SHARYGIN 2022 Problem 85

An incircle of triangle $ABC$ touches $AB$ , $BC$ , $AC$ at points $C_1$ , $A_1$ , $B_1$ respectively. Let $A'$ be the reflection of $A_1$ about $B_1C_1$ , point $C'$ is defined similarly. Lines $A'C_1$ and $C'A_1$ meet at point $D$ . Prove that $BD \parallel AC$ .