# 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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