ZA-SAMO 2022 Problem 4

Let $ABC$ be a triangle with $AB < AC$ . A point $P$ on the circumcircle of $ABC$ (on the same side of $BC$ as $A$ ) is chosen in such a way that $BP = CP$ . Let $BP$ and the angle bisector of $\angle BAC$ intersect at $Q$ , and let the line through $Q$ and parallel to $BC$ intersect $AC$ at $R$ . Prove that $BR = CR$ .