BMO Round 2, 2024

Let $ABC$ be an acute-angled triangle with $AB > AC$. Let $P$ be the intersection of the tangents to the circumcircle of $ABC$ at $B$ and $C$. The line through the midpoints of line segments $PB$ and $PC$ meets lines $AB$ and $AC$ at $X$ and $Y$ respectively.

Prove that the quadrilateral $AXPY$ is cyclic.