KZ-MO 2008 Problem 5

Let $\triangle ABC$ be a triangle and let $K$ be some point on the side $AB$ , so that the tangent line from $K$ to the incircle of $\triangle ABC$ intersects the ray $AC$ at $L$ . Assume that $\omega$ is tangent to sides $AB$ and $AC$ , and to the circumcircle of $\triangle AKL$ . Prove that $\omega$ is tangent to the circumcircle of $\triangle ABC$ as well.