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Algebra Level 3

\[\begin{cases} x^{2}+y^{2}=z^{2} \\ xy=z \\ (x+y)^{2} = 80 \end{cases} \]

If the above system of equations holds true for some positive real numbers \(x\), \(y\), and \(z\), then find

\[\large \frac{z^{4}}{\left(\frac{x^3y+xy^3}{z^3}\right)} . \]

Hint: Use factorization.

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