Do not use calculus!

Algebra Level 3

Let

\(A = {\frac{(x + \frac{1}{x})^{6} - (x^{6} + \frac{1}{x^{6}}) - 2}{(x + \frac{1}{x})^{3} + (x^{3} + \frac{1}{x^{3}})}}\)

Find \(min(A)\).

Clarifications: \(x\) is a positive real number.

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