TopsOJ Summer 2025 - Team Round - Problem 14

ABC is an acute triangle with AB=13,BC=14,CA=15. P is a point in the interior of ABC such that AP=\sqrt{65} and \angle BPC=90^{\circ}+\frac{1}{2}\angle BAC. Extend AP to meet BC at X. The tangent to the circumcircle of BPC at P intersects BC at Y. Given that XY=\frac{p}{q} where p,q are coprime, compute p+q.