NZ-NZMO 2020 Problem 6

Let $\triangle ABC$ be an acute triangle with $AB > AC$ . Let $P$ be the foot of the altitude from $C$ to $AB$ and let $Q$ be the foot of the altitude from $B$ to $AC$ . Let $X$ be the intersection of $PQ$ and $BC$ . Let the intersection of the circumcircles of triangle $\triangle AXC$ and triangle $\triangle PQC$ be distinct points: $C$ and $Y$ . Prove that $PY$ bisects $AX$ .