ISL 2022 G4

Let $A B C$ be an acute-angled triangle with $A C>A B$, let $O$ be its circumcentre, and let $D$ be a point on the segment $B C$. The line through $D$ perpendicular to $B C$ intersects the lines $A O, A C$ and $A B$ at $W, X$ and $Y$, respectively. The circumcircles of triangles $A X Y$ and $A B C$ intersect again at $Z \neq A$. Prove that if $O W=O D$, then $D Z$ is tangent to the circle $A X Y$. (United Kingdom)