An algebra problem by Sourabh Jangid

Algebra Level 5

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

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 \(m+n\).