SHARYGIN 2016 Problem 16

Let $BB_1$ and $CC_1$ be altitudes of triangle $ABC$ . The tangents to the circumcircle of $AB_1C_1$ at $B_1$ and $C_1$ meet AB and $AC$ at points $M$ and $N$ respectively. Prove that the common point of circles $AMN$ and $AB_1C_1$ distinct from $A$ lies on the Euler line of $ABC$ .