Inequalities till the very end

Level pending

Consider \( x,y,z \in \mathbb{R^{+}} \) that satisfy \( x^4 + y^4 + z^4 = 1 \). Suppose the minimum value of

\( \frac{x^3}{1-x^8} + \frac{y^3}{1-y^8} + \frac{z^3}{1-z^8} \)

is \( \mathfrak{P} \). Find the value of \( \lceil \mathfrak{P}^4 \rceil \).

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