SHARYGIN 2018 Problem 7

Let $E$ be a common point of circles $w_1$ and $w_2$. Let $AB$ be a common tangent to these circles, and $CD$ be a line parallel to $AB$, such that $A$ and $C$ lie on $w_1$, $B$ and $D$ lie on $w_2$. The circles $ABE$ and $CDE$ meet for the second time at point $F$. Prove that $F$ bisects one of arcs $CD$ of circle $CDE$.