PL-OM 2016 Problem 2

Let $ABCD$ be a quadrilateral circumscribed on the circle $\omega$ with center $I$ . Assume $\angle BAD+ \angle ADC <\pi$ . Let $M, \ N$ be points of tangency of $\omega$ with $AB, \ CD$ respectively. Consider a point $K \in MN$ such that $AK=AM$ . Prove that $ID$ bisects the segment $KN$ .