IR-TST 2018 Problem 13

Two circles $\omega_1(O)$ and $\omega_2$ intersect each other at $A,B$ ,and $O$ lies on $\omega_2$ . Let $S$ be a point on $AB$ such that $OS\perp AB$ . Line $OS$ intersects $\omega_2$ at $P$ (other than $O$ ). The bisector of $\hat{ASP}$ intersects $\omega_1$ at $L$ ( $A$ and $L$ are on the same side of the line $OP$ ). Let $K$ be a point on $\omega_2$ such that $PS=PK$ ( $A$ and $K$ are on the same side of the line $OP$ ). Prove that $SL=KL$ .

Proposed by Ali Zamani