RU-ALLRUSSIAN 2018 Problem 92

Circle $\omega$ is tangent to sides $AB, AC$ of triangle $ABC$ . A circle $\Omega$ touches the side $AC$ and line $AB$ (produced beyond $B$ ), and touches $\omega$ at a point $L$ on side $BC$ . Line $AL$ meets $\omega, \Omega$ again at $K, M$ . It turned out that $KB \parallel CM$ . Prove that $\triangle LCM$ is isosceles.