SHARYGIN 2012 Problem 16

Given right-angled triangle $ABC$ with hypothenuse $AB$ . Let $M$ be the midpoint of $AB$ and $O$ be the center of circumcircle $\omega$ of triangle $CMB$ . Line $AC$ meets $\omega$ for the second time in point $K$ . Segment $KO$ meets the circumcircle of triangle $ABC$ in point $L$ . Prove that segments $AL$ and $KM$ meet on the circumcircle of triangle $ACM$ .