NZ-NZMO 2024 Problem 6

Let $\omega$ be the incircle of scalene triangle $ABC$ . Let $\omega$ be tangent to $AB$ and $AC$ at points $X$ and $Y$ . Construct points $X'$ and $Y'$ on line segments $AB$ and $AC$ respectively such that $AX' = XB$ and $AY' = YC$ . Let line $CX'$ intersects $\omega$ at points $P, Q$ such that $P$ is closer to $C$ than $Q$ . Also let $R$ be the intersection of lines $CX'$ and $BY'$ . Prove that $CP = RX'$ .