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

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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