An algebra problem by Sourabh Jangid

$\begin{cases} x = \sqrt{y^2 - \frac1{16} } + \sqrt{z^2 - \frac1{16}} \\ y = \sqrt{z^2 - \frac1{25} } + \sqrt{x^2 - \frac1{25}} \\ z = \sqrt{x^2 - \frac1{36} } + \sqrt{y^2 - \frac1{36}} \\ \end{cases}$

Let $x,y$ and $z$ be real numbers satisfying the system of equations above. If the value of $x+y+z$ can be expressed as $\dfrac m{\sqrt n}$, where $m$ and $n$ are positive integers with $n$ square-free, find $m+n$.

This problem is from AIME 2006.