AIME 2024a Problem 10

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