2024 AIME II Problem 12

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.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.