HMMT 2020 T3

Let $A B C$ be a triangle inscribed in a circle $\omega$ and $\ell$ be the tangent to $\omega$ at $A$. The line through $B$ parallel to $A C$ meets $\ell$ at $P$, and the line through $C$ parallel to $A B$ meets $\ell$ at $Q$. The circumcircles of $A B P$ and $A C Q$ meet at $S \neq A$. Show that $A S$ bisects $B C$.