OIM 2020 Problem 6

Let $ABC$ be an acute, scalene triangle. Let $H$ be the orthocenter and $O$ be the circumcenter of triangle $ABC$ , and let $P$ be a point interior to the segment $HO.$ The circle with center $P$ and radius $PA$ intersects the lines $AB$ and $AC$ again at $R$ and $S$ , respectively. Denote by $Q$ the symmetric point of $P$ with respect to the perpendicular bisector of $BC$ . Prove that points $P$ , $Q$ , $R$ and $S$ lie on the same circle.