BALKAN 2022 Problem 1

Problem. Let $A B C$ be an acute triangle such that $C A \neq C B$ with circumcircle $\omega$ and circumcentre $O$. Let $t_{A}$ and $t_{B}$ be the tangents to $\omega$ at $A$ and $B$ respectively, which meet at $X$. Let $Y$ be the foot of the perpendicular from $O$ onto the line segment $C X$. The line through $C$ parallel to line $A B$ meets $t_{A}$ at $Z$. Prove that the line $Y Z$ passes through the midpoint of the line segment $A C$.