Advanced Number Theory · Hard problem

In $\triangle ABC$, $AB = 10$, $BC = 14$, and $CA = 12$. Let $M$ be the midpoint of $\overline{BC}$. The circle passing through $A$, $B$, and $M$ meets line $AC$ again at a point $P$ (other than $A$). Then $AP = \dfrac{p}{q}$ for relatively prime positive integers $p, q$. Find $p+q$.