OMCC 2008 Problem 6

Let $ABC$ be an acute triangle. Take points $P$ and $Q$ inside $AB$ and $AC$ , respectively, such that $BPQC$ is cyclic. The circumcircle of $ABQ$ intersects $BC$ again in $S$ and the circumcircle of $APC$ intersects $BC$ again in $R$ , $PR$ and $QS$ intersect again in $L$ . Prove that the intersection of $AL$ and $BC$ does not depend on the selection of $P$ and $Q$ .