OSN 2014 Problem 6

Let $ABC$ be a triangle. Suppose $D$ is on $BC$ such that $AD$ bisects $\angle BAC$ . Suppose $M$ is on $AB$ such that $\angle MDA = \angle ABC$ , and $N$ is on $AC$ such that $\angle NDA = \angle ACB$ . If $AD$ and $MN$ intersect on $P$ , prove that $AD^3 = AB \cdot AC \cdot AP$ .