Is logarithms required?

Algebra Level 3

If \(2^{x}=4^{y}=8^{z}\) and \(xyz= 288\), with \( \frac{1}{2x} + \frac{1}{4y}+ \frac{1}{8z}= \frac{m}{n}\) for coprime positive integers \(m \) and \(n\), find \(m+n\).

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