SHARYGIN 2014 Problem 19

Two circles $\omega_1$ and $\omega_2$ touch externally at point $P$. Let $A$ be a point of $\omega_2$ not lying on the line through the centers of the circles, and $AB, AC$ be the tangents to $\omega_1$. Lines $BP, CP$ meet $\omega_2$ for the second time at points $E$ and $F$. Prove that line $EF$, the tangent to $\omega_2$ at point $A$ and the common tangent at $P$ concur.