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Algebra Level 3

\[\large \frac{\left( x + \frac{1}{x} \right)^6 - (x^6 + \frac{1}{x^6}) - 2}{\left( x + \frac{1}{x} \right)^3 + (x^3 + \frac{1}{x^3})}\]

Find the minimum value of the expression above such that \(x\) is positive.

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