SHARYGIN 2019 Problem 15

The incircle $\omega$ of triangle $ABC$ touches the sides $BC$ , $CA$ and $AB$ at points $D$ , $E$ and $F$ respectively. The perpendicular from $E$ to $DF$ meets $BC$ at point $X$ , and the perpendicular from $F$ to $DE$ meets $BC$ at point $Y$ . The segment $AD$ meets $\omega$ for the second time at point $Z$ . Prove that the circumcircle of the triangle $XYZ$ touches $\omega$ .