The points $A, B$ and $C$ lie on the surface of a sphere with center $O$ and radius 20.  It is given that $AB=13, BC=14, CA=15,$ and that the distance from $O$ to triangle $ABC$ is $\frac{m\sqrt{n}}k,$ where $m, n,$ and $k$ are positive integers, $m$ and $k$ are relatively prime, and $n$ is not divisible by the square of any prime.  Find $m+n+k.$