Tasty problem to be digested

Algebra Level 4

\[\large \sum_{\text{cyc}} \frac {x^3}{1 - x^8} \]

If \(x\), \(y\) and \(z\) are positive numbers such that \(x^4 + y^4 + z^4 = 1\), find the minimum value of the sum above.

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