TH-TMO 2022 Problem 1

Let $BC$ be a chord of a circle $\Gamma$ and $A$ be a point inside $\Gamma$ such that $\angle BAC$ is acute. Outside $\triangle ABC$ , construct two isosceles triangles $\triangle ACP$ and $\triangle ABR$ such that $\angle ACP$ and $\angle ABR$ are right angles. Let lines $BA$ and $CA$ meet $\Gamma$ again at points $E$ and $F$ , respectively. Let lines $EP$ and $FR$ meet $\Gamma$ again at points $X$ and $Y$ , respectively. Prove that $BX=CY$ .