# It is indeed a Geometry Problem!

Geometry Level 5

\begin{cases} \begin{align}xy+yz+zx&=1 \\\\ \dfrac{6}{x+\frac{1}{x}}=\dfrac{8}{y+\frac{1}{y}}&=\dfrac{10}{z+\frac{1}{z}}\end{align} \end{cases}

$$x,y,$$ and $$z$$ are real numbers satisfying the system of equations above.

If $$x^2+y^2+z^2 = \frac{m}{n}$$, where $$m$$ and $$n$$ are coprime positive integers, find $$m+n$$.

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