KMO 2022 Problem 2

In a scalene triangle $ABC$ , let the angle bisector of $A$ meets side $BC$ at $D$ . Let $E, F$ be the circumcenter of the triangles $ABD$ and $ADC$ , respectively. Suppose that the circumcircles of the triangles $BDE$ and $DCF$ intersect at $P(\neq D)$ , and denote by $O, X, Y$ the circumcenters of the triangles $ABC, BDE, DCF$ , respectively. Prove that $OP$ and $XY$ are parallel.