HMMT 2024 G9

Let $A B C$ be a triangle. Let $X$ be the point on side $\overline{A B}$ such that $\angle B X C=60^{\circ}$. Let $P$ be the point on segment $\overline{C X}$ such that $B P \perp A C$. Given that $A B=6, A C=7$, and $B P=4$, compute $C P$.