There are two distinct real numbers, \(x\) and \(y\). For every real number \(z\) that is equal to neither \(x\) nor \(y\), they satisfy:

\[\frac{x^3+16}{x}=\frac{y^3+16}{y}\neq\frac{z^3+16}{z}.\]

Find the value of \((x-y)^2\).

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