CN-LEAGUE 2014 Problem 2

Let $ABC$ be an acute triangle such that $\angle BAC \neq 60^\circ$ . Let $D,E$ be points such that $BD,CE$ are tangent to the circumcircle of $ABC$ and $BD=CE=BC$ ( $A$ is on one side of line $BC$ and $D,E$ are on the other side). Let $F,G$ be intersections of line $DE$ and lines $AB,AC$ . Let $M$ be intersection of $CF$ and $BD$ , and $N$ be intersection of $CE$ and $BG$ . Prove that $AM=AN$ .