CN-TST 2021 Problem 5

Given a triangle $ABC$ , a circle $\Omega$ is tangent to $AB,AC$ at $B,C,$ respectively. Point $D$ is the midpoint of $AC$ , $O$ is the circumcenter of triangle $ABC$ . A circle $\Gamma$ passing through $A,C$ intersects the minor arc $BC$ on $\Omega$ at $P$ , and intersects $AB$ at $Q$ . It is known that the midpoint $R$ of minor arc $PQ$ satisfies that $CR \perp AB$ . Ray $PQ$ intersects line $AC$ at $L$ , $M$ is the midpoint of $AL$ , $N$ is the midpoint of $DR$ , and $X$ is the projection of $M$ onto $ON$ . Prove that the circumcircle of triangle $DNX$ passes through the center of $\Gamma$ .