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)