CGMO 2016 Problem 2

In $\triangle ABC, BC=a, CA=b, AB=c,$ and $\Gamma$ is its circumcircle. $(1)$ Determine a necessary and sufficient condition on $a,b$ and $c$ if there exists a unique point $P(P\neq B, P\neq C)$ on the arc $BC$ of $\Gamma$ not passing through point $A$ such that $PA=PB+PC$ . $(2)$ Let $P$ be the unique point stated in $(1)$ . If $AP$ bisects $BC$ , prove that $\angle BAC<60^{\circ}$ .