IGO 2019 A4

Given an acute non-isosceles triangle $ABC$ with circumcircle $\Gamma$ . $M$ is the midpoint of segment $BC$ and $N$ is the midpoint of arc $BC$ of $\Gamma$ (the one that doesn't contain $A$ ). $X$ and $Y$ are points on $\Gamma$ such that $BX\parallel CY\parallel AM$ . Assume there exists point $Z$ on segment $BC$ such that circumcircle of triangle $XYZ$ is tangent to $BC$ . Let $\omega$ be the circumcircle of triangle $ZMN$ . Line $AM$ meets $\omega$ for the second time at $P$ . Let $K$ be a point on $\omega$ such that $KN\parallel AM$ , $\omega_b$ be a circle that passes through $B$ , $X$ and tangents to $BC$ and $\omega_c$ be a circle that passes through $C$ , $Y$ and tangents to $BC$ . Prove that circle with center $K$ and radius $KP$ is tangent to 3 circles $\omega_b$ , $\omega_c$ and $\Gamma$ .

Proposed by Tran Quan - Vietnam