BALTIC-WAY 2020 Problem 12

Let $A B C$ be a triangle with circumcircle $\omega$. The internal angle bisectors of $\angle A B C$ and $\angle A C B$ intersect $\omega$ at $X \neq B$ and $Y \neq C$, respectively. Let $K$ be a point on $C X$ such that $\angle K A C=90^{\circ}$. Similarly, let $L$ be a point on $B Y$ such that $\angle L A B=90^{\circ}$. Let $S$ be the midpoint of $\operatorname{arc} C A B$ of $\omega$. Prove that $S K=S L$.