2024 AIME I Problem 10

Let \triangle ABC have side lengths AB = 5, BC = 9, and CA = 10. The tangents to the circumcircle of \triangle ABC at B and C intersect at point D, and \overline{AD} intersects the circumcircle at P \ne A. The length of AP is equal to \frac{m}{n}, where m and n are relatively prime integers. Find m + n.

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.