HMMT 2021 U26

Let triangle $A B C$ have incircle $\omega$, which touches $B C, C A$, and $A B$ at $D, E$, and $F$, respectively. Then, let $\omega_{1}$ and $\omega_{2}$ be circles tangent to $A D$ and internally tangent to $\omega$ at $E$ and $F$, respectively. Let $P$ be the intersection of line $E F$ and the line passing through the centers of $\omega_{1}$ and $\omega_{2}$. If $\omega_{1}$ and $\omega_{2}$ have radii 5 and 6 , respectively, compute $P E \cdot P F$.