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Algebra Level 4

\[f(x,y,z) = 2x^2+2y^2-2z^2+\dfrac{7}{xy}+\dfrac{1}{z}\]

There exists three pairwise distinct numbers \(a,b\) and \(c\) that satisfies

\[f(a,b,c) = f(b,c,a) = f(c,a,b)\]

Given that \(abc = \dfrac{m}{n}\), where \(m\) and \(n\) are positive coprime integers, find the value of \(m+n\)


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