E_R=Y -> L=E_C=Y_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 5,
          "answer_text": "L=E_C=E_R=Y -> L=Y_C=E_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=SROY.jpg\nGoal state image: L=E_C=SROY_R=E.jpg\nInitial state: L=E_C=E_R=SROY\nGoal state: L=E_C=SROY_R=E\nAnswer the following questions.\n\nQuestion (depth 3): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=SROY.jpg\nGoal state image: L=E_C=SROY_R=E.jpg\nInitial state: L=E_C=E_R=SROY\nGoal state: L=E_C=SROY_R=E\nAnswer the following questions.\n\nReasoning (depth 3): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        }
      ],
      "full_problem_prompt": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=SROY.jpg\nGoal state image: L=E_C=SROY_R=E.jpg\nInitial state: L=E_C=E_R=SROY\nGoal state: L=E_C=SROY_R=E\nAnswer the following questions."
    },
    {
      "problem_id": "L=LSROY_C=E_R=E__TO__L=E_C=LSROY_R=E",
      "n_rings": 5,
      "source_pole": "L",
      "target_pole": "C",
      "aux_pole": "R",
      "initial_state": "L=LSROY_C=E_R=E",
      "goal_state": "L=E_C=LSROY_R=E",
      "initial_image": "L=LSROY_C=E_R=E.jpg",
      "goal_image": "L=E_C=LSROY_R=E.jpg",
      "questions": [
        {
          "qid": "L=LSROY_C=E_R=E__TO__L=E_C=LSROY_R=E__Q1",
          "depth": 1,
          "substack_size": 4,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=SROY_C=E_R=E",
              "to_state": "L=E_C=SROY_R=E",
              "text": "L=SROY_C=E_R=E -> L=E_C=SROY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=SROY_C=E_R=E",
              "to_state": "L=E_C=E_R=SROY",
              "text": "L=SROY_C=E_R=E -> L=E_C=E_R=SROY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=SROY_R=E",
              "to_state": "L=SROY_C=E_R=E",
              "text": "L=E_C=SROY_R=E -> L=SROY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=SROY_R=E",
              "to_state": "L=E_C=E_R=SROY",
              "text": "L=E_C=SROY_R=E -> L=E_C=E_R=SROY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=SROY",
              "to_state": "L=SROY_C=E_R=E",
              "text": "L=E_C=E_R=SROY -> L=SROY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=SROY",
              "to_state": "L=E_C=SROY_R=E",
              "text": "L=E_C=E_R=SROY -> L=E_C=SROY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 2,
          "answer_text": "L=SROY_C=E_R=E -> L=E_C=E_R=SROY",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=LSROY_C=E_R=E.jpg\nGoal state image: L=E_C=LSROY_R=E.jpg\nInitial state: L=LSROY_C=E_R=E\nGoal state: L=E_C=LSROY_R=E\nAnswer the following questions.\n\nQuestion (depth 1): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=SROY_C=E_R=E -> L=E_C=SROY_R=E\n2. L=SROY_C=E_R=E -> L=E_C=E_R=SROY\n3. L=E_C=SROY_R=E -> L=SROY_C=E_R=E\n4. L=E_C=SROY_R=E -> L=E_C=E_R=SROY\n5. L=E_C=E_R=SROY -> L=SROY_C=E_R=E\n6. L=E_C=E_R=SROY -> L=E_C=SROY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=LSROY_C=E_R=E.jpg\nGoal state image: L=E_C=LSROY_R=E.jpg\nInitial state: L=LSROY_C=E_R=E\nGoal state: L=E_C=LSROY_R=E\nAnswer the following questions.\n\nReasoning (depth 1): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=SROY_C=E_R=E -> L=E_C=SROY_R=E\n2. L=SROY_C=E_R=E -> L=E_C=E_R=SROY\n3. L=E_C=SROY_R=E -> L=SROY_C=E_R=E\n4. L=E_C=SROY_R=E -> L=E_C=E_R=SROY\n5. L=E_C=E_R=SROY -> L=SROY_C=E_R=E\n6. L=E_C=E_R=SROY -> L=E_C=SROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=SROY_C=E_R=E -> L=E_C=SROY_R=E\n2. L=SROY_C=E_R=E -> L=E_C=E_R=SROY\n3. L=E_C=SROY_R=E -> L=SROY_C=E_R=E\n4. L=E_C=SROY_R=E -> L=E_C=E_R=SROY\n5. L=E_C=E_R=SROY -> L=SROY_C=E_R=E\n6. L=E_C=E_R=SROY -> L=E_C=SROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=LSROY_C=E_R=E__TO__L=E_C=LSROY_R=E__Q2",
          "depth": 2,
          "substack_size": 3,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=ROY_C=E_R=E",
              "to_state": "L=E_C=ROY_R=E",
              "text": "L=ROY_C=E_R=E -> L=E_C=ROY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=ROY_C=E_R=E",
              "to_state": "L=E_C=E_R=ROY",
              "text": "L=ROY_C=E_R=E -> L=E_C=E_R=ROY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=ROY_R=E",
              "to_state": "L=ROY_C=E_R=E",
              "text": "L=E_C=ROY_R=E -> L=ROY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=ROY_R=E",
              "to_state": "L=E_C=E_R=ROY",
              "text": "L=E_C=ROY_R=E -> L=E_C=E_R=ROY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=ROY",
              "to_state": "L=ROY_C=E_R=E",
              "text": "L=E_C=E_R=ROY -> L=ROY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=ROY",
              "to_state": "L=E_C=ROY_R=E",
              "text": "L=E_C=E_R=ROY -> L=E_C=ROY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 1,
          "answer_text": "L=ROY_C=E_R=E -> L=E_C=ROY_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=LSROY_C=E_R=E.jpg\nGoal state image: L=E_C=LSROY_R=E.jpg\nInitial state: L=LSROY_C=E_R=E\nGoal state: L=E_C=LSROY_R=E\nAnswer the following questions.\n\nQuestion (depth 2): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=LSROY_C=E_R=E.jpg\nGoal state image: L=E_C=LSROY_R=E.jpg\nInitial state: L=LSROY_C=E_R=E\nGoal state: L=E_C=LSROY_R=E\nAnswer the following questions.\n\nReasoning (depth 2): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=LSROY_C=E_R=E__TO__L=E_C=LSROY_R=E__Q3",
          "depth": 3,
          "substack_size": 2,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=OY_C=E_R=E -> L=E_C=OY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=OY_C=E_R=E -> L=E_C=E_R=OY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=OY_R=E -> L=OY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=E_C=OY_R=E -> L=E_C=E_R=OY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=E_R=OY -> L=OY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=E_C=E_R=OY -> L=E_C=OY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 2,
          "answer_text": "L=OY_C=E_R=E -> L=E_C=E_R=OY",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=LSROY_C=E_R=E.jpg\nGoal state image: L=E_C=LSROY_R=E.jpg\nInitial state: L=LSROY_C=E_R=E\nGoal state: L=E_C=LSROY_R=E\nAnswer the following questions.\n\nQuestion (depth 3): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=LSROY_C=E_R=E.jpg\nGoal state image: L=E_C=LSROY_R=E.jpg\nInitial state: L=LSROY_C=E_R=E\nGoal state: L=E_C=LSROY_R=E\nAnswer the following questions.\n\nReasoning (depth 3): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=LSROY_C=E_R=E__TO__L=E_C=LSROY_R=E__Q4",
          "depth": 4,
          "substack_size": 1,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=Y_C=E_R=E -> L=E_C=Y_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=Y_C=E_R=E -> L=E_C=E_R=Y",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=Y_R=E -> L=Y_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=E_C=Y_R=E -> L=E_C=E_R=Y",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=E_R=Y -> L=Y_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=E_C=E_R=Y -> L=E_C=Y_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 1,
          "answer_text": "L=Y_C=E_R=E -> L=E_C=Y_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=LSROY_C=E_R=E.jpg\nGoal state image: L=E_C=LSROY_R=E.jpg\nInitial state: L=LSROY_C=E_R=E\nGoal state: L=E_C=LSROY_R=E\nAnswer the following questions.\n\nQuestion (depth 4): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=LSROY_C=E_R=E.jpg\nGoal state image: L=E_C=LSROY_R=E.jpg\nInitial state: L=LSROY_C=E_R=E\nGoal state: L=E_C=LSROY_R=E\nAnswer the following questions.\n\nReasoning (depth 4): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        }
      ],
      "full_problem_prompt": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=LSROY_C=E_R=E.jpg\nGoal state image: L=E_C=LSROY_R=E.jpg\nInitial state: L=LSROY_C=E_R=E\nGoal state: L=E_C=LSROY_R=E\nAnswer the following questions."
    },
    {
      "problem_id": "L=LSROY_C=E_R=E__TO__L=E_C=E_R=LSROY",
      "n_rings": 5,
      "source_pole": "L",
      "target_pole": "R",
      "aux_pole": "C",
      "initial_state": "L=LSROY_C=E_R=E",
      "goal_state": "L=E_C=E_R=LSROY",
      "initial_image": "L=LSROY_C=E_R=E.jpg",
      "goal_image": "L=E_C=E_R=LSROY.jpg",
      "questions": [
        {
          "qid": "L=LSROY_C=E_R=E__TO__L=E_C=E_R=LSROY__Q1",
          "depth": 1,
          "substack_size": 4,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=SROY_C=E_R=E",
              "to_state": "L=E_C=SROY_R=E",
              "text": "L=SROY_C=E_R=E -> L=E_C=SROY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=SROY_C=E_R=E",
              "to_state": "L=E_C=E_R=SROY",
              "text": "L=SROY_C=E_R=E -> L=E_C=E_R=SROY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=SROY_R=E",
              "to_state": "L=SROY_C=E_R=E",
              "text": "L=E_C=SROY_R=E -> L=SROY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=SROY_R=E",
              "to_state": "L=E_C=E_R=SROY",
              "text": "L=E_C=SROY_R=E -> L=E_C=E_R=SROY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=SROY",
              "to_state": "L=SROY_C=E_R=E",
              "text": "L=E_C=E_R=SROY -> L=SROY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=SROY",
              "to_state": "L=E_C=SROY_R=E",
              "text": "L=E_C=E_R=SROY -> L=E_C=SROY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 1,
          "answer_text": "L=SROY_C=E_R=E -> L=E_C=SROY_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=LSROY_C=E_R=E.jpg\nGoal state image: L=E_C=E_R=LSROY.jpg\nInitial state: L=LSROY_C=E_R=E\nGoal state: L=E_C=E_R=LSROY\nAnswer the following questions.\n\nQuestion (depth 1): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=SROY_C=E_R=E -> L=E_C=SROY_R=E\n2. L=SROY_C=E_R=E -> L=E_C=E_R=SROY\n3. L=E_C=SROY_R=E -> L=SROY_C=E_R=E\n4. L=E_C=SROY_R=E -> L=E_C=E_R=SROY\n5. L=E_C=E_R=SROY -> L=SROY_C=E_R=E\n6. L=E_C=E_R=SROY -> L=E_C=SROY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=LSROY_C=E_R=E.jpg\nGoal state image: L=E_C=E_R=LSROY.jpg\nInitial state: L=LSROY_C=E_R=E\nGoal state: L=E_C=E_R=LSROY\nAnswer the following questions.\n\nReasoning (depth 1): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=SROY_C=E_R=E -> L=E_C=SROY_R=E\n2. L=SROY_C=E_R=E -> L=E_C=E_R=SROY\n3. L=E_C=SROY_R=E -> L=SROY_C=E_R=E\n4. L=E_C=SROY_R=E -> L=E_C=E_R=SROY\n5. L=E_C=E_R=SROY -> L=SROY_C=E_R=E\n6. L=E_C=E_R=SROY -> L=E_C=SROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=SROY_C=E_R=E -> L=E_C=SROY_R=E\n2. L=SROY_C=E_R=E -> L=E_C=E_R=SROY\n3. L=E_C=SROY_R=E -> L=SROY_C=E_R=E\n4. L=E_C=SROY_R=E -> L=E_C=E_R=SROY\n5. L=E_C=E_R=SROY -> L=SROY_C=E_R=E\n6. L=E_C=E_R=SROY -> L=E_C=SROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=LSROY_C=E_R=E__TO__L=E_C=E_R=LSROY__Q2",
          "depth": 2,
          "substack_size": 3,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=ROY_C=E_R=E",
              "to_state": "L=E_C=ROY_R=E",
              "text": "L=ROY_C=E_R=E -> L=E_C=ROY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=ROY_C=E_R=E",
              "to_state": "L=E_C=E_R=ROY",
              "text": "L=ROY_C=E_R=E -> L=E_C=E_R=ROY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=ROY_R=E",
              "to_state": "L=ROY_C=E_R=E",
              "text": "L=E_C=ROY_R=E -> L=ROY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=ROY_R=E",
              "to_state": "L=E_C=E_R=ROY",
              "text": "L=E_C=ROY_R=E -> L=E_C=E_R=ROY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=ROY",
              "to_state": "L=ROY_C=E_R=E",
              "text": "L=E_C=E_R=ROY -> L=ROY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=ROY",
              "to_state": "L=E_C=ROY_R=E",
              "text": "L=E_C=E_R=ROY -> L=E_C=ROY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 2,
          "answer_text": "L=ROY_C=E_R=E -> L=E_C=E_R=ROY",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=LSROY_C=E_R=E.jpg\nGoal state image: L=E_C=E_R=LSROY.jpg\nInitial state: L=LSROY_C=E_R=E\nGoal state: L=E_C=E_R=LSROY\nAnswer the following questions.\n\nQuestion (depth 2): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=LSROY_C=E_R=E.jpg\nGoal state image: L=E_C=E_R=LSROY.jpg\nInitial state: L=LSROY_C=E_R=E\nGoal state: L=E_C=E_R=LSROY\nAnswer the following questions.\n\nReasoning (depth 2): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=LSROY_C=E_R=E__TO__L=E_C=E_R=LSROY__Q3",
          "depth": 3,
          "substack_size": 2,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=OY_C=E_R=E -> L=E_C=OY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=OY_C=E_R=E -> L=E_C=E_R=OY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=OY_R=E -> L=OY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=E_C=OY_R=E -> L=E_C=E_R=OY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=E_R=OY -> L=OY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=E_C=E_R=OY -> L=E_C=OY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 1,
          "answer_text": "L=OY_C=E_R=E -> L=E_C=OY_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=LSROY_C=E_R=E.jpg\nGoal state image: L=E_C=E_R=LSROY.jpg\nInitial state: L=LSROY_C=E_R=E\nGoal state: L=E_C=E_R=LSROY\nAnswer the following questions.\n\nQuestion (depth 3): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=LSROY_C=E_R=E.jpg\nGoal state image: L=E_C=E_R=LSROY.jpg\nInitial state: L=LSROY_C=E_R=E\nGoal state: L=E_C=E_R=LSROY\nAnswer the following questions.\n\nReasoning (depth 3): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=LSROY_C=E_R=E__TO__L=E_C=E_R=LSROY__Q4",
          "depth": 4,
          "substack_size": 1,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=Y_C=E_R=E -> L=E_C=Y_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=Y_C=E_R=E -> L=E_C=E_R=Y",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=Y_R=E -> L=Y_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=E_C=Y_R=E -> L=E_C=E_R=Y",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=E_R=Y -> L=Y_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=E_C=E_R=Y -> L=E_C=Y_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 2,
          "answer_text": "L=Y_C=E_R=E -> L=E_C=E_R=Y",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=LSROY_C=E_R=E.jpg\nGoal state image: L=E_C=E_R=LSROY.jpg\nInitial state: L=LSROY_C=E_R=E\nGoal state: L=E_C=E_R=LSROY\nAnswer the following questions.\n\nQuestion (depth 4): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=LSROY_C=E_R=E.jpg\nGoal state image: L=E_C=E_R=LSROY.jpg\nInitial state: L=LSROY_C=E_R=E\nGoal state: L=E_C=E_R=LSROY\nAnswer the following questions.\n\nReasoning (depth 4): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        }
      ],
      "full_problem_prompt": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=LSROY_C=E_R=E.jpg\nGoal state image: L=E_C=E_R=LSROY.jpg\nInitial state: L=LSROY_C=E_R=E\nGoal state: L=E_C=E_R=LSROY\nAnswer the following questions."
