USAJMO 2012 Question 1

In triangle $ABC$, let $P\in AB$, $Q\in AC$ satisfy $AP=AQ$. Let $S,R$ be distinct points on $BC$, with $S$ between $B$ and $R$, such that $\angle BPS=\angle PRS$ and $\angle CQR=\angle QSR$. Prove that $P,Q,R,S$ lie on one circle.

Proposed by Sungyoon Kim, Inseok Seo (credit in the supplied notes).