IR-MO-R2 2019 Problem 2

$ABC$ is an isosceles triangle ( $AB=AC$ ). Point $X$ is an arbitrary point on $BC$ . $Z \in AC$ and $Y \in AB$ such that $\angle BXY = \angle ZXC$ . A line parallel to $YZ$ passes through $B$ and cuts $XZ$ at $T$ . Prove that $AT$ bisects $\angle A$ .