Can we assume that \(x=y=z\)?

Algebra Level 5

\[\large \frac{zxy}{(1+5z)(4z+3x)(5x+6y)(y+18)}\]

Suppose the maximum value of the expression above for non-negative reals \(x\), \(y\) and \(z\) is of the form \(\dfrac{1}{r}\) for some \(r\in \mathbb{R}\). Find \(r\).

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