SHARYGIN 2021 Problem 14

Let $\gamma_A, \gamma_B, \gamma_C$ be excircles of triangle $ABC$ , touching the sides $BC$ , $CA$ , $AB$ respectively. Let $l_A$ denote the common external tangent to $\gamma_B$ and $\gamma_C$ distinct from $BC$ . Define $l_B, l_C$ similarly. The tangent from a point $P$ of $l_A$ to $\gamma_B$ distinct from $l_A$ meets $l_C$ at point $X$ . Similarly the tangent from $P$ to $\gamma_C$ meets $l_B$ at $Y$ . Prove that $XY$ touches $\gamma_A$ .