    },
    {
      "problem_id": "L=E_C=LSROY_R=E__TO__L=LSROY_C=E_R=E",
      "n_rings": 5,
      "source_pole": "C",
      "target_pole": "L",
      "aux_pole": "R",
      "initial_state": "L=E_C=LSROY_R=E",
      "goal_state": "L=LSROY_C=E_R=E",
      "initial_image": "L=E_C=LSROY_R=E.jpg",
      "goal_image": "L=LSROY_C=E_R=E.jpg",
      "questions": [
        {
          "qid": "L=E_C=LSROY_R=E__TO__L=LSROY_C=E_R=E__Q1",
          "depth": 1,
          "substack_size": 4,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=SROY_C=E_R=E",
              "to_state": "L=E_C=SROY_R=E",
              "text": "L=SROY_C=E_R=E -> L=E_C=SROY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=SROY_C=E_R=E",
              "to_state": "L=E_C=E_R=SROY",
              "text": "L=SROY_C=E_R=E -> L=E_C=E_R=SROY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=SROY_R=E",
              "to_state": "L=SROY_C=E_R=E",
              "text": "L=E_C=SROY_R=E -> L=SROY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=SROY_R=E",
              "to_state": "L=E_C=E_R=SROY",
              "text": "L=E_C=SROY_R=E -> L=E_C=E_R=SROY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=SROY",
              "to_state": "L=SROY_C=E_R=E",
              "text": "L=E_C=E_R=SROY -> L=SROY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=SROY",
              "to_state": "L=E_C=SROY_R=E",
              "text": "L=E_C=E_R=SROY -> L=E_C=SROY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 4,
          "answer_text": "L=E_C=SROY_R=E -> L=E_C=E_R=SROY",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=LSROY_R=E.jpg\nGoal state image: L=LSROY_C=E_R=E.jpg\nInitial state: L=E_C=LSROY_R=E\nGoal state: L=LSROY_C=E_R=E\nAnswer the following questions.\n\nQuestion (depth 1): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=SROY_C=E_R=E -> L=E_C=SROY_R=E\n2. L=SROY_C=E_R=E -> L=E_C=E_R=SROY\n3. L=E_C=SROY_R=E -> L=SROY_C=E_R=E\n4. L=E_C=SROY_R=E -> L=E_C=E_R=SROY\n5. L=E_C=E_R=SROY -> L=SROY_C=E_R=E\n6. L=E_C=E_R=SROY -> L=E_C=SROY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=LSROY_R=E.jpg\nGoal state image: L=LSROY_C=E_R=E.jpg\nInitial state: L=E_C=LSROY_R=E\nGoal state: L=LSROY_C=E_R=E\nAnswer the following questions.\n\nReasoning (depth 1): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=SROY_C=E_R=E -> L=E_C=SROY_R=E\n2. L=SROY_C=E_R=E -> L=E_C=E_R=SROY\n3. L=E_C=SROY_R=E -> L=SROY_C=E_R=E\n4. L=E_C=SROY_R=E -> L=E_C=E_R=SROY\n5. L=E_C=E_R=SROY -> L=SROY_C=E_R=E\n6. L=E_C=E_R=SROY -> L=E_C=SROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=SROY_C=E_R=E -> L=E_C=SROY_R=E\n2. L=SROY_C=E_R=E -> L=E_C=E_R=SROY\n3. L=E_C=SROY_R=E -> L=SROY_C=E_R=E\n4. L=E_C=SROY_R=E -> L=E_C=E_R=SROY\n5. L=E_C=E_R=SROY -> L=SROY_C=E_R=E\n6. L=E_C=E_R=SROY -> L=E_C=SROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=LSROY_R=E__TO__L=LSROY_C=E_R=E__Q2",
          "depth": 2,
          "substack_size": 3,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=ROY_C=E_R=E",
              "to_state": "L=E_C=ROY_R=E",
              "text": "L=ROY_C=E_R=E -> L=E_C=ROY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=ROY_C=E_R=E",
              "to_state": "L=E_C=E_R=ROY",
              "text": "L=ROY_C=E_R=E -> L=E_C=E_R=ROY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=ROY_R=E",
              "to_state": "L=ROY_C=E_R=E",
              "text": "L=E_C=ROY_R=E -> L=ROY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=ROY_R=E",
              "to_state": "L=E_C=E_R=ROY",
              "text": "L=E_C=ROY_R=E -> L=E_C=E_R=ROY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=ROY",
              "to_state": "L=ROY_C=E_R=E",
              "text": "L=E_C=E_R=ROY -> L=ROY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=ROY",
              "to_state": "L=E_C=ROY_R=E",
              "text": "L=E_C=E_R=ROY -> L=E_C=ROY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 3,
          "answer_text": "L=E_C=ROY_R=E -> L=ROY_C=E_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=LSROY_R=E.jpg\nGoal state image: L=LSROY_C=E_R=E.jpg\nInitial state: L=E_C=LSROY_R=E\nGoal state: L=LSROY_C=E_R=E\nAnswer the following questions.\n\nQuestion (depth 2): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=LSROY_R=E.jpg\nGoal state image: L=LSROY_C=E_R=E.jpg\nInitial state: L=E_C=LSROY_R=E\nGoal state: L=LSROY_C=E_R=E\nAnswer the following questions.\n\nReasoning (depth 2): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=LSROY_R=E__TO__L=LSROY_C=E_R=E__Q3",
          "depth": 3,
          "substack_size": 2,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=OY_C=E_R=E -> L=E_C=OY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=OY_C=E_R=E -> L=E_C=E_R=OY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=OY_R=E -> L=OY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=E_C=OY_R=E -> L=E_C=E_R=OY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=E_R=OY -> L=OY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=E_C=E_R=OY -> L=E_C=OY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 4,
          "answer_text": "L=E_C=OY_R=E -> L=E_C=E_R=OY",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=LSROY_R=E.jpg\nGoal state image: L=LSROY_C=E_R=E.jpg\nInitial state: L=E_C=LSROY_R=E\nGoal state: L=LSROY_C=E_R=E\nAnswer the following questions.\n\nQuestion (depth 3): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=LSROY_R=E.jpg\nGoal state image: L=LSROY_C=E_R=E.jpg\nInitial state: L=E_C=LSROY_R=E\nGoal state: L=LSROY_C=E_R=E\nAnswer the following questions.\n\nReasoning (depth 3): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=LSROY_R=E__TO__L=LSROY_C=E_R=E__Q4",
          "depth": 4,
          "substack_size": 1,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=Y_C=E_R=E -> L=E_C=Y_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=Y_C=E_R=E -> L=E_C=E_R=Y",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=Y_R=E -> L=Y_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=E_C=Y_R=E -> L=E_C=E_R=Y",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=E_R=Y -> L=Y_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=E_C=E_R=Y -> L=E_C=Y_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 3,
          "answer_text": "L=E_C=Y_R=E -> L=Y_C=E_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=LSROY_R=E.jpg\nGoal state image: L=LSROY_C=E_R=E.jpg\nInitial state: L=E_C=LSROY_R=E\nGoal state: L=LSROY_C=E_R=E\nAnswer the following questions.\n\nQuestion (depth 4): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=LSROY_R=E.jpg\nGoal state image: L=LSROY_C=E_R=E.jpg\nInitial state: L=E_C=LSROY_R=E\nGoal state: L=LSROY_C=E_R=E\nAnswer the following questions.\n\nReasoning (depth 4): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        }
      ],
      "full_problem_prompt": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=LSROY_R=E.jpg\nGoal state image: L=LSROY_C=E_R=E.jpg\nInitial state: L=E_C=LSROY_R=E\nGoal state: L=LSROY_C=E_R=E\nAnswer the following questions."
    },
    {
      "problem_id": "L=E_C=LSROY_R=E__TO__L=E_C=E_R=LSROY",
      "n_rings": 5,
      "source_pole": "C",
      "target_pole": "R",
      "aux_pole": "L",
      "initial_state": "L=E_C=LSROY_R=E",
      "goal_state": "L=E_C=E_R=LSROY",
      "initial_image": "L=E_C=LSROY_R=E.jpg",
      "goal_image": "L=E_C=E_R=LSROY.jpg",
      "questions": [
        {
          "qid": "L=E_C=LSROY_R=E__TO__L=E_C=E_R=LSROY__Q1",
          "depth": 1,
          "substack_size": 4,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=SROY_C=E_R=E",
              "to_state": "L=E_C=SROY_R=E",
              "text": "L=SROY_C=E_R=E -> L=E_C=SROY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=SROY_C=E_R=E",
              "to_state": "L=E_C=E_R=SROY",
              "text": "L=SROY_C=E_R=E -> L=E_C=E_R=SROY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=SROY_R=E",
              "to_state": "L=SROY_C=E_R=E",
              "text": "L=E_C=SROY_R=E -> L=SROY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=SROY_R=E",
              "to_state": "L=E_C=E_R=SROY",
              "text": "L=E_C=SROY_R=E -> L=E_C=E_R=SROY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=SROY",
              "to_state": "L=SROY_C=E_R=E",
              "text": "L=E_C=E_R=SROY -> L=SROY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=SROY",
              "to_state": "L=E_C=SROY_R=E",
              "text": "L=E_C=E_R=SROY -> L=E_C=SROY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 3,
          "answer_text": "L=E_C=SROY_R=E -> L=SROY_C=E_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=LSROY_R=E.jpg\nGoal state image: L=E_C=E_R=LSROY.jpg\nInitial state: L=E_C=LSROY_R=E\nGoal state: L=E_C=E_R=LSROY\nAnswer the following questions.\n\nQuestion (depth 1): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=SROY_C=E_R=E -> L=E_C=SROY_R=E\n2. L=SROY_C=E_R=E -> L=E_C=E_R=SROY\n3. L=E_C=SROY_R=E -> L=SROY_C=E_R=E\n4. L=E_C=SROY_R=E -> L=E_C=E_R=SROY\n5. L=E_C=E_R=SROY -> L=SROY_C=E_R=E\n6. L=E_C=E_R=SROY -> L=E_C=SROY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=LSROY_R=E.jpg\nGoal state image: L=E_C=E_R=LSROY.jpg\nInitial state: L=E_C=LSROY_R=E\nGoal state: L=E_C=E_R=LSROY\nAnswer the following questions.\n\nReasoning (depth 1): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=SROY_C=E_R=E -> L=E_C=SROY_R=E\n2. L=SROY_C=E_R=E -> L=E_C=E_R=SROY\n3. L=E_C=SROY_R=E -> L=SROY_C=E_R=E\n4. L=E_C=SROY_R=E -> L=E_C=E_R=SROY\n5. L=E_C=E_R=SROY -> L=SROY_C=E_R=E\n6. L=E_C=E_R=SROY -> L=E_C=SROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=SROY_C=E_R=E -> L=E_C=SROY_R=E\n2. L=SROY_C=E_R=E -> L=E_C=E_R=SROY\n3. L=E_C=SROY_R=E -> L=SROY_C=E_R=E\n4. L=E_C=SROY_R=E -> L=E_C=E_R=SROY\n5. L=E_C=E_R=SROY -> L=SROY_C=E_R=E\n6. L=E_C=E_R=SROY -> L=E_C=SROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=LSROY_R=E__TO__L=E_C=E_R=LSROY__Q2",
          "depth": 2,
          "substack_size": 3,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=ROY_C=E_R=E",
              "to_state": "L=E_C=ROY_R=E",
              "text": "L=ROY_C=E_R=E -> L=E_C=ROY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=ROY_C=E_R=E",
              "to_state": "L=E_C=E_R=ROY",
              "text": "L=ROY_C=E_R=E -> L=E_C=E_R=ROY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=ROY_R=E",
              "to_state": "L=ROY_C=E_R=E",
              "text": "L=E_C=ROY_R=E -> L=ROY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=ROY_R=E",
              "to_state": "L=E_C=E_R=ROY",
              "text": "L=E_C=ROY_R=E -> L=E_C=E_R=ROY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=ROY",
              "to_state": "L=ROY_C=E_R=E",
              "text": "L=E_C=E_R=ROY -> L=ROY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=ROY",
              "to_state": "L=E_C=ROY_R=E",
              "text": "L=E_C=E_R=ROY -> L=E_C=ROY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 4,
          "answer_text": "L=E_C=ROY_R=E -> L=E_C=E_R=ROY",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=LSROY_R=E.jpg\nGoal state image: L=E_C=E_R=LSROY.jpg\nInitial state: L=E_C=LSROY_R=E\nGoal state: L=E_C=E_R=LSROY\nAnswer the following questions.\n\nQuestion (depth 2): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=LSROY_R=E.jpg\nGoal state image: L=E_C=E_R=LSROY.jpg\nInitial state: L=E_C=LSROY_R=E\nGoal state: L=E_C=E_R=LSROY\nAnswer the following questions.\n\nReasoning (depth 2): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=LSROY_R=E__TO__L=E_C=E_R=LSROY__Q3",
          "depth": 3,
          "substack_size": 2,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=OY_C=E_R=E -> L=E_C=OY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=OY_C=E_R=E -> L=E_C=E_R=OY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=OY_R=E -> L=OY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=E_C=OY_R=E -> L=E_C=E_R=OY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=E_R=OY -> L=OY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=E_C=E_R=OY -> L=E_C=OY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 3,
          "answer_text": "L=E_C=OY_R=E -> L=OY_C=E_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=LSROY_R=E.jpg\nGoal state image: L=E_C=E_R=LSROY.jpg\nInitial state: L=E_C=LSROY_R=E\nGoal state: L=E_C=E_R=LSROY\nAnswer the following questions.\n\nQuestion (depth 3): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=LSROY_R=E.jpg\nGoal state image: L=E_C=E_R=LSROY.jpg\nInitial state: L=E_C=LSROY_R=E\nGoal state: L=E_C=E_R=LSROY\nAnswer the following questions.\n\nReasoning (depth 3): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=LSROY_R=E__TO__L=E_C=E_R=LSROY__Q4",
          "depth": 4,
          "substack_size": 1,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=Y_C=E_R=E -> L=E_C=Y_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=Y_C=E_R=E -> L=E_C=E_R=Y",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=Y_R=E -> L=Y_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=E_C=Y_R=E -> L=E_C=E_R=Y",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=E_R=Y -> L=Y_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=E_C=E_R=Y -> L=E_C=Y_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 4,
          "answer_text": "L=E_C=Y_R=E -> L=E_C=E_R=Y",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=LSROY_R=E.jpg\nGoal state image: L=E_C=E_R=LSROY.jpg\nInitial state: L=E_C=LSROY_R=E\nGoal state: L=E_C=E_R=LSROY\nAnswer the following questions.\n\nQuestion (depth 4): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=LSROY_R=E.jpg\nGoal state image: L=E_C=E_R=LSROY.jpg\nInitial state: L=E_C=LSROY_R=E\nGoal state: L=E_C=E_R=LSROY\nAnswer the following questions.\n\nReasoning (depth 4): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        }
      ],
      "full_problem_prompt": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=LSROY_R=E.jpg\nGoal state image: L=E_C=E_R=LSROY.jpg\nInitial state: L=E_C=LSROY_R=E\nGoal state: L=E_C=E_R=LSROY\nAnswer the following questions."
