Half way square root

Algebra Level 2

If the following holds for any numbers \(x\) and \(y\), what is the maximum value of \(abc\sqrt{def}\): \[4x+9y+16=a^2 x+b^2 y+c^2=\frac{x}{d}+\frac{y}{e}+\frac{1}{f}?\]

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