CGMO 2021 Problem 2

In acute triangle $ABC$ ( $AB \neq AC$ ), $I$ is its incenter and $J$ is the $A$ -excenter. $X, Y$ are on minor arcs $\widehat{AB}$ and $\widehat{AC}$ respectively such that $\angle{AXI}=\angle{AYJ}=90^{\circ}$ . $K$ is on line $BC$ such that $KI=KJ$ . Proof that line $AK$ bisects $\overline{XY}$ .