2006 AIME II Problem 15

Given that $x$, $y$, and $z$ are real numbers that satisfy:

$x=\sqrt{y^2-\frac{1}{16}}+\sqrt{z^2-\frac{1}{16}}$
$y=\sqrt{z^2-\frac{1}{25}}+\sqrt{x^2-\frac{1}{25}}$
$z=\sqrt{x^2-\frac{1}{36}}+\sqrt{y^2-\frac{1}{36}}$

and that $x+y+z=\frac{m}{\sqrt{n}}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime, find $m+n$.

Leading zeroes must be inputted, so if your answer is 34, then input 034. Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.