{"task": {"agent_timeout": 3000, "task": "aime_78", "verifier_timeout": 3000, "instruction": "  Solve the following problem. Reason step by step, create the file `/app/answer.txt`, and put your final answer there.\n  Your answer should be a single integer from 1 to 999 inclusive.\n\n    Let \\(O=(0,0)\\), \\(A=\\left(\\tfrac{1}{2},0\\right)\\), and \\(B=\\left(0,\\tfrac{\\sqrt{3}}{2}\\right)\\) be points in the coordinate plane. Let \\(\\mathcal{F}\\) be the family of segments \\(\\overline{PQ}\\) of unit length lying in the first quadrant with \\(P\\) on the \\(x\\)-axis and \\(Q\\) on the \\(y\\)-axis. There is a unique point \\(C\\) on \\(\\overline{AB}\\), distinct from \\(A\\) and \\(B\\),  that does not belong to any segment from \\(\\mathcal{F}\\) other than \\(\\overline{AB}\\). Then \\(OC^2=\\tfrac{p}{q}\\), where \\(p\\) and \\(q\\) are relatively prime positive integers. Find \\(p+q\\).\n", "memory": "", "runnable": false, "difficulty": "unknown", "language": "", "cpus": "", "instruction_truncated": false, "category": "reasoning", "compose": false, "has_solution": true, "oracle": null, "docker_image": "", "taskset": "aime", "tags": ["math", "AIME"]}, "runs": []}