SHARYGIN 2014 Problem 21

Let $ABCD$ be a circumscribed quadrilateral. Its incircle $\omega$ touches sides $BC$ and $DA$ at points $E$ and $F$ respectively. It is known that lines $AB, FE$ and $CD$ concur. The circumcircles of triangles $AED$ and $BFC$ meet $\omega$ for the second time at points $E_1$ and $F_1$. Prove that $EF \parallel E_1F_1$.