RU-ALLRUSSIAN 2016 Problem 97

In triangle $ABC$ , $AB<AC$ and $\omega$ is incirle.The $A$ -excircle is tangent to $BC$ at $A^\prime$ .Point $X$ lies on $AA^\prime$ such that segment $A^\prime X$ doesn't intersect with $\omega$ .The tangents from $X$ to $\omega$ intersect with $BC$ at $Y,Z$ .Prove that the sum $XY+XZ$ not depends to point $X$ .(Mitrofanov)