HMMT 2017 U10

Let $A B C$ be a triangle in the plane with $A B=13, B C=14, A C=15$. Let $M_{n}$ denote the smallest possible value of $\left(A P^{n}+B P^{n}+C P^{n}\right)^{\frac{1}{n}}$ over all points $P$ in the plane. Find $\lim _{n \rightarrow \infty} M_{n}$.