    },
    {
      "problem_id": "L=E_C=E_R=LSROY__TO__L=LSROY_C=E_R=E",
      "n_rings": 5,
      "source_pole": "R",
      "target_pole": "L",
      "aux_pole": "C",
      "initial_state": "L=E_C=E_R=LSROY",
      "goal_state": "L=LSROY_C=E_R=E",
      "initial_image": "L=E_C=E_R=LSROY.jpg",
      "goal_image": "L=LSROY_C=E_R=E.jpg",
      "questions": [
        {
          "qid": "L=E_C=E_R=LSROY__TO__L=LSROY_C=E_R=E__Q1",
          "depth": 1,
          "substack_size": 4,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=SROY_C=E_R=E",
              "to_state": "L=E_C=SROY_R=E",
              "text": "L=SROY_C=E_R=E -> L=E_C=SROY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=SROY_C=E_R=E",
              "to_state": "L=E_C=E_R=SROY",
              "text": "L=SROY_C=E_R=E -> L=E_C=E_R=SROY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=SROY_R=E",
              "to_state": "L=SROY_C=E_R=E",
              "text": "L=E_C=SROY_R=E -> L=SROY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=SROY_R=E",
              "to_state": "L=E_C=E_R=SROY",
              "text": "L=E_C=SROY_R=E -> L=E_C=E_R=SROY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=SROY",
              "to_state": "L=SROY_C=E_R=E",
              "text": "L=E_C=E_R=SROY -> L=SROY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=SROY",
              "to_state": "L=E_C=SROY_R=E",
              "text": "L=E_C=E_R=SROY -> L=E_C=SROY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 6,
          "answer_text": "L=E_C=E_R=SROY -> L=E_C=SROY_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=LSROY.jpg\nGoal state image: L=LSROY_C=E_R=E.jpg\nInitial state: L=E_C=E_R=LSROY\nGoal state: L=LSROY_C=E_R=E\nAnswer the following questions.\n\nQuestion (depth 1): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=SROY_C=E_R=E -> L=E_C=SROY_R=E\n2. L=SROY_C=E_R=E -> L=E_C=E_R=SROY\n3. L=E_C=SROY_R=E -> L=SROY_C=E_R=E\n4. L=E_C=SROY_R=E -> L=E_C=E_R=SROY\n5. L=E_C=E_R=SROY -> L=SROY_C=E_R=E\n6. L=E_C=E_R=SROY -> L=E_C=SROY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=LSROY.jpg\nGoal state image: L=LSROY_C=E_R=E.jpg\nInitial state: L=E_C=E_R=LSROY\nGoal state: L=LSROY_C=E_R=E\nAnswer the following questions.\n\nReasoning (depth 1): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=SROY_C=E_R=E -> L=E_C=SROY_R=E\n2. L=SROY_C=E_R=E -> L=E_C=E_R=SROY\n3. L=E_C=SROY_R=E -> L=SROY_C=E_R=E\n4. L=E_C=SROY_R=E -> L=E_C=E_R=SROY\n5. L=E_C=E_R=SROY -> L=SROY_C=E_R=E\n6. L=E_C=E_R=SROY -> L=E_C=SROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=SROY_C=E_R=E -> L=E_C=SROY_R=E\n2. L=SROY_C=E_R=E -> L=E_C=E_R=SROY\n3. L=E_C=SROY_R=E -> L=SROY_C=E_R=E\n4. L=E_C=SROY_R=E -> L=E_C=E_R=SROY\n5. L=E_C=E_R=SROY -> L=SROY_C=E_R=E\n6. L=E_C=E_R=SROY -> L=E_C=SROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=E_R=LSROY__TO__L=LSROY_C=E_R=E__Q2",
          "depth": 2,
          "substack_size": 3,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=ROY_C=E_R=E",
              "to_state": "L=E_C=ROY_R=E",
              "text": "L=ROY_C=E_R=E -> L=E_C=ROY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=ROY_C=E_R=E",
              "to_state": "L=E_C=E_R=ROY",
              "text": "L=ROY_C=E_R=E -> L=E_C=E_R=ROY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=ROY_R=E",
              "to_state": "L=ROY_C=E_R=E",
              "text": "L=E_C=ROY_R=E -> L=ROY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=ROY_R=E",
              "to_state": "L=E_C=E_R=ROY",
              "text": "L=E_C=ROY_R=E -> L=E_C=E_R=ROY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=ROY",
              "to_state": "L=ROY_C=E_R=E",
              "text": "L=E_C=E_R=ROY -> L=ROY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=ROY",
              "to_state": "L=E_C=ROY_R=E",
              "text": "L=E_C=E_R=ROY -> L=E_C=ROY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 5,
          "answer_text": "L=E_C=E_R=ROY -> L=ROY_C=E_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=LSROY.jpg\nGoal state image: L=LSROY_C=E_R=E.jpg\nInitial state: L=E_C=E_R=LSROY\nGoal state: L=LSROY_C=E_R=E\nAnswer the following questions.\n\nQuestion (depth 2): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=LSROY.jpg\nGoal state image: L=LSROY_C=E_R=E.jpg\nInitial state: L=E_C=E_R=LSROY\nGoal state: L=LSROY_C=E_R=E\nAnswer the following questions.\n\nReasoning (depth 2): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=E_R=LSROY__TO__L=LSROY_C=E_R=E__Q3",
          "depth": 3,
          "substack_size": 2,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=OY_C=E_R=E -> L=E_C=OY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=OY_C=E_R=E -> L=E_C=E_R=OY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=OY_R=E -> L=OY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=E_C=OY_R=E -> L=E_C=E_R=OY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=E_R=OY -> L=OY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=E_C=E_R=OY -> L=E_C=OY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 6,
          "answer_text": "L=E_C=E_R=OY -> L=E_C=OY_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=LSROY.jpg\nGoal state image: L=LSROY_C=E_R=E.jpg\nInitial state: L=E_C=E_R=LSROY\nGoal state: L=LSROY_C=E_R=E\nAnswer the following questions.\n\nQuestion (depth 3): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=LSROY.jpg\nGoal state image: L=LSROY_C=E_R=E.jpg\nInitial state: L=E_C=E_R=LSROY\nGoal state: L=LSROY_C=E_R=E\nAnswer the following questions.\n\nReasoning (depth 3): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=E_R=LSROY__TO__L=LSROY_C=E_R=E__Q4",
          "depth": 4,
          "substack_size": 1,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=Y_C=E_R=E -> L=E_C=Y_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=Y_C=E_R=E -> L=E_C=E_R=Y",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=Y_R=E -> L=Y_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=E_C=Y_R=E -> L=E_C=E_R=Y",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=E_R=Y -> L=Y_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=E_C=E_R=Y -> L=E_C=Y_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 5,
          "answer_text": "L=E_C=E_R=Y -> L=Y_C=E_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=LSROY.jpg\nGoal state image: L=LSROY_C=E_R=E.jpg\nInitial state: L=E_C=E_R=LSROY\nGoal state: L=LSROY_C=E_R=E\nAnswer the following questions.\n\nQuestion (depth 4): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=LSROY.jpg\nGoal state image: L=LSROY_C=E_R=E.jpg\nInitial state: L=E_C=E_R=LSROY\nGoal state: L=LSROY_C=E_R=E\nAnswer the following questions.\n\nReasoning (depth 4): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        }
      ],
      "full_problem_prompt": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=LSROY.jpg\nGoal state image: L=LSROY_C=E_R=E.jpg\nInitial state: L=E_C=E_R=LSROY\nGoal state: L=LSROY_C=E_R=E\nAnswer the following questions."
    },
    {
      "problem_id": "L=E_C=E_R=LSROY__TO__L=E_C=LSROY_R=E",
      "n_rings": 5,
      "source_pole": "R",
      "target_pole": "C",
      "aux_pole": "L",
      "initial_state": "L=E_C=E_R=LSROY",
      "goal_state": "L=E_C=LSROY_R=E",
      "initial_image": "L=E_C=E_R=LSROY.jpg",
      "goal_image": "L=E_C=LSROY_R=E.jpg",
      "questions": [
        {
          "qid": "L=E_C=E_R=LSROY__TO__L=E_C=LSROY_R=E__Q1",
          "depth": 1,
          "substack_size": 4,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=SROY_C=E_R=E",
              "to_state": "L=E_C=SROY_R=E",
              "text": "L=SROY_C=E_R=E -> L=E_C=SROY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=SROY_C=E_R=E",
              "to_state": "L=E_C=E_R=SROY",
              "text": "L=SROY_C=E_R=E -> L=E_C=E_R=SROY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=SROY_R=E",
              "to_state": "L=SROY_C=E_R=E",
              "text": "L=E_C=SROY_R=E -> L=SROY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=SROY_R=E",
              "to_state": "L=E_C=E_R=SROY",
              "text": "L=E_C=SROY_R=E -> L=E_C=E_R=SROY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=SROY",
              "to_state": "L=SROY_C=E_R=E",
              "text": "L=E_C=E_R=SROY -> L=SROY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=SROY",
              "to_state": "L=E_C=SROY_R=E",
              "text": "L=E_C=E_R=SROY -> L=E_C=SROY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 5,
          "answer_text": "L=E_C=E_R=SROY -> L=SROY_C=E_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=LSROY.jpg\nGoal state image: L=E_C=LSROY_R=E.jpg\nInitial state: L=E_C=E_R=LSROY\nGoal state: L=E_C=LSROY_R=E\nAnswer the following questions.\n\nQuestion (depth 1): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=SROY_C=E_R=E -> L=E_C=SROY_R=E\n2. L=SROY_C=E_R=E -> L=E_C=E_R=SROY\n3. L=E_C=SROY_R=E -> L=SROY_C=E_R=E\n4. L=E_C=SROY_R=E -> L=E_C=E_R=SROY\n5. L=E_C=E_R=SROY -> L=SROY_C=E_R=E\n6. L=E_C=E_R=SROY -> L=E_C=SROY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=LSROY.jpg\nGoal state image: L=E_C=LSROY_R=E.jpg\nInitial state: L=E_C=E_R=LSROY\nGoal state: L=E_C=LSROY_R=E\nAnswer the following questions.\n\nReasoning (depth 1): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=SROY_C=E_R=E -> L=E_C=SROY_R=E\n2. L=SROY_C=E_R=E -> L=E_C=E_R=SROY\n3. L=E_C=SROY_R=E -> L=SROY_C=E_R=E\n4. L=E_C=SROY_R=E -> L=E_C=E_R=SROY\n5. L=E_C=E_R=SROY -> L=SROY_C=E_R=E\n6. L=E_C=E_R=SROY -> L=E_C=SROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=SROY_C=E_R=E -> L=E_C=SROY_R=E\n2. L=SROY_C=E_R=E -> L=E_C=E_R=SROY\n3. L=E_C=SROY_R=E -> L=SROY_C=E_R=E\n4. L=E_C=SROY_R=E -> L=E_C=E_R=SROY\n5. L=E_C=E_R=SROY -> L=SROY_C=E_R=E\n6. L=E_C=E_R=SROY -> L=E_C=SROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=E_R=LSROY__TO__L=E_C=LSROY_R=E__Q2",
          "depth": 2,
          "substack_size": 3,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=ROY_C=E_R=E",
              "to_state": "L=E_C=ROY_R=E",
              "text": "L=ROY_C=E_R=E -> L=E_C=ROY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=ROY_C=E_R=E",
              "to_state": "L=E_C=E_R=ROY",
              "text": "L=ROY_C=E_R=E -> L=E_C=E_R=ROY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=ROY_R=E",
              "to_state": "L=ROY_C=E_R=E",
              "text": "L=E_C=ROY_R=E -> L=ROY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=ROY_R=E",
              "to_state": "L=E_C=E_R=ROY",
              "text": "L=E_C=ROY_R=E -> L=E_C=E_R=ROY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=ROY",
              "to_state": "L=ROY_C=E_R=E",
              "text": "L=E_C=E_R=ROY -> L=ROY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=ROY",
              "to_state": "L=E_C=ROY_R=E",
              "text": "L=E_C=E_R=ROY -> L=E_C=ROY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 6,
          "answer_text": "L=E_C=E_R=ROY -> L=E_C=ROY_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=LSROY.jpg\nGoal state image: L=E_C=LSROY_R=E.jpg\nInitial state: L=E_C=E_R=LSROY\nGoal state: L=E_C=LSROY_R=E\nAnswer the following questions.\n\nQuestion (depth 2): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=LSROY.jpg\nGoal state image: L=E_C=LSROY_R=E.jpg\nInitial state: L=E_C=E_R=LSROY\nGoal state: L=E_C=LSROY_R=E\nAnswer the following questions.\n\nReasoning (depth 2): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=E_R=LSROY__TO__L=E_C=LSROY_R=E__Q3",
          "depth": 3,
          "substack_size": 2,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=OY_C=E_R=E -> L=E_C=OY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=OY_C=E_R=E -> L=E_C=E_R=OY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=OY_R=E -> L=OY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=E_C=OY_R=E -> L=E_C=E_R=OY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=E_R=OY -> L=OY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=E_C=E_R=OY -> L=E_C=OY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 5,
          "answer_text": "L=E_C=E_R=OY -> L=OY_C=E_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=LSROY.jpg\nGoal state image: L=E_C=LSROY_R=E.jpg\nInitial state: L=E_C=E_R=LSROY\nGoal state: L=E_C=LSROY_R=E\nAnswer the following questions.\n\nQuestion (depth 3): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=LSROY.jpg\nGoal state image: L=E_C=LSROY_R=E.jpg\nInitial state: L=E_C=E_R=LSROY\nGoal state: L=E_C=LSROY_R=E\nAnswer the following questions.\n\nReasoning (depth 3): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=E_R=LSROY__TO__L=E_C=LSROY_R=E__Q4",
          "depth": 4,
          "substack_size": 1,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=Y_C=E_R=E -> L=E_C=Y_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=Y_C=E_R=E -> L=E_C=E_R=Y",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=Y_R=E -> L=Y_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=E_C=Y_R=E -> L=E_C=E_R=Y",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=E_R=Y -> L=Y_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=E_C=E_R=Y -> L=E_C=Y_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 6,
          "answer_text": "L=E_C=E_R=Y -> L=E_C=Y_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=LSROY.jpg\nGoal state image: L=E_C=LSROY_R=E.jpg\nInitial state: L=E_C=E_R=LSROY\nGoal state: L=E_C=LSROY_R=E\nAnswer the following questions.\n\nQuestion (depth 4): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=LSROY.jpg\nGoal state image: L=E_C=LSROY_R=E.jpg\nInitial state: L=E_C=E_R=LSROY\nGoal state: L=E_C=LSROY_R=E\nAnswer the following questions.\n\nReasoning (depth 4): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        }
      ],
      "full_problem_prompt": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=LSROY.jpg\nGoal state image: L=E_C=LSROY_R=E.jpg\nInitial state: L=E_C=E_R=LSROY\nGoal state: L=E_C=LSROY_R=E\nAnswer the following questions."
