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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