TUYMAADA 2024 S6

The triangle $ABC$ is given. On the arc $BC$ of its circumscribed circle, which does not contain point $A$ , the variable point $X$ is selected, and on the rays $XB$ and $XC$ , the variable points $Y$ and $Z$ , respectively, so that $XA = XY = XZ$ . Prove that the line $YZ$ passes through a fixed point. Proposed by A. Kuznetsov