    },
    {
      "problem_id": "L=GLSROY_C=E_R=E__TO__L=E_C=GLSROY_R=E",
      "n_rings": 6,
      "source_pole": "L",
      "target_pole": "C",
      "aux_pole": "R",
      "initial_state": "L=GLSROY_C=E_R=E",
      "goal_state": "L=E_C=GLSROY_R=E",
      "initial_image": "L=GLSROY_C=E_R=E.jpg",
      "goal_image": "L=E_C=GLSROY_R=E.jpg",
      "questions": [
        {
          "qid": "L=GLSROY_C=E_R=E__TO__L=E_C=GLSROY_R=E__Q1",
          "depth": 1,
          "substack_size": 5,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=LSROY_C=E_R=E",
              "to_state": "L=E_C=LSROY_R=E",
              "text": "L=LSROY_C=E_R=E -> L=E_C=LSROY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=LSROY_C=E_R=E",
              "to_state": "L=E_C=E_R=LSROY",
              "text": "L=LSROY_C=E_R=E -> L=E_C=E_R=LSROY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=LSROY_R=E",
              "to_state": "L=LSROY_C=E_R=E",
              "text": "L=E_C=LSROY_R=E -> L=LSROY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=LSROY_R=E",
              "to_state": "L=E_C=E_R=LSROY",
              "text": "L=E_C=LSROY_R=E -> L=E_C=E_R=LSROY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=LSROY",
              "to_state": "L=LSROY_C=E_R=E",
              "text": "L=E_C=E_R=LSROY -> L=LSROY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=LSROY",
              "to_state": "L=E_C=LSROY_R=E",
              "text": "L=E_C=E_R=LSROY -> L=E_C=LSROY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 2,
          "answer_text": "L=LSROY_C=E_R=E -> L=E_C=E_R=LSROY",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=GLSROY_C=E_R=E.jpg\nGoal state image: L=E_C=GLSROY_R=E.jpg\nInitial state: L=GLSROY_C=E_R=E\nGoal state: L=E_C=GLSROY_R=E\nAnswer the following questions.\n\nQuestion (depth 1): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=LSROY_C=E_R=E -> L=E_C=LSROY_R=E\n2. L=LSROY_C=E_R=E -> L=E_C=E_R=LSROY\n3. L=E_C=LSROY_R=E -> L=LSROY_C=E_R=E\n4. L=E_C=LSROY_R=E -> L=E_C=E_R=LSROY\n5. L=E_C=E_R=LSROY -> L=LSROY_C=E_R=E\n6. L=E_C=E_R=LSROY -> L=E_C=LSROY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=GLSROY_C=E_R=E.jpg\nGoal state image: L=E_C=GLSROY_R=E.jpg\nInitial state: L=GLSROY_C=E_R=E\nGoal state: L=E_C=GLSROY_R=E\nAnswer the following questions.\n\nReasoning (depth 1): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=LSROY_C=E_R=E -> L=E_C=LSROY_R=E\n2. L=LSROY_C=E_R=E -> L=E_C=E_R=LSROY\n3. L=E_C=LSROY_R=E -> L=LSROY_C=E_R=E\n4. L=E_C=LSROY_R=E -> L=E_C=E_R=LSROY\n5. L=E_C=E_R=LSROY -> L=LSROY_C=E_R=E\n6. L=E_C=E_R=LSROY -> L=E_C=LSROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=LSROY_C=E_R=E -> L=E_C=LSROY_R=E\n2. L=LSROY_C=E_R=E -> L=E_C=E_R=LSROY\n3. L=E_C=LSROY_R=E -> L=LSROY_C=E_R=E\n4. L=E_C=LSROY_R=E -> L=E_C=E_R=LSROY\n5. L=E_C=E_R=LSROY -> L=LSROY_C=E_R=E\n6. L=E_C=E_R=LSROY -> L=E_C=LSROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=GLSROY_C=E_R=E__TO__L=E_C=GLSROY_R=E__Q2",
          "depth": 2,
          "substack_size": 4,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=SROY_C=E_R=E",
              "to_state": "L=E_C=SROY_R=E",
              "text": "L=SROY_C=E_R=E -> L=E_C=SROY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=SROY_C=E_R=E",
              "to_state": "L=E_C=E_R=SROY",
              "text": "L=SROY_C=E_R=E -> L=E_C=E_R=SROY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=SROY_R=E",
              "to_state": "L=SROY_C=E_R=E",
              "text": "L=E_C=SROY_R=E -> L=SROY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=SROY_R=E",
              "to_state": "L=E_C=E_R=SROY",
              "text": "L=E_C=SROY_R=E -> L=E_C=E_R=SROY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=SROY",
              "to_state": "L=SROY_C=E_R=E",
              "text": "L=E_C=E_R=SROY -> L=SROY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=SROY",
              "to_state": "L=E_C=SROY_R=E",
              "text": "L=E_C=E_R=SROY -> L=E_C=SROY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 1,
          "answer_text": "L=SROY_C=E_R=E -> L=E_C=SROY_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=GLSROY_C=E_R=E.jpg\nGoal state image: L=E_C=GLSROY_R=E.jpg\nInitial state: L=GLSROY_C=E_R=E\nGoal state: L=E_C=GLSROY_R=E\nAnswer the following questions.\n\nQuestion (depth 2): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=SROY_C=E_R=E -> L=E_C=SROY_R=E\n2. L=SROY_C=E_R=E -> L=E_C=E_R=SROY\n3. L=E_C=SROY_R=E -> L=SROY_C=E_R=E\n4. L=E_C=SROY_R=E -> L=E_C=E_R=SROY\n5. L=E_C=E_R=SROY -> L=SROY_C=E_R=E\n6. L=E_C=E_R=SROY -> L=E_C=SROY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=GLSROY_C=E_R=E.jpg\nGoal state image: L=E_C=GLSROY_R=E.jpg\nInitial state: L=GLSROY_C=E_R=E\nGoal state: L=E_C=GLSROY_R=E\nAnswer the following questions.\n\nReasoning (depth 2): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=SROY_C=E_R=E -> L=E_C=SROY_R=E\n2. L=SROY_C=E_R=E -> L=E_C=E_R=SROY\n3. L=E_C=SROY_R=E -> L=SROY_C=E_R=E\n4. L=E_C=SROY_R=E -> L=E_C=E_R=SROY\n5. L=E_C=E_R=SROY -> L=SROY_C=E_R=E\n6. L=E_C=E_R=SROY -> L=E_C=SROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=SROY_C=E_R=E -> L=E_C=SROY_R=E\n2. L=SROY_C=E_R=E -> L=E_C=E_R=SROY\n3. L=E_C=SROY_R=E -> L=SROY_C=E_R=E\n4. L=E_C=SROY_R=E -> L=E_C=E_R=SROY\n5. L=E_C=E_R=SROY -> L=SROY_C=E_R=E\n6. L=E_C=E_R=SROY -> L=E_C=SROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=GLSROY_C=E_R=E__TO__L=E_C=GLSROY_R=E__Q3",
          "depth": 3,
          "substack_size": 3,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=ROY_C=E_R=E",
              "to_state": "L=E_C=ROY_R=E",
              "text": "L=ROY_C=E_R=E -> L=E_C=ROY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=ROY_C=E_R=E",
              "to_state": "L=E_C=E_R=ROY",
              "text": "L=ROY_C=E_R=E -> L=E_C=E_R=ROY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=ROY_R=E",
              "to_state": "L=ROY_C=E_R=E",
              "text": "L=E_C=ROY_R=E -> L=ROY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=ROY_R=E",
              "to_state": "L=E_C=E_R=ROY",
              "text": "L=E_C=ROY_R=E -> L=E_C=E_R=ROY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=ROY",
              "to_state": "L=ROY_C=E_R=E",
              "text": "L=E_C=E_R=ROY -> L=ROY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=ROY",
              "to_state": "L=E_C=ROY_R=E",
              "text": "L=E_C=E_R=ROY -> L=E_C=ROY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 2,
          "answer_text": "L=ROY_C=E_R=E -> L=E_C=E_R=ROY",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=GLSROY_C=E_R=E.jpg\nGoal state image: L=E_C=GLSROY_R=E.jpg\nInitial state: L=GLSROY_C=E_R=E\nGoal state: L=E_C=GLSROY_R=E\nAnswer the following questions.\n\nQuestion (depth 3): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=GLSROY_C=E_R=E.jpg\nGoal state image: L=E_C=GLSROY_R=E.jpg\nInitial state: L=GLSROY_C=E_R=E\nGoal state: L=E_C=GLSROY_R=E\nAnswer the following questions.\n\nReasoning (depth 3): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=GLSROY_C=E_R=E__TO__L=E_C=GLSROY_R=E__Q4",
          "depth": 4,
          "substack_size": 2,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=OY_C=E_R=E -> L=E_C=OY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=OY_C=E_R=E -> L=E_C=E_R=OY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=OY_R=E -> L=OY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=E_C=OY_R=E -> L=E_C=E_R=OY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=E_R=OY -> L=OY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=E_C=E_R=OY -> L=E_C=OY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 1,
          "answer_text": "L=OY_C=E_R=E -> L=E_C=OY_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=GLSROY_C=E_R=E.jpg\nGoal state image: L=E_C=GLSROY_R=E.jpg\nInitial state: L=GLSROY_C=E_R=E\nGoal state: L=E_C=GLSROY_R=E\nAnswer the following questions.\n\nQuestion (depth 4): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=GLSROY_C=E_R=E.jpg\nGoal state image: L=E_C=GLSROY_R=E.jpg\nInitial state: L=GLSROY_C=E_R=E\nGoal state: L=E_C=GLSROY_R=E\nAnswer the following questions.\n\nReasoning (depth 4): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=GLSROY_C=E_R=E__TO__L=E_C=GLSROY_R=E__Q5",
          "depth": 5,
          "substack_size": 1,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=Y_C=E_R=E -> L=E_C=Y_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=Y_C=E_R=E -> L=E_C=E_R=Y",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=Y_R=E -> L=Y_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=E_C=Y_R=E -> L=E_C=E_R=Y",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=E_R=Y -> L=Y_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=E_C=E_R=Y -> L=E_C=Y_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 2,
          "answer_text": "L=Y_C=E_R=E -> L=E_C=E_R=Y",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=GLSROY_C=E_R=E.jpg\nGoal state image: L=E_C=GLSROY_R=E.jpg\nInitial state: L=GLSROY_C=E_R=E\nGoal state: L=E_C=GLSROY_R=E\nAnswer the following questions.\n\nQuestion (depth 5): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=GLSROY_C=E_R=E.jpg\nGoal state image: L=E_C=GLSROY_R=E.jpg\nInitial state: L=GLSROY_C=E_R=E\nGoal state: L=E_C=GLSROY_R=E\nAnswer the following questions.\n\nReasoning (depth 5): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        }
      ],
      "full_problem_prompt": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=GLSROY_C=E_R=E.jpg\nGoal state image: L=E_C=GLSROY_R=E.jpg\nInitial state: L=GLSROY_C=E_R=E\nGoal state: L=E_C=GLSROY_R=E\nAnswer the following questions."
