# 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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