SHARYGIN 2012 Problem 22

A circle $\omega$ with center $I$ is inscribed into a segment of the disk, formed by an arc and a chord $AB$ . Point $M$ is the midpoint of this arc $AB$ , and point $N$ is the midpoint of the complementary arc. The tangents from $N$ touch $\omega$ in points $C$ and $D$ . The opposite sidelines $AC$ and $BD$ of quadrilateral $ABCD$ meet in point $X$ , and the diagonals of $ABCD$ meet in point $Y$ . Prove that points $X, Y, I$ and $M$ are collinear.