RU-239-OPEN 2023 Problem 2

The excircles of triangle $ABC$ touch its sides $BC$ , $CA$ , and $AB$ at points $A_1$ , $B_1$ , and $C_1$ , respectively. Let $B_2$ and $C_2$ be the midpoints of segments $BB_1$ and $CC_1$ , respectively. Line $B_2C_2$ intersects line $BC$ at point $W$ . Prove that $AW = A_1W$ .