HMMT 2014 G10

Let $A B C$ be a triangle with $A B=13, B C=14$, and $C A=15$. Let $\Gamma$ be the circumcircle of $A B C$, let $O$ be its circumcenter, and let $M$ be the midpoint of minor arc $\widehat{B C}$. Circle $\omega_{1}$ is internally tangent to $\Gamma$ at $A$, and circle $\omega_{2}$, centered at $M$, is externally tangent to $\omega_{1}$ at a point $T$. Ray $A T$ meets segment $B C$ at point $S$, such that $B S-C S=4 / 15$. Find the radius of $\omega_{2}$.