IGO 2022 I5

Let $ABCD$ be a quadrilateral inscribed in a circle $\omega$ with center $O$ . Let $P$ be the intersection of two diagonals $AC$ and $BD$ . Let $Q$ be a point lying on the segment $OP$ . Let $E$ and $F$ be the orthogonal projections of $Q$ on the lines $AD$ and $BC$ , respectively. The points $M$ and $N$ lie on the circumcircle of triangle $QEF$ such that $QM \parallel AC$ and $QN \parallel BD$ . Prove that the two lines $ME$ and $NF$ meet on the perpendicular bisector of segment $CD$ .

Proposed by Tran Quang Hung, Vietnam