SHARYGIN 2023 Problem 11

Let $H$ be the orthocenter of an acute-angled triangle $ABC$ ; $E$ , $F$ be points on $AB, AC$ respectively, such that $AEHF$ is a parallelogram; $X, Y$ be the common points of the line $EF$ and the circumcircle $\omega$ of triangle $ABC$ ; $Z$ be the point of $\omega$ opposite to $A$ . Prove that $H$ is the orthocenter of triangle $XYZ$ .