    },
    {
      "problem_id": "L=GLSROY_C=E_R=E__TO__L=E_C=E_R=GLSROY",
      "n_rings": 6,
      "source_pole": "L",
      "target_pole": "R",
      "aux_pole": "C",
      "initial_state": "L=GLSROY_C=E_R=E",
      "goal_state": "L=E_C=E_R=GLSROY",
      "initial_image": "L=GLSROY_C=E_R=E.jpg",
      "goal_image": "L=E_C=E_R=GLSROY.jpg",
      "questions": [
        {
          "qid": "L=GLSROY_C=E_R=E__TO__L=E_C=E_R=GLSROY__Q1",
          "depth": 1,
          "substack_size": 5,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=LSROY_C=E_R=E",
              "to_state": "L=E_C=LSROY_R=E",
              "text": "L=LSROY_C=E_R=E -> L=E_C=LSROY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=LSROY_C=E_R=E",
              "to_state": "L=E_C=E_R=LSROY",
              "text": "L=LSROY_C=E_R=E -> L=E_C=E_R=LSROY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=LSROY_R=E",
              "to_state": "L=LSROY_C=E_R=E",
              "text": "L=E_C=LSROY_R=E -> L=LSROY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=LSROY_R=E",
              "to_state": "L=E_C=E_R=LSROY",
              "text": "L=E_C=LSROY_R=E -> L=E_C=E_R=LSROY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=LSROY",
              "to_state": "L=LSROY_C=E_R=E",
              "text": "L=E_C=E_R=LSROY -> L=LSROY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=LSROY",
              "to_state": "L=E_C=LSROY_R=E",
              "text": "L=E_C=E_R=LSROY -> L=E_C=LSROY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 1,
          "answer_text": "L=LSROY_C=E_R=E -> L=E_C=LSROY_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=GLSROY_C=E_R=E.jpg\nGoal state image: L=E_C=E_R=GLSROY.jpg\nInitial state: L=GLSROY_C=E_R=E\nGoal state: L=E_C=E_R=GLSROY\nAnswer the following questions.\n\nQuestion (depth 1): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=LSROY_C=E_R=E -> L=E_C=LSROY_R=E\n2. L=LSROY_C=E_R=E -> L=E_C=E_R=LSROY\n3. L=E_C=LSROY_R=E -> L=LSROY_C=E_R=E\n4. L=E_C=LSROY_R=E -> L=E_C=E_R=LSROY\n5. L=E_C=E_R=LSROY -> L=LSROY_C=E_R=E\n6. L=E_C=E_R=LSROY -> L=E_C=LSROY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=GLSROY_C=E_R=E.jpg\nGoal state image: L=E_C=E_R=GLSROY.jpg\nInitial state: L=GLSROY_C=E_R=E\nGoal state: L=E_C=E_R=GLSROY\nAnswer the following questions.\n\nReasoning (depth 1): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=LSROY_C=E_R=E -> L=E_C=LSROY_R=E\n2. L=LSROY_C=E_R=E -> L=E_C=E_R=LSROY\n3. L=E_C=LSROY_R=E -> L=LSROY_C=E_R=E\n4. L=E_C=LSROY_R=E -> L=E_C=E_R=LSROY\n5. L=E_C=E_R=LSROY -> L=LSROY_C=E_R=E\n6. L=E_C=E_R=LSROY -> L=E_C=LSROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=LSROY_C=E_R=E -> L=E_C=LSROY_R=E\n2. L=LSROY_C=E_R=E -> L=E_C=E_R=LSROY\n3. L=E_C=LSROY_R=E -> L=LSROY_C=E_R=E\n4. L=E_C=LSROY_R=E -> L=E_C=E_R=LSROY\n5. L=E_C=E_R=LSROY -> L=LSROY_C=E_R=E\n6. L=E_C=E_R=LSROY -> L=E_C=LSROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=GLSROY_C=E_R=E__TO__L=E_C=E_R=GLSROY__Q2",
          "depth": 2,
          "substack_size": 4,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=SROY_C=E_R=E",
              "to_state": "L=E_C=SROY_R=E",
              "text": "L=SROY_C=E_R=E -> L=E_C=SROY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=SROY_C=E_R=E",
              "to_state": "L=E_C=E_R=SROY",
              "text": "L=SROY_C=E_R=E -> L=E_C=E_R=SROY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=SROY_R=E",
              "to_state": "L=SROY_C=E_R=E",
              "text": "L=E_C=SROY_R=E -> L=SROY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=SROY_R=E",
              "to_state": "L=E_C=E_R=SROY",
              "text": "L=E_C=SROY_R=E -> L=E_C=E_R=SROY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=SROY",
              "to_state": "L=SROY_C=E_R=E",
              "text": "L=E_C=E_R=SROY -> L=SROY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=SROY",
              "to_state": "L=E_C=SROY_R=E",
              "text": "L=E_C=E_R=SROY -> L=E_C=SROY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 2,
          "answer_text": "L=SROY_C=E_R=E -> L=E_C=E_R=SROY",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=GLSROY_C=E_R=E.jpg\nGoal state image: L=E_C=E_R=GLSROY.jpg\nInitial state: L=GLSROY_C=E_R=E\nGoal state: L=E_C=E_R=GLSROY\nAnswer the following questions.\n\nQuestion (depth 2): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=SROY_C=E_R=E -> L=E_C=SROY_R=E\n2. L=SROY_C=E_R=E -> L=E_C=E_R=SROY\n3. L=E_C=SROY_R=E -> L=SROY_C=E_R=E\n4. L=E_C=SROY_R=E -> L=E_C=E_R=SROY\n5. L=E_C=E_R=SROY -> L=SROY_C=E_R=E\n6. L=E_C=E_R=SROY -> L=E_C=SROY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=GLSROY_C=E_R=E.jpg\nGoal state image: L=E_C=E_R=GLSROY.jpg\nInitial state: L=GLSROY_C=E_R=E\nGoal state: L=E_C=E_R=GLSROY\nAnswer the following questions.\n\nReasoning (depth 2): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=SROY_C=E_R=E -> L=E_C=SROY_R=E\n2. L=SROY_C=E_R=E -> L=E_C=E_R=SROY\n3. L=E_C=SROY_R=E -> L=SROY_C=E_R=E\n4. L=E_C=SROY_R=E -> L=E_C=E_R=SROY\n5. L=E_C=E_R=SROY -> L=SROY_C=E_R=E\n6. L=E_C=E_R=SROY -> L=E_C=SROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=SROY_C=E_R=E -> L=E_C=SROY_R=E\n2. L=SROY_C=E_R=E -> L=E_C=E_R=SROY\n3. L=E_C=SROY_R=E -> L=SROY_C=E_R=E\n4. L=E_C=SROY_R=E -> L=E_C=E_R=SROY\n5. L=E_C=E_R=SROY -> L=SROY_C=E_R=E\n6. L=E_C=E_R=SROY -> L=E_C=SROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=GLSROY_C=E_R=E__TO__L=E_C=E_R=GLSROY__Q3",
          "depth": 3,
          "substack_size": 3,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=ROY_C=E_R=E",
              "to_state": "L=E_C=ROY_R=E",
              "text": "L=ROY_C=E_R=E -> L=E_C=ROY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=ROY_C=E_R=E",
              "to_state": "L=E_C=E_R=ROY",
              "text": "L=ROY_C=E_R=E -> L=E_C=E_R=ROY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=ROY_R=E",
              "to_state": "L=ROY_C=E_R=E",
              "text": "L=E_C=ROY_R=E -> L=ROY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=ROY_R=E",
              "to_state": "L=E_C=E_R=ROY",
              "text": "L=E_C=ROY_R=E -> L=E_C=E_R=ROY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=ROY",
              "to_state": "L=ROY_C=E_R=E",
              "text": "L=E_C=E_R=ROY -> L=ROY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=ROY",
              "to_state": "L=E_C=ROY_R=E",
              "text": "L=E_C=E_R=ROY -> L=E_C=ROY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 1,
          "answer_text": "L=ROY_C=E_R=E -> L=E_C=ROY_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=GLSROY_C=E_R=E.jpg\nGoal state image: L=E_C=E_R=GLSROY.jpg\nInitial state: L=GLSROY_C=E_R=E\nGoal state: L=E_C=E_R=GLSROY\nAnswer the following questions.\n\nQuestion (depth 3): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=GLSROY_C=E_R=E.jpg\nGoal state image: L=E_C=E_R=GLSROY.jpg\nInitial state: L=GLSROY_C=E_R=E\nGoal state: L=E_C=E_R=GLSROY\nAnswer the following questions.\n\nReasoning (depth 3): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=GLSROY_C=E_R=E__TO__L=E_C=E_R=GLSROY__Q4",
          "depth": 4,
          "substack_size": 2,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=OY_C=E_R=E -> L=E_C=OY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=OY_C=E_R=E -> L=E_C=E_R=OY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=OY_R=E -> L=OY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=E_C=OY_R=E -> L=E_C=E_R=OY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=E_R=OY -> L=OY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=E_C=E_R=OY -> L=E_C=OY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 2,
          "answer_text": "L=OY_C=E_R=E -> L=E_C=E_R=OY",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=GLSROY_C=E_R=E.jpg\nGoal state image: L=E_C=E_R=GLSROY.jpg\nInitial state: L=GLSROY_C=E_R=E\nGoal state: L=E_C=E_R=GLSROY\nAnswer the following questions.\n\nQuestion (depth 4): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=GLSROY_C=E_R=E.jpg\nGoal state image: L=E_C=E_R=GLSROY.jpg\nInitial state: L=GLSROY_C=E_R=E\nGoal state: L=E_C=E_R=GLSROY\nAnswer the following questions.\n\nReasoning (depth 4): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=GLSROY_C=E_R=E__TO__L=E_C=E_R=GLSROY__Q5",
          "depth": 5,
          "substack_size": 1,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=Y_C=E_R=E -> L=E_C=Y_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=Y_C=E_R=E -> L=E_C=E_R=Y",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=Y_R=E -> L=Y_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=E_C=Y_R=E -> L=E_C=E_R=Y",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=E_R=Y -> L=Y_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=E_C=E_R=Y -> L=E_C=Y_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 1,
          "answer_text": "L=Y_C=E_R=E -> L=E_C=Y_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=GLSROY_C=E_R=E.jpg\nGoal state image: L=E_C=E_R=GLSROY.jpg\nInitial state: L=GLSROY_C=E_R=E\nGoal state: L=E_C=E_R=GLSROY\nAnswer the following questions.\n\nQuestion (depth 5): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=GLSROY_C=E_R=E.jpg\nGoal state image: L=E_C=E_R=GLSROY.jpg\nInitial state: L=GLSROY_C=E_R=E\nGoal state: L=E_C=E_R=GLSROY\nAnswer the following questions.\n\nReasoning (depth 5): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        }
      ],
      "full_problem_prompt": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=GLSROY_C=E_R=E.jpg\nGoal state image: L=E_C=E_R=GLSROY.jpg\nInitial state: L=GLSROY_C=E_R=E\nGoal state: L=E_C=E_R=GLSROY\nAnswer the following questions."
    },
    {
      "problem_id": "L=E_C=GLSROY_R=E__TO__L=GLSROY_C=E_R=E",
      "n_rings": 6,
      "source_pole": "C",
      "target_pole": "L",
      "aux_pole": "R",
      "initial_state": "L=E_C=GLSROY_R=E",
      "goal_state": "L=GLSROY_C=E_R=E",
      "initial_image": "L=E_C=GLSROY_R=E.jpg",
      "goal_image": "L=GLSROY_C=E_R=E.jpg",
      "questions": [
        {
          "qid": "L=E_C=GLSROY_R=E__TO__L=GLSROY_C=E_R=E__Q1",
          "depth": 1,
          "substack_size": 5,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=LSROY_C=E_R=E",
              "to_state": "L=E_C=LSROY_R=E",
              "text": "L=LSROY_C=E_R=E -> L=E_C=LSROY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=LSROY_C=E_R=E",
              "to_state": "L=E_C=E_R=LSROY",
              "text": "L=LSROY_C=E_R=E -> L=E_C=E_R=LSROY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=LSROY_R=E",
              "to_state": "L=LSROY_C=E_R=E",
              "text": "L=E_C=LSROY_R=E -> L=LSROY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=LSROY_R=E",
              "to_state": "L=E_C=E_R=LSROY",
              "text": "L=E_C=LSROY_R=E -> L=E_C=E_R=LSROY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=LSROY",
              "to_state": "L=LSROY_C=E_R=E",
              "text": "L=E_C=E_R=LSROY -> L=LSROY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=LSROY",
              "to_state": "L=E_C=LSROY_R=E",
              "text": "L=E_C=E_R=LSROY -> L=E_C=LSROY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 4,
          "answer_text": "L=E_C=LSROY_R=E -> L=E_C=E_R=LSROY",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=GLSROY_R=E.jpg\nGoal state image: L=GLSROY_C=E_R=E.jpg\nInitial state: L=E_C=GLSROY_R=E\nGoal state: L=GLSROY_C=E_R=E\nAnswer the following questions.\n\nQuestion (depth 1): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=LSROY_C=E_R=E -> L=E_C=LSROY_R=E\n2. L=LSROY_C=E_R=E -> L=E_C=E_R=LSROY\n3. L=E_C=LSROY_R=E -> L=LSROY_C=E_R=E\n4. L=E_C=LSROY_R=E -> L=E_C=E_R=LSROY\n5. L=E_C=E_R=LSROY -> L=LSROY_C=E_R=E\n6. L=E_C=E_R=LSROY -> L=E_C=LSROY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=GLSROY_R=E.jpg\nGoal state image: L=GLSROY_C=E_R=E.jpg\nInitial state: L=E_C=GLSROY_R=E\nGoal state: L=GLSROY_C=E_R=E\nAnswer the following questions.\n\nReasoning (depth 1): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=LSROY_C=E_R=E -> L=E_C=LSROY_R=E\n2. L=LSROY_C=E_R=E -> L=E_C=E_R=LSROY\n3. L=E_C=LSROY_R=E -> L=LSROY_C=E_R=E\n4. L=E_C=LSROY_R=E -> L=E_C=E_R=LSROY\n5. L=E_C=E_R=LSROY -> L=LSROY_C=E_R=E\n6. L=E_C=E_R=LSROY -> L=E_C=LSROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=LSROY_C=E_R=E -> L=E_C=LSROY_R=E\n2. L=LSROY_C=E_R=E -> L=E_C=E_R=LSROY\n3. L=E_C=LSROY_R=E -> L=LSROY_C=E_R=E\n4. L=E_C=LSROY_R=E -> L=E_C=E_R=LSROY\n5. L=E_C=E_R=LSROY -> L=LSROY_C=E_R=E\n6. L=E_C=E_R=LSROY -> L=E_C=LSROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=GLSROY_R=E__TO__L=GLSROY_C=E_R=E__Q2",
          "depth": 2,
          "substack_size": 4,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=SROY_C=E_R=E",
              "to_state": "L=E_C=SROY_R=E",
              "text": "L=SROY_C=E_R=E -> L=E_C=SROY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=SROY_C=E_R=E",
              "to_state": "L=E_C=E_R=SROY",
              "text": "L=SROY_C=E_R=E -> L=E_C=E_R=SROY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=SROY_R=E",
              "to_state": "L=SROY_C=E_R=E",
              "text": "L=E_C=SROY_R=E -> L=SROY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=SROY_R=E",
              "to_state": "L=E_C=E_R=SROY",
              "text": "L=E_C=SROY_R=E -> L=E_C=E_R=SROY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=SROY",
              "to_state": "L=SROY_C=E_R=E",
              "text": "L=E_C=E_R=SROY -> L=SROY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=SROY",
              "to_state": "L=E_C=SROY_R=E",
              "text": "L=E_C=E_R=SROY -> L=E_C=SROY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 3,
          "answer_text": "L=E_C=SROY_R=E -> L=SROY_C=E_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=GLSROY_R=E.jpg\nGoal state image: L=GLSROY_C=E_R=E.jpg\nInitial state: L=E_C=GLSROY_R=E\nGoal state: L=GLSROY_C=E_R=E\nAnswer the following questions.\n\nQuestion (depth 2): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=SROY_C=E_R=E -> L=E_C=SROY_R=E\n2. L=SROY_C=E_R=E -> L=E_C=E_R=SROY\n3. L=E_C=SROY_R=E -> L=SROY_C=E_R=E\n4. L=E_C=SROY_R=E -> L=E_C=E_R=SROY\n5. L=E_C=E_R=SROY -> L=SROY_C=E_R=E\n6. L=E_C=E_R=SROY -> L=E_C=SROY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=GLSROY_R=E.jpg\nGoal state image: L=GLSROY_C=E_R=E.jpg\nInitial state: L=E_C=GLSROY_R=E\nGoal state: L=GLSROY_C=E_R=E\nAnswer the following questions.\n\nReasoning (depth 2): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=SROY_C=E_R=E -> L=E_C=SROY_R=E\n2. L=SROY_C=E_R=E -> L=E_C=E_R=SROY\n3. L=E_C=SROY_R=E -> L=SROY_C=E_R=E\n4. L=E_C=SROY_R=E -> L=E_C=E_R=SROY\n5. L=E_C=E_R=SROY -> L=SROY_C=E_R=E\n6. L=E_C=E_R=SROY -> L=E_C=SROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=SROY_C=E_R=E -> L=E_C=SROY_R=E\n2. L=SROY_C=E_R=E -> L=E_C=E_R=SROY\n3. L=E_C=SROY_R=E -> L=SROY_C=E_R=E\n4. L=E_C=SROY_R=E -> L=E_C=E_R=SROY\n5. L=E_C=E_R=SROY -> L=SROY_C=E_R=E\n6. L=E_C=E_R=SROY -> L=E_C=SROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=GLSROY_R=E__TO__L=GLSROY_C=E_R=E__Q3",
