{"task": {"agent_timeout": 600, "task": "omnimath_1637", "verifier_timeout": 60, "instruction": "# Mathematical Problem\n\nLet $A B C$ be a triangle with $A B=7, B C=9$, and $C A=4$. Let $D$ be the point such that $A B \\| C D$ and $C A \\| B D$. Let $R$ be a point within triangle $B C D$. Lines $\\ell$ and $m$ going through $R$ are parallel to $C A$ and $A B$ respectively. Line $\\ell$ meets $A B$ and $B C$ at $P$ and $P^{\\prime}$ respectively, and $m$ meets $C A$ and $B C$ at $Q$ and $Q^{\\prime}$ respectively. If $S$ denotes the largest possible sum of the areas of triangles $B P P^{\\prime}, R P^{\\prime} Q^{\\prime}$, and $C Q Q^{\\prime}$, determine the value of $S^{2}$.\n\n## Instructions\n\nSolve the mathematical problem above and write your final answer to `/workspace/answer.txt`.\n\n**Important**: Write only your final answer to the file, not the full solution process.\n\n### Guidelines\n\n- Provide your final numerical answer or mathematical expression\n- Write the answer as plain text (no special formatting needed)\n- Be precise and clear in your answer\n- The answer should directly respond to what the problem asks for\n\n### Example\n\nIf the problem asks \"What is 2 + 2?\", your answer file should contain:\n\n```\n4\n```\n\nYour answer will be evaluated against the correct solution using an automated grading system.\n", "memory": "", "runnable": false, "difficulty": "hard", "language": "", "cpus": "", "instruction_truncated": false, "category": "math", "compose": false, "has_solution": true, "oracle": null, "docker_image": "", "taskset": "omnimath", "tags": ["omni-math"]}, "runs": []}