DE-MO 2017 Problem 2

Let $ABC$ be a triangle such that $\vert AB\vert \ne \vert AC\vert$ . Prove that there exists a point $D \ne A$ on its circumcircle satisfying the following property: For any points $M, N$ outside the circumcircle on the rays $AB$ and $AC$ , respectively, satisfying $\vert BM\vert=\vert CN\vert$ , the circumcircle of $AMN$ passes through $D$ .