USAJMO 2013 Question 3

In triangle $ABC$, points $P,Q,R$ lie on $BC,CA,AB$, respectively. Let $\omega_A,\omega_B,\omega_C$ be the circumcircles of $AQR,BRP,CPQ$. Segment $AP$ meets these circles again at $X,Y,Z$, respectively. Prove $YX/XZ=BP/PC$.

Proposed by Zuming Feng (as credited in the supplied compilation).