SILK-ROAD 2020 Problem 2

The triangle $ABC$ is inscribed in the circle $\omega$ . Points $K, L, M$ are marked on the sides $AB, BC, CA$ , respectively, and $CM \cdot CL = AM \cdot BL$ . Ray $LK$ intersects line $AC$ at point $P$ . The common chord of the circle $\omega$ and the circumscribed circle of the triangle $KMP$ meets the segment $AM$ at the point $S$ . Prove that $SK \parallel BC$ .