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

\(x,y,\) 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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