ZA-SAMO 2017 Problem 5

Let $ABC$ be a triangle with circumcircle $\Gamma$ . Let $D$ be a point on segment $BC$ such that $\angle BAD = \angle DAC$ , and let $M$ and $N$ be points on segments $BD$ and $CD$ , respectively, such that $\angle MAD = \angle DAN$ . Let $S, P$ and $Q$ (all different from $A$ ) be the intersections of the rays $AD$ , $AM$ and $AN$ with $\Gamma$ , respectively.

Show that the intersection of $SM$ and $QD$ lies on $\Gamma$ .