BALTIC-WAY 2020 Problem 14

An acute triangle $A B C$ is given and let $H$ be its orthocenter. Let $\omega$ be the circle through $B, C$ and $H$, and let $\Gamma$ be the circle with diameter $A H$. Let $X \neq H$ be the other intersection point of $\omega$ and $\Gamma$, and let $\gamma$ be the reflection of $\Gamma$ over $A X$.

Suppose $\gamma$ and $\omega$ intersect again at $Y \neq X$, and line $A H$ and $\omega$ intersect again at $Z \neq H$. Show that the circle through $A, Y, Z$ passes through the midpoint of segment $B C$.