          "depth": 3,
          "substack_size": 3,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=ROY_C=E_R=E",
              "to_state": "L=E_C=ROY_R=E",
              "text": "L=ROY_C=E_R=E -> L=E_C=ROY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=ROY_C=E_R=E",
              "to_state": "L=E_C=E_R=ROY",
              "text": "L=ROY_C=E_R=E -> L=E_C=E_R=ROY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=ROY_R=E",
              "to_state": "L=ROY_C=E_R=E",
              "text": "L=E_C=ROY_R=E -> L=ROY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=ROY_R=E",
              "to_state": "L=E_C=E_R=ROY",
              "text": "L=E_C=ROY_R=E -> L=E_C=E_R=ROY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=ROY",
              "to_state": "L=ROY_C=E_R=E",
              "text": "L=E_C=E_R=ROY -> L=ROY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=ROY",
              "to_state": "L=E_C=ROY_R=E",
              "text": "L=E_C=E_R=ROY -> L=E_C=ROY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 4,
          "answer_text": "L=E_C=ROY_R=E -> L=E_C=E_R=ROY",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=GLSROY_R=E.jpg\nGoal state image: L=GLSROY_C=E_R=E.jpg\nInitial state: L=E_C=GLSROY_R=E\nGoal state: L=GLSROY_C=E_R=E\nAnswer the following questions.\n\nQuestion (depth 3): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=GLSROY_R=E.jpg\nGoal state image: L=GLSROY_C=E_R=E.jpg\nInitial state: L=E_C=GLSROY_R=E\nGoal state: L=GLSROY_C=E_R=E\nAnswer the following questions.\n\nReasoning (depth 3): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=GLSROY_R=E__TO__L=GLSROY_C=E_R=E__Q4",
          "depth": 4,
          "substack_size": 2,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=OY_C=E_R=E -> L=E_C=OY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=OY_C=E_R=E -> L=E_C=E_R=OY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=OY_R=E -> L=OY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=E_C=OY_R=E -> L=E_C=E_R=OY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=E_R=OY -> L=OY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=E_C=E_R=OY -> L=E_C=OY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 3,
          "answer_text": "L=E_C=OY_R=E -> L=OY_C=E_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=GLSROY_R=E.jpg\nGoal state image: L=GLSROY_C=E_R=E.jpg\nInitial state: L=E_C=GLSROY_R=E\nGoal state: L=GLSROY_C=E_R=E\nAnswer the following questions.\n\nQuestion (depth 4): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=GLSROY_R=E.jpg\nGoal state image: L=GLSROY_C=E_R=E.jpg\nInitial state: L=E_C=GLSROY_R=E\nGoal state: L=GLSROY_C=E_R=E\nAnswer the following questions.\n\nReasoning (depth 4): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=GLSROY_R=E__TO__L=GLSROY_C=E_R=E__Q5",
          "depth": 5,
          "substack_size": 1,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=Y_C=E_R=E -> L=E_C=Y_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=Y_C=E_R=E -> L=E_C=E_R=Y",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=Y_R=E -> L=Y_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=E_C=Y_R=E -> L=E_C=E_R=Y",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=E_R=Y -> L=Y_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=E_C=E_R=Y -> L=E_C=Y_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 4,
          "answer_text": "L=E_C=Y_R=E -> L=E_C=E_R=Y",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=GLSROY_R=E.jpg\nGoal state image: L=GLSROY_C=E_R=E.jpg\nInitial state: L=E_C=GLSROY_R=E\nGoal state: L=GLSROY_C=E_R=E\nAnswer the following questions.\n\nQuestion (depth 5): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=GLSROY_R=E.jpg\nGoal state image: L=GLSROY_C=E_R=E.jpg\nInitial state: L=E_C=GLSROY_R=E\nGoal state: L=GLSROY_C=E_R=E\nAnswer the following questions.\n\nReasoning (depth 5): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        }
      ],
      "full_problem_prompt": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=GLSROY_R=E.jpg\nGoal state image: L=GLSROY_C=E_R=E.jpg\nInitial state: L=E_C=GLSROY_R=E\nGoal state: L=GLSROY_C=E_R=E\nAnswer the following questions."
    },
    {
      "problem_id": "L=E_C=GLSROY_R=E__TO__L=E_C=E_R=GLSROY",
      "n_rings": 6,
      "source_pole": "C",
      "target_pole": "R",
      "aux_pole": "L",
      "initial_state": "L=E_C=GLSROY_R=E",
      "goal_state": "L=E_C=E_R=GLSROY",
      "initial_image": "L=E_C=GLSROY_R=E.jpg",
      "goal_image": "L=E_C=E_R=GLSROY.jpg",
      "questions": [
        {
          "qid": "L=E_C=GLSROY_R=E__TO__L=E_C=E_R=GLSROY__Q1",
          "depth": 1,
          "substack_size": 5,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=LSROY_C=E_R=E",
              "to_state": "L=E_C=LSROY_R=E",
              "text": "L=LSROY_C=E_R=E -> L=E_C=LSROY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=LSROY_C=E_R=E",
              "to_state": "L=E_C=E_R=LSROY",
              "text": "L=LSROY_C=E_R=E -> L=E_C=E_R=LSROY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=LSROY_R=E",
              "to_state": "L=LSROY_C=E_R=E",
              "text": "L=E_C=LSROY_R=E -> L=LSROY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=LSROY_R=E",
              "to_state": "L=E_C=E_R=LSROY",
              "text": "L=E_C=LSROY_R=E -> L=E_C=E_R=LSROY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=LSROY",
              "to_state": "L=LSROY_C=E_R=E",
              "text": "L=E_C=E_R=LSROY -> L=LSROY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=LSROY",
              "to_state": "L=E_C=LSROY_R=E",
              "text": "L=E_C=E_R=LSROY -> L=E_C=LSROY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 3,
          "answer_text": "L=E_C=LSROY_R=E -> L=LSROY_C=E_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=GLSROY_R=E.jpg\nGoal state image: L=E_C=E_R=GLSROY.jpg\nInitial state: L=E_C=GLSROY_R=E\nGoal state: L=E_C=E_R=GLSROY\nAnswer the following questions.\n\nQuestion (depth 1): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=LSROY_C=E_R=E -> L=E_C=LSROY_R=E\n2. L=LSROY_C=E_R=E -> L=E_C=E_R=LSROY\n3. L=E_C=LSROY_R=E -> L=LSROY_C=E_R=E\n4. L=E_C=LSROY_R=E -> L=E_C=E_R=LSROY\n5. L=E_C=E_R=LSROY -> L=LSROY_C=E_R=E\n6. L=E_C=E_R=LSROY -> L=E_C=LSROY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=GLSROY_R=E.jpg\nGoal state image: L=E_C=E_R=GLSROY.jpg\nInitial state: L=E_C=GLSROY_R=E\nGoal state: L=E_C=E_R=GLSROY\nAnswer the following questions.\n\nReasoning (depth 1): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=LSROY_C=E_R=E -> L=E_C=LSROY_R=E\n2. L=LSROY_C=E_R=E -> L=E_C=E_R=LSROY\n3. L=E_C=LSROY_R=E -> L=LSROY_C=E_R=E\n4. L=E_C=LSROY_R=E -> L=E_C=E_R=LSROY\n5. L=E_C=E_R=LSROY -> L=LSROY_C=E_R=E\n6. L=E_C=E_R=LSROY -> L=E_C=LSROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=LSROY_C=E_R=E -> L=E_C=LSROY_R=E\n2. L=LSROY_C=E_R=E -> L=E_C=E_R=LSROY\n3. L=E_C=LSROY_R=E -> L=LSROY_C=E_R=E\n4. L=E_C=LSROY_R=E -> L=E_C=E_R=LSROY\n5. L=E_C=E_R=LSROY -> L=LSROY_C=E_R=E\n6. L=E_C=E_R=LSROY -> L=E_C=LSROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=GLSROY_R=E__TO__L=E_C=E_R=GLSROY__Q2",
          "depth": 2,
          "substack_size": 4,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=SROY_C=E_R=E",
              "to_state": "L=E_C=SROY_R=E",
              "text": "L=SROY_C=E_R=E -> L=E_C=SROY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=SROY_C=E_R=E",
              "to_state": "L=E_C=E_R=SROY",
              "text": "L=SROY_C=E_R=E -> L=E_C=E_R=SROY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=SROY_R=E",
              "to_state": "L=SROY_C=E_R=E",
              "text": "L=E_C=SROY_R=E -> L=SROY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=SROY_R=E",
              "to_state": "L=E_C=E_R=SROY",
              "text": "L=E_C=SROY_R=E -> L=E_C=E_R=SROY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=SROY",
              "to_state": "L=SROY_C=E_R=E",
              "text": "L=E_C=E_R=SROY -> L=SROY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=SROY",
              "to_state": "L=E_C=SROY_R=E",
              "text": "L=E_C=E_R=SROY -> L=E_C=SROY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 4,
          "answer_text": "L=E_C=SROY_R=E -> L=E_C=E_R=SROY",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=GLSROY_R=E.jpg\nGoal state image: L=E_C=E_R=GLSROY.jpg\nInitial state: L=E_C=GLSROY_R=E\nGoal state: L=E_C=E_R=GLSROY\nAnswer the following questions.\n\nQuestion (depth 2): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=SROY_C=E_R=E -> L=E_C=SROY_R=E\n2. L=SROY_C=E_R=E -> L=E_C=E_R=SROY\n3. L=E_C=SROY_R=E -> L=SROY_C=E_R=E\n4. L=E_C=SROY_R=E -> L=E_C=E_R=SROY\n5. L=E_C=E_R=SROY -> L=SROY_C=E_R=E\n6. L=E_C=E_R=SROY -> L=E_C=SROY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=GLSROY_R=E.jpg\nGoal state image: L=E_C=E_R=GLSROY.jpg\nInitial state: L=E_C=GLSROY_R=E\nGoal state: L=E_C=E_R=GLSROY\nAnswer the following questions.\n\nReasoning (depth 2): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=SROY_C=E_R=E -> L=E_C=SROY_R=E\n2. L=SROY_C=E_R=E -> L=E_C=E_R=SROY\n3. L=E_C=SROY_R=E -> L=SROY_C=E_R=E\n4. L=E_C=SROY_R=E -> L=E_C=E_R=SROY\n5. L=E_C=E_R=SROY -> L=SROY_C=E_R=E\n6. L=E_C=E_R=SROY -> L=E_C=SROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=SROY_C=E_R=E -> L=E_C=SROY_R=E\n2. L=SROY_C=E_R=E -> L=E_C=E_R=SROY\n3. L=E_C=SROY_R=E -> L=SROY_C=E_R=E\n4. L=E_C=SROY_R=E -> L=E_C=E_R=SROY\n5. L=E_C=E_R=SROY -> L=SROY_C=E_R=E\n6. L=E_C=E_R=SROY -> L=E_C=SROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=GLSROY_R=E__TO__L=E_C=E_R=GLSROY__Q3",
          "depth": 3,
          "substack_size": 3,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=ROY_C=E_R=E",
              "to_state": "L=E_C=ROY_R=E",
              "text": "L=ROY_C=E_R=E -> L=E_C=ROY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=ROY_C=E_R=E",
              "to_state": "L=E_C=E_R=ROY",
              "text": "L=ROY_C=E_R=E -> L=E_C=E_R=ROY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=ROY_R=E",
              "to_state": "L=ROY_C=E_R=E",
              "text": "L=E_C=ROY_R=E -> L=ROY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=ROY_R=E",
              "to_state": "L=E_C=E_R=ROY",
              "text": "L=E_C=ROY_R=E -> L=E_C=E_R=ROY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=ROY",
              "to_state": "L=ROY_C=E_R=E",
              "text": "L=E_C=E_R=ROY -> L=ROY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=ROY",
              "to_state": "L=E_C=ROY_R=E",
              "text": "L=E_C=E_R=ROY -> L=E_C=ROY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 3,
          "answer_text": "L=E_C=ROY_R=E -> L=ROY_C=E_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=GLSROY_R=E.jpg\nGoal state image: L=E_C=E_R=GLSROY.jpg\nInitial state: L=E_C=GLSROY_R=E\nGoal state: L=E_C=E_R=GLSROY\nAnswer the following questions.\n\nQuestion (depth 3): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=GLSROY_R=E.jpg\nGoal state image: L=E_C=E_R=GLSROY.jpg\nInitial state: L=E_C=GLSROY_R=E\nGoal state: L=E_C=E_R=GLSROY\nAnswer the following questions.\n\nReasoning (depth 3): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=GLSROY_R=E__TO__L=E_C=E_R=GLSROY__Q4",
          "depth": 4,
          "substack_size": 2,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=OY_C=E_R=E -> L=E_C=OY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=OY_C=E_R=E -> L=E_C=E_R=OY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=OY_R=E -> L=OY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=E_C=OY_R=E -> L=E_C=E_R=OY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=E_R=OY -> L=OY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=E_C=E_R=OY -> L=E_C=OY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 4,
          "answer_text": "L=E_C=OY_R=E -> L=E_C=E_R=OY",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=GLSROY_R=E.jpg\nGoal state image: L=E_C=E_R=GLSROY.jpg\nInitial state: L=E_C=GLSROY_R=E\nGoal state: L=E_C=E_R=GLSROY\nAnswer the following questions.\n\nQuestion (depth 4): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=GLSROY_R=E.jpg\nGoal state image: L=E_C=E_R=GLSROY.jpg\nInitial state: L=E_C=GLSROY_R=E\nGoal state: L=E_C=E_R=GLSROY\nAnswer the following questions.\n\nReasoning (depth 4): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=GLSROY_R=E__TO__L=E_C=E_R=GLSROY__Q5",
          "depth": 5,
          "substack_size": 1,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=Y_C=E_R=E -> L=E_C=Y_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=Y_C=E_R=E -> L=E_C=E_R=Y",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=Y_R=E -> L=Y_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=E_C=Y_R=E -> L=E_C=E_R=Y",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=E_R=Y -> L=Y_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=E_C=E_R=Y -> L=E_C=Y_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 3,
          "answer_text": "L=E_C=Y_R=E -> L=Y_C=E_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=GLSROY_R=E.jpg\nGoal state image: L=E_C=E_R=GLSROY.jpg\nInitial state: L=E_C=GLSROY_R=E\nGoal state: L=E_C=E_R=GLSROY\nAnswer the following questions.\n\nQuestion (depth 5): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=GLSROY_R=E.jpg\nGoal state image: L=E_C=E_R=GLSROY.jpg\nInitial state: L=E_C=GLSROY_R=E\nGoal state: L=E_C=E_R=GLSROY\nAnswer the following questions.\n\nReasoning (depth 5): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        }
      ],
      "full_problem_prompt": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=GLSROY_R=E.jpg\nGoal state image: L=E_C=E_R=GLSROY.jpg\nInitial state: L=E_C=GLSROY_R=E\nGoal state: L=E_C=E_R=GLSROY\nAnswer the following questions."
