# 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 $$m+n$$.

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