Question 25
In the triangle $ABC$, $AB=585$, $BC=520$, $CA=455$. Let $P,Q$ be points on the side $BC$, and $R\ne A$ the intersection of the line $AQ$ with the circumcircle $\omega$ of the triangle $ABC$. Suppose $PR$ is parallel to $AC$ and the circumcircle of the triangle $PQR$ is tangent to $\omega$ at $R$. Find $PQ$.