SHARYGIN 2021 Problem 6

Three circles $\Gamma_1,\Gamma_2,\Gamma_3$ are inscribed into an angle(the radius of $\Gamma_1$ is the minimal, and the radius of $\Gamma_3$ is the maximal) in such a way that $\Gamma_2$ touches $\Gamma_1$ and $\Gamma_3$ at points $A$ and $B$ respectively. Let $\ell$ be a tangent to $A$ to $\Gamma_1$ . Consider circles $\omega$ touching $\Gamma_1$ and $\ell$ . Find the locus of meeting points of common internal tangents to $\omega$ and $\Gamma_3$ .