HMMT November 2022 U30

Let $A B C$ be a triangle with $A B=8, A C=12$, and $B C=5$. Let $M$ be the second intersection of the internal angle bisector of $\angle B A C$ with the circumcircle of $A B C$. Let $\omega$ be the circle centered at $M$ tangent to $A B$ and $A C$. The tangents to $\omega$ from $B$ and $C$, other than $A B$ and $A C$ respectively, intersect at a point $D$. Compute $A D$.