PL-OM 2022 Problem 5

Let $ABC$ be a triangle satisfying $AB<AC$ . Let $M$ be the midpoint of $BC$ . A point $P$ lies on the segment $AB$ with $AP>PB$ . A point $Q$ lies on the segment $AC$ with $\angle MPA = \angle AQM$ . The perpendicular bisectors of $BC$ and $PQ$ intersect at $S$ . Prove that $\angle BAC + \angle QSP = \angle QMP$ .