    },
    {
      "problem_id": "L=E_C=E_R=GLSROY__TO__L=GLSROY_C=E_R=E",
      "n_rings": 6,
      "source_pole": "R",
      "target_pole": "L",
      "aux_pole": "C",
      "initial_state": "L=E_C=E_R=GLSROY",
      "goal_state": "L=GLSROY_C=E_R=E",
      "initial_image": "L=E_C=E_R=GLSROY.jpg",
      "goal_image": "L=GLSROY_C=E_R=E.jpg",
      "questions": [
        {
          "qid": "L=E_C=E_R=GLSROY__TO__L=GLSROY_C=E_R=E__Q1",
          "depth": 1,
          "substack_size": 5,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=LSROY_C=E_R=E",
              "to_state": "L=E_C=LSROY_R=E",
              "text": "L=LSROY_C=E_R=E -> L=E_C=LSROY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=LSROY_C=E_R=E",
              "to_state": "L=E_C=E_R=LSROY",
              "text": "L=LSROY_C=E_R=E -> L=E_C=E_R=LSROY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=LSROY_R=E",
              "to_state": "L=LSROY_C=E_R=E",
              "text": "L=E_C=LSROY_R=E -> L=LSROY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=LSROY_R=E",
              "to_state": "L=E_C=E_R=LSROY",
              "text": "L=E_C=LSROY_R=E -> L=E_C=E_R=LSROY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=LSROY",
              "to_state": "L=LSROY_C=E_R=E",
              "text": "L=E_C=E_R=LSROY -> L=LSROY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=LSROY",
              "to_state": "L=E_C=LSROY_R=E",
              "text": "L=E_C=E_R=LSROY -> L=E_C=LSROY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 6,
          "answer_text": "L=E_C=E_R=LSROY -> L=E_C=LSROY_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=GLSROY.jpg\nGoal state image: L=GLSROY_C=E_R=E.jpg\nInitial state: L=E_C=E_R=GLSROY\nGoal state: L=GLSROY_C=E_R=E\nAnswer the following questions.\n\nQuestion (depth 1): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=LSROY_C=E_R=E -> L=E_C=LSROY_R=E\n2. L=LSROY_C=E_R=E -> L=E_C=E_R=LSROY\n3. L=E_C=LSROY_R=E -> L=LSROY_C=E_R=E\n4. L=E_C=LSROY_R=E -> L=E_C=E_R=LSROY\n5. L=E_C=E_R=LSROY -> L=LSROY_C=E_R=E\n6. L=E_C=E_R=LSROY -> L=E_C=LSROY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=GLSROY.jpg\nGoal state image: L=GLSROY_C=E_R=E.jpg\nInitial state: L=E_C=E_R=GLSROY\nGoal state: L=GLSROY_C=E_R=E\nAnswer the following questions.\n\nReasoning (depth 1): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=LSROY_C=E_R=E -> L=E_C=LSROY_R=E\n2. L=LSROY_C=E_R=E -> L=E_C=E_R=LSROY\n3. L=E_C=LSROY_R=E -> L=LSROY_C=E_R=E\n4. L=E_C=LSROY_R=E -> L=E_C=E_R=LSROY\n5. L=E_C=E_R=LSROY -> L=LSROY_C=E_R=E\n6. L=E_C=E_R=LSROY -> L=E_C=LSROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=LSROY_C=E_R=E -> L=E_C=LSROY_R=E\n2. L=LSROY_C=E_R=E -> L=E_C=E_R=LSROY\n3. L=E_C=LSROY_R=E -> L=LSROY_C=E_R=E\n4. L=E_C=LSROY_R=E -> L=E_C=E_R=LSROY\n5. L=E_C=E_R=LSROY -> L=LSROY_C=E_R=E\n6. L=E_C=E_R=LSROY -> L=E_C=LSROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=E_R=GLSROY__TO__L=GLSROY_C=E_R=E__Q2",
          "depth": 2,
          "substack_size": 4,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=SROY_C=E_R=E",
              "to_state": "L=E_C=SROY_R=E",
              "text": "L=SROY_C=E_R=E -> L=E_C=SROY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=SROY_C=E_R=E",
              "to_state": "L=E_C=E_R=SROY",
              "text": "L=SROY_C=E_R=E -> L=E_C=E_R=SROY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=SROY_R=E",
              "to_state": "L=SROY_C=E_R=E",
              "text": "L=E_C=SROY_R=E -> L=SROY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=SROY_R=E",
              "to_state": "L=E_C=E_R=SROY",
              "text": "L=E_C=SROY_R=E -> L=E_C=E_R=SROY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=SROY",
              "to_state": "L=SROY_C=E_R=E",
              "text": "L=E_C=E_R=SROY -> L=SROY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=SROY",
              "to_state": "L=E_C=SROY_R=E",
              "text": "L=E_C=E_R=SROY -> L=E_C=SROY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 5,
          "answer_text": "L=E_C=E_R=SROY -> L=SROY_C=E_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=GLSROY.jpg\nGoal state image: L=GLSROY_C=E_R=E.jpg\nInitial state: L=E_C=E_R=GLSROY\nGoal state: L=GLSROY_C=E_R=E\nAnswer the following questions.\n\nQuestion (depth 2): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=SROY_C=E_R=E -> L=E_C=SROY_R=E\n2. L=SROY_C=E_R=E -> L=E_C=E_R=SROY\n3. L=E_C=SROY_R=E -> L=SROY_C=E_R=E\n4. L=E_C=SROY_R=E -> L=E_C=E_R=SROY\n5. L=E_C=E_R=SROY -> L=SROY_C=E_R=E\n6. L=E_C=E_R=SROY -> L=E_C=SROY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=GLSROY.jpg\nGoal state image: L=GLSROY_C=E_R=E.jpg\nInitial state: L=E_C=E_R=GLSROY\nGoal state: L=GLSROY_C=E_R=E\nAnswer the following questions.\n\nReasoning (depth 2): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=SROY_C=E_R=E -> L=E_C=SROY_R=E\n2. L=SROY_C=E_R=E -> L=E_C=E_R=SROY\n3. L=E_C=SROY_R=E -> L=SROY_C=E_R=E\n4. L=E_C=SROY_R=E -> L=E_C=E_R=SROY\n5. L=E_C=E_R=SROY -> L=SROY_C=E_R=E\n6. L=E_C=E_R=SROY -> L=E_C=SROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=SROY_C=E_R=E -> L=E_C=SROY_R=E\n2. L=SROY_C=E_R=E -> L=E_C=E_R=SROY\n3. L=E_C=SROY_R=E -> L=SROY_C=E_R=E\n4. L=E_C=SROY_R=E -> L=E_C=E_R=SROY\n5. L=E_C=E_R=SROY -> L=SROY_C=E_R=E\n6. L=E_C=E_R=SROY -> L=E_C=SROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=E_R=GLSROY__TO__L=GLSROY_C=E_R=E__Q3",
          "depth": 3,
          "substack_size": 3,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=ROY_C=E_R=E",
              "to_state": "L=E_C=ROY_R=E",
              "text": "L=ROY_C=E_R=E -> L=E_C=ROY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=ROY_C=E_R=E",
              "to_state": "L=E_C=E_R=ROY",
              "text": "L=ROY_C=E_R=E -> L=E_C=E_R=ROY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=ROY_R=E",
              "to_state": "L=ROY_C=E_R=E",
              "text": "L=E_C=ROY_R=E -> L=ROY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=ROY_R=E",
              "to_state": "L=E_C=E_R=ROY",
              "text": "L=E_C=ROY_R=E -> L=E_C=E_R=ROY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=ROY",
              "to_state": "L=ROY_C=E_R=E",
              "text": "L=E_C=E_R=ROY -> L=ROY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=ROY",
              "to_state": "L=E_C=ROY_R=E",
              "text": "L=E_C=E_R=ROY -> L=E_C=ROY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 6,
          "answer_text": "L=E_C=E_R=ROY -> L=E_C=ROY_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=GLSROY.jpg\nGoal state image: L=GLSROY_C=E_R=E.jpg\nInitial state: L=E_C=E_R=GLSROY\nGoal state: L=GLSROY_C=E_R=E\nAnswer the following questions.\n\nQuestion (depth 3): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=GLSROY.jpg\nGoal state image: L=GLSROY_C=E_R=E.jpg\nInitial state: L=E_C=E_R=GLSROY\nGoal state: L=GLSROY_C=E_R=E\nAnswer the following questions.\n\nReasoning (depth 3): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=E_R=GLSROY__TO__L=GLSROY_C=E_R=E__Q4",
          "depth": 4,
          "substack_size": 2,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=OY_C=E_R=E -> L=E_C=OY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=OY_C=E_R=E -> L=E_C=E_R=OY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=OY_R=E -> L=OY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=E_C=OY_R=E -> L=E_C=E_R=OY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=E_R=OY -> L=OY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=E_C=E_R=OY -> L=E_C=OY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 5,
          "answer_text": "L=E_C=E_R=OY -> L=OY_C=E_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=GLSROY.jpg\nGoal state image: L=GLSROY_C=E_R=E.jpg\nInitial state: L=E_C=E_R=GLSROY\nGoal state: L=GLSROY_C=E_R=E\nAnswer the following questions.\n\nQuestion (depth 4): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=GLSROY.jpg\nGoal state image: L=GLSROY_C=E_R=E.jpg\nInitial state: L=E_C=E_R=GLSROY\nGoal state: L=GLSROY_C=E_R=E\nAnswer the following questions.\n\nReasoning (depth 4): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=E_R=GLSROY__TO__L=GLSROY_C=E_R=E__Q5",
          "depth": 5,
          "substack_size": 1,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=Y_C=E_R=E -> L=E_C=Y_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=Y_C=E_R=E -> L=E_C=E_R=Y",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=Y_R=E -> L=Y_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=E_C=Y_R=E -> L=E_C=E_R=Y",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=E_R=Y -> L=Y_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=E_C=E_R=Y -> L=E_C=Y_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 6,
          "answer_text": "L=E_C=E_R=Y -> L=E_C=Y_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=GLSROY.jpg\nGoal state image: L=GLSROY_C=E_R=E.jpg\nInitial state: L=E_C=E_R=GLSROY\nGoal state: L=GLSROY_C=E_R=E\nAnswer the following questions.\n\nQuestion (depth 5): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=GLSROY.jpg\nGoal state image: L=GLSROY_C=E_R=E.jpg\nInitial state: L=E_C=E_R=GLSROY\nGoal state: L=GLSROY_C=E_R=E\nAnswer the following questions.\n\nReasoning (depth 5): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        }
      ],
      "full_problem_prompt": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=GLSROY.jpg\nGoal state image: L=GLSROY_C=E_R=E.jpg\nInitial state: L=E_C=E_R=GLSROY\nGoal state: L=GLSROY_C=E_R=E\nAnswer the following questions."
