Awesome Inequality

Algebra Level 5

\[\large \left(\dfrac{x}{y}+\dfrac{y}{z}+\dfrac{z}{x}\right)\left(\dfrac{y}{x}+\dfrac{z}{y}+\dfrac{x}{z}\right) \]

Let \(S\) be the minimum value of the above expression for positive reals \(x,y\) and \(z\) satisfying \[\left(\dfrac{x}{y}+\dfrac{y}{z}+\dfrac{z}{x}\right)+\left(\dfrac{y}{x}+\dfrac{z}{y}+\dfrac{x}{z}\right)=8\; . \]

Given that \(S\) can be expressed as \(a\sqrt{b}+c\) where \(a,b\) and \(c\) are integers and \(b\) square-free, find \(a+b+c\).

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