OMM 2017 Problem 5

On a circle $\Gamma$ , points $A, B, N, C, D, M$ are chosen in a clockwise order in such a way that $N$ and $M$ are the midpoints of clockwise arcs $BC$ and $AD$ respectively. Let $P$ be the intersection of $AC$ and $BD$ , and let $Q$ be a point on line $MB$ such that $PQ$ is perpendicular to $MN$ . Point $R$ is chosen on segment $MC$ such that $QB = RC$ , prove that the midpoint of $QR$ lies on $AC$ .