    },
    {
      "problem_id": "L=E_C=E_R=GLSROY__TO__L=E_C=GLSROY_R=E",
      "n_rings": 6,
      "source_pole": "R",
      "target_pole": "C",
      "aux_pole": "L",
      "initial_state": "L=E_C=E_R=GLSROY",
      "goal_state": "L=E_C=GLSROY_R=E",
      "initial_image": "L=E_C=E_R=GLSROY.jpg",
      "goal_image": "L=E_C=GLSROY_R=E.jpg",
      "questions": [
        {
          "qid": "L=E_C=E_R=GLSROY__TO__L=E_C=GLSROY_R=E__Q1",
          "depth": 1,
          "substack_size": 5,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=LSROY_C=E_R=E",
              "to_state": "L=E_C=LSROY_R=E",
              "text": "L=LSROY_C=E_R=E -> L=E_C=LSROY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=LSROY_C=E_R=E",
              "to_state": "L=E_C=E_R=LSROY",
              "text": "L=LSROY_C=E_R=E -> L=E_C=E_R=LSROY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=LSROY_R=E",
              "to_state": "L=LSROY_C=E_R=E",
              "text": "L=E_C=LSROY_R=E -> L=LSROY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=LSROY_R=E",
              "to_state": "L=E_C=E_R=LSROY",
              "text": "L=E_C=LSROY_R=E -> L=E_C=E_R=LSROY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=LSROY",
              "to_state": "L=LSROY_C=E_R=E",
              "text": "L=E_C=E_R=LSROY -> L=LSROY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=LSROY",
              "to_state": "L=E_C=LSROY_R=E",
              "text": "L=E_C=E_R=LSROY -> L=E_C=LSROY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 5,
          "answer_text": "L=E_C=E_R=LSROY -> L=LSROY_C=E_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=GLSROY.jpg\nGoal state image: L=E_C=GLSROY_R=E.jpg\nInitial state: L=E_C=E_R=GLSROY\nGoal state: L=E_C=GLSROY_R=E\nAnswer the following questions.\n\nQuestion (depth 1): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=LSROY_C=E_R=E -> L=E_C=LSROY_R=E\n2. L=LSROY_C=E_R=E -> L=E_C=E_R=LSROY\n3. L=E_C=LSROY_R=E -> L=LSROY_C=E_R=E\n4. L=E_C=LSROY_R=E -> L=E_C=E_R=LSROY\n5. L=E_C=E_R=LSROY -> L=LSROY_C=E_R=E\n6. L=E_C=E_R=LSROY -> L=E_C=LSROY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=GLSROY.jpg\nGoal state image: L=E_C=GLSROY_R=E.jpg\nInitial state: L=E_C=E_R=GLSROY\nGoal state: L=E_C=GLSROY_R=E\nAnswer the following questions.\n\nReasoning (depth 1): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=LSROY_C=E_R=E -> L=E_C=LSROY_R=E\n2. L=LSROY_C=E_R=E -> L=E_C=E_R=LSROY\n3. L=E_C=LSROY_R=E -> L=LSROY_C=E_R=E\n4. L=E_C=LSROY_R=E -> L=E_C=E_R=LSROY\n5. L=E_C=E_R=LSROY -> L=LSROY_C=E_R=E\n6. L=E_C=E_R=LSROY -> L=E_C=LSROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=LSROY_C=E_R=E -> L=E_C=LSROY_R=E\n2. L=LSROY_C=E_R=E -> L=E_C=E_R=LSROY\n3. L=E_C=LSROY_R=E -> L=LSROY_C=E_R=E\n4. L=E_C=LSROY_R=E -> L=E_C=E_R=LSROY\n5. L=E_C=E_R=LSROY -> L=LSROY_C=E_R=E\n6. L=E_C=E_R=LSROY -> L=E_C=LSROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=E_R=GLSROY__TO__L=E_C=GLSROY_R=E__Q2",
          "depth": 2,
          "substack_size": 4,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=SROY_C=E_R=E",
              "to_state": "L=E_C=SROY_R=E",
              "text": "L=SROY_C=E_R=E -> L=E_C=SROY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=SROY_C=E_R=E",
              "to_state": "L=E_C=E_R=SROY",
              "text": "L=SROY_C=E_R=E -> L=E_C=E_R=SROY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=SROY_R=E",
              "to_state": "L=SROY_C=E_R=E",
              "text": "L=E_C=SROY_R=E -> L=SROY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=SROY_R=E",
              "to_state": "L=E_C=E_R=SROY",
              "text": "L=E_C=SROY_R=E -> L=E_C=E_R=SROY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=SROY",
              "to_state": "L=SROY_C=E_R=E",
              "text": "L=E_C=E_R=SROY -> L=SROY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=SROY",
              "to_state": "L=E_C=SROY_R=E",
              "text": "L=E_C=E_R=SROY -> L=E_C=SROY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 6,
          "answer_text": "L=E_C=E_R=SROY -> L=E_C=SROY_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=GLSROY.jpg\nGoal state image: L=E_C=GLSROY_R=E.jpg\nInitial state: L=E_C=E_R=GLSROY\nGoal state: L=E_C=GLSROY_R=E\nAnswer the following questions.\n\nQuestion (depth 2): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=SROY_C=E_R=E -> L=E_C=SROY_R=E\n2. L=SROY_C=E_R=E -> L=E_C=E_R=SROY\n3. L=E_C=SROY_R=E -> L=SROY_C=E_R=E\n4. L=E_C=SROY_R=E -> L=E_C=E_R=SROY\n5. L=E_C=E_R=SROY -> L=SROY_C=E_R=E\n6. L=E_C=E_R=SROY -> L=E_C=SROY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=GLSROY.jpg\nGoal state image: L=E_C=GLSROY_R=E.jpg\nInitial state: L=E_C=E_R=GLSROY\nGoal state: L=E_C=GLSROY_R=E\nAnswer the following questions.\n\nReasoning (depth 2): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=SROY_C=E_R=E -> L=E_C=SROY_R=E\n2. L=SROY_C=E_R=E -> L=E_C=E_R=SROY\n3. L=E_C=SROY_R=E -> L=SROY_C=E_R=E\n4. L=E_C=SROY_R=E -> L=E_C=E_R=SROY\n5. L=E_C=E_R=SROY -> L=SROY_C=E_R=E\n6. L=E_C=E_R=SROY -> L=E_C=SROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=SROY_C=E_R=E -> L=E_C=SROY_R=E\n2. L=SROY_C=E_R=E -> L=E_C=E_R=SROY\n3. L=E_C=SROY_R=E -> L=SROY_C=E_R=E\n4. L=E_C=SROY_R=E -> L=E_C=E_R=SROY\n5. L=E_C=E_R=SROY -> L=SROY_C=E_R=E\n6. L=E_C=E_R=SROY -> L=E_C=SROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=E_R=GLSROY__TO__L=E_C=GLSROY_R=E__Q3",
          "depth": 3,
          "substack_size": 3,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=ROY_C=E_R=E",
              "to_state": "L=E_C=ROY_R=E",
              "text": "L=ROY_C=E_R=E -> L=E_C=ROY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=ROY_C=E_R=E",
              "to_state": "L=E_C=E_R=ROY",
              "text": "L=ROY_C=E_R=E -> L=E_C=E_R=ROY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=ROY_R=E",
              "to_state": "L=ROY_C=E_R=E",
              "text": "L=E_C=ROY_R=E -> L=ROY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=ROY_R=E",
              "to_state": "L=E_C=E_R=ROY",
              "text": "L=E_C=ROY_R=E -> L=E_C=E_R=ROY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=ROY",
              "to_state": "L=ROY_C=E_R=E",
              "text": "L=E_C=E_R=ROY -> L=ROY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=ROY",
              "to_state": "L=E_C=ROY_R=E",
              "text": "L=E_C=E_R=ROY -> L=E_C=ROY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 5,
          "answer_text": "L=E_C=E_R=ROY -> L=ROY_C=E_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=GLSROY.jpg\nGoal state image: L=E_C=GLSROY_R=E.jpg\nInitial state: L=E_C=E_R=GLSROY\nGoal state: L=E_C=GLSROY_R=E\nAnswer the following questions.\n\nQuestion (depth 3): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=GLSROY.jpg\nGoal state image: L=E_C=GLSROY_R=E.jpg\nInitial state: L=E_C=E_R=GLSROY\nGoal state: L=E_C=GLSROY_R=E\nAnswer the following questions.\n\nReasoning (depth 3): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=ROY_C=E_R=E -> L=E_C=ROY_R=E\n2. L=ROY_C=E_R=E -> L=E_C=E_R=ROY\n3. L=E_C=ROY_R=E -> L=ROY_C=E_R=E\n4. L=E_C=ROY_R=E -> L=E_C=E_R=ROY\n5. L=E_C=E_R=ROY -> L=ROY_C=E_R=E\n6. L=E_C=E_R=ROY -> L=E_C=ROY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=E_R=GLSROY__TO__L=E_C=GLSROY_R=E__Q4",
          "depth": 4,
          "substack_size": 2,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=OY_C=E_R=E -> L=E_C=OY_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=OY_C=E_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=OY_C=E_R=E -> L=E_C=E_R=OY",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=OY_R=E -> L=OY_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=OY_R=E",
              "to_state": "L=E_C=E_R=OY",
              "text": "L=E_C=OY_R=E -> L=E_C=E_R=OY",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=OY_C=E_R=E",
              "text": "L=E_C=E_R=OY -> L=OY_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=OY",
              "to_state": "L=E_C=OY_R=E",
              "text": "L=E_C=E_R=OY -> L=E_C=OY_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 6,
          "answer_text": "L=E_C=E_R=OY -> L=E_C=OY_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=GLSROY.jpg\nGoal state image: L=E_C=GLSROY_R=E.jpg\nInitial state: L=E_C=E_R=GLSROY\nGoal state: L=E_C=GLSROY_R=E\nAnswer the following questions.\n\nQuestion (depth 4): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=GLSROY.jpg\nGoal state image: L=E_C=GLSROY_R=E.jpg\nInitial state: L=E_C=E_R=GLSROY\nGoal state: L=E_C=GLSROY_R=E\nAnswer the following questions.\n\nReasoning (depth 4): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=OY_C=E_R=E -> L=E_C=OY_R=E\n2. L=OY_C=E_R=E -> L=E_C=E_R=OY\n3. L=E_C=OY_R=E -> L=OY_C=E_R=E\n4. L=E_C=OY_R=E -> L=E_C=E_R=OY\n5. L=E_C=E_R=OY -> L=OY_C=E_R=E\n6. L=E_C=E_R=OY -> L=E_C=OY_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        },
        {
          "qid": "L=E_C=E_R=GLSROY__TO__L=E_C=GLSROY_R=E__Q5",
          "depth": 5,
          "substack_size": 1,
          "prompt": "Which sub-problem must be solved first to move the bottom ring of the current problem?",
          "options": [
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=Y_C=E_R=E -> L=E_C=Y_R=E",
              "src_pole": "L",
              "tgt_pole": "C"
            },
            {
              "from_state": "L=Y_C=E_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=Y_C=E_R=E -> L=E_C=E_R=Y",
              "src_pole": "L",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=Y_R=E -> L=Y_C=E_R=E",
              "src_pole": "C",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=Y_R=E",
              "to_state": "L=E_C=E_R=Y",
              "text": "L=E_C=Y_R=E -> L=E_C=E_R=Y",
              "src_pole": "C",
              "tgt_pole": "R"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=Y_C=E_R=E",
              "text": "L=E_C=E_R=Y -> L=Y_C=E_R=E",
              "src_pole": "R",
              "tgt_pole": "L"
            },
            {
              "from_state": "L=E_C=E_R=Y",
              "to_state": "L=E_C=Y_R=E",
              "text": "L=E_C=E_R=Y -> L=E_C=Y_R=E",
              "src_pole": "R",
              "tgt_pole": "C"
            },
            {
              "from_state": "",
              "to_state": "",
              "text": "Doesn't exist",
              "src_pole": "",
              "tgt_pole": ""
            }
          ],
          "answer_index": 5,
          "answer_text": "L=E_C=E_R=Y -> L=Y_C=E_R=E",
          "reason_prompt": "Explain why this sub-problem is required. Write a short justification.",
          "reason_response": "",
          "full_prompt_main": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=GLSROY.jpg\nGoal state image: L=E_C=GLSROY_R=E.jpg\nInitial state: L=E_C=E_R=GLSROY\nGoal state: L=E_C=GLSROY_R=E\nAnswer the following questions.\n\nQuestion (depth 5): Which sub-problem must be solved first to move the bottom ring of the current problem?\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nRespond with a single option number.",
          "full_prompt_reason": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=GLSROY.jpg\nGoal state image: L=E_C=GLSROY_R=E.jpg\nInitial state: L=E_C=E_R=GLSROY\nGoal state: L=E_C=GLSROY_R=E\nAnswer the following questions.\n\nReasoning (depth 5): Explain why this sub-problem is required. Write a short justification.\nWrite a short justification.",
          "prompt_turn0": "Problem setup:\n(See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.\n\nState notation:\nWe represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order:\nG, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings):\nL=GLSROY_C=E_R=E\nThis means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nYou are given two images.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text.",
          "prompt_followup": "Use the same two images from the previous turn.\nImage 1 is the Initial state.\nImage 2 is the Goal state.\n\nOptions:\n1. L=Y_C=E_R=E -> L=E_C=Y_R=E\n2. L=Y_C=E_R=E -> L=E_C=E_R=Y\n3. L=E_C=Y_R=E -> L=Y_C=E_R=E\n4. L=E_C=Y_R=E -> L=E_C=E_R=Y\n5. L=E_C=E_R=Y -> L=Y_C=E_R=E\n6. L=E_C=E_R=Y -> L=E_C=Y_R=E\n7. Doesn't exist\n\nTask:\n1) Select the sub-problem that must be solved first. Answer with a single option number.\n2) Explain why the chosen sub-problem is required. Write one or two sentences.\n\nOutput format:\nReturn ONLY a single JSON object on one line with this schema:\n{\"mcq\": <number>, \"reason\": \"<text>\"}\n- mcq: integer option number\n- reason: 1-2 sentences\nDo not output any extra text."
        }
      ],
      "full_problem_prompt": "Problem setup: (See setup image: ./frames/setup.jpg)\nThe scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change.\n\nState notation: We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.\n\nRing identifiers and size order: G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.\n\nExample (6 rings): L=GLSROY_C=E_R=E. This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty.\n\nInitial state image: L=E_C=E_R=GLSROY.jpg\nGoal state image: L=E_C=GLSROY_R=E.jpg\nInitial state: L=E_C=E_R=GLSROY\nGoal state: L=E_C=GLSROY_R=E\nAnswer the following questions."
    }
  ],
  "question_header": {
    "setup_image": "./frames/setup.jpg",
    "problem_setup": "The scene shows a tabletop Tower-of-Hanoi style setup recorded from an egocentric viewpoint. Three poles are placed left to right and remain fixed across the dataset. The poles are denoted as L, C, and R for left, center, and right from the viewer’s perspective. Up to six rings are used. Each ring has a unique color identifier and a unique size rank. The environment is controlled. The background is covered by a gray enclosure. Hands wear sky-blue gloves to reduce appearance variation. The camera viewpoint is not mirrored or flipped. The pole order will not change. Each move relocates exactly one ring. Moving two or more rings at once is not allowed.",
    "state_notation": "We represent a configuration as L=..._C=..._R=.... For each pole, ring identifiers appear in bottom-to-top order. E denotes an empty pole.",
    "ring_identifiers_and_size_order": "G, L, S, R, O, Y correspond to Green, Lime, Sky, Red, Orange, and Yellow. The size order is G > L > S > R > O > Y. G is the largest ring and Y is the smallest ring.",
    "example_6_rings": {
      "state": "L=GLSROY_C=E_R=E",
      "explanation": "This means all six rings are stacked on the left pole. G is at the bottom and Y is at the top. The center and right poles are empty."
    }
  }
}