US-TST 2014 Problem 1

Let $A B C$ be an acute triangle, and let $X$ be a variable interior point on the minor $\operatorname{arc} B C$ of its circumcircle. Let $P$ and $Q$ be the feet of the perpendiculars from $X$ to lines $C A$ and $C B$, respectively. Let $R$ be the intersection of line $P Q$ and the perpendicular from $B$ to $A C$. Let $\ell$ be the line through $P$ parallel to $X R$. Prove that as $X$ varies along minor arc $B C$, the line $\ell$ always passes through a fixed point.