The Other AM-GM (Part 2)

Algebra Level 4

Let \( w, \space x, \space y, \) and \(z\) be positive real numbers, such that \( \dfrac{1}{w} + \dfrac{1}{x} + \dfrac{1}{z} = 5 - \dfrac{1}{y}\). The minimum value of \( w^{4}x^{3}y^{2} z\) is a form of \( \dfrac{a}{b} \), where \(a \) and \( b\) are positive coprime integers. If \( \sqrt[3]{\dfrac{a}{b} } = \dfrac{c}{d} \), where \(c \) and \(d \) are coprime positive integers, find \(b-a+d-c\).

Note: Should you only use AM-GM?

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