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 \(m+n.\)

(第二十四届AIME2 2006 第15题)