OME 2016 Problem 3

In the circumscircle of a triangle $ABC$ , let $A_1$ be the point diametrically opposed to the vertex $A$ . Let $A'$ the intersection point of $AA'$ and $BC$ . The perpendicular to the line $AA'$ from $A'$ meets the sides $AB$ and $AC$ at $M$ and $N$ , respectively. Prove that the points $A,M,A_1$ and $N$ lie on a circle which has the center on the height from $A$ of the triangle $ABC$ .