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Geometry Level 5

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 the length of the altitude to the hypotenuse can be expressed in the form \(\dfrac{m}{n}\) for coprime positive integers \(m\) and \(n\), find \(m+n\).

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