Let $ABCD$ be a quadrilateral, such that $AB = BC = CD.$ There are points $X, Y$ on rays $CA, BD,$ respectively, such that $BX = CY.$ Let $P, Q, R, S$ be the midpoints of segments $BX, CY ,$ $XD, YA,$ respectively. Prove that points $P, Q, R, S$ lie on a circle.