A right triangle has legs \(a\) and \(b\), and hypotenuse \(c\).

The length of the median to the hypotenuse is \(\sqrt[3]{a^{3}b+b^{3}a}\).

If \(\mathcal{H}\) is the minimum possible value of \(c\), find the value of \(16 \mathcal{H}\).

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