BG-MO 2016 Problem 5

Let $\triangle {ABC}$ be isosceles triangle with $AC=BC$ . The point $D$ lies on the extension of $AC$ beyond $C$ and is that $AC>CD$ . The angular bisector of $\angle BCD$ intersects $BD$ at point $N$ and let $M$ be the midpoint of $BD$ . The tangent at $M$ to the circumcircle of triangle $AMD$ intersects the side $BC$ at point $P$ . Prove that points $A,P,M$ and $N$ lie on a circle.