Question 5
Line $\ell$ intersects sides $BC,AD$ of cyclic quadrilateral $ABCD$ at interior points $R,S$, respectively, and intersects ray $DC$ beyond $C$ at $Q$, and ray $BA$ beyond $A$ at $P$. Circumcircles of $QCR,QDS$ intersect at $N\ne Q$, while circumcircles of $PAS,PBR$ intersect at $M\ne P$. Let $MP$ and $NQ$ meet at $X$, $AB$ and $CD$ meet at $K$, and $BC$ and $AD$ meet at $L$. Prove that $X$ lies on $KL$.
AI-generated configuration (the line $KL$ is not added):
AI-generated configuration (the line $KL$ is not added):