A weird system of equation

Geometry Level 5

\[ \begin{eqnarray} x&=&\sqrt{y^2-\dfrac{1}{16}}+\sqrt{z^2-\dfrac{1}{16}} \\ y&=&\sqrt{z^2-\dfrac{1}{25}}+\sqrt{x^2-\dfrac{1}{25}} \\ z&=&\sqrt{x^2-\dfrac{1}{36}}+\sqrt{y^2-\dfrac{1}{36}} \\ \end{eqnarray} \]

Given that \(x, y \) and \(z\) are real numbers that satisfy the system of equations above. If \(x+y+z=\dfrac{m}{\sqrt{n}}\) where \(m, n\) are positive integers and \(n\) is square-free, find the value of \(m+n\).

This problem is not original
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