Two Circles Tangent at a Vertex

Triangle $ABC$ has side lengths $AB=7$, $BC=8$, $CA=9$. A circle $\omega_1$ passes through $B$ and is tangent to the line $AC$ at $A$; a circle $\omega_2$ passes through $C$ and is tangent to the line $AB$ at $A$. Let $K$ be the intersection point of $\omega_1$ and $\omega_2$ other than $A$. Given that $AK$, written as a fraction in lowest terms, equals $\dfrac pq$, find $p+q$.