CONO-SUR 2021 Problem 6

Let $ABC$ be a scalene triangle with circle $\Gamma$ . Let $P,Q,R,S$ distinct points on the $BC$ side, in that order, such that $\angle BAP = \angle CAS$ and $\angle BAQ = \angle CAR$ . Let $U, V, W, Z$ be the intersections, distinct from $A$ , of the $AP, AQ, AR$ and $AS$ with $\Gamma$ , respectively. Let $X = UQ \cap SW$ , $Y = PV \cap ZR$ , $T = UR \cap VS$ and $K = PW \cap ZQ$ . Suppose that the points $M$ and $N$ are well determined, such that $M = KX \cap TY$ and $N = TX \cap KY$ . Show that $M, N, A$ are collinear.