Giant expression

Algebra Level 3

\[\large \frac{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}+\frac{1}{f}}{\frac{1}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}+\frac{1}{\frac{1}{b}+\frac{1}{c}+\frac{1}{d}}+\frac{1}{\frac{1}{c}+\frac{1}{d}+\frac{1}{e}}+\frac{1}{\frac{1}{d}+\frac{1}{e}+\frac{1}{f}}+\frac{1}{\frac{1}{e}+\frac{1}{f}+\frac{1}{a}}+\frac{1}{\frac{1}{f}+\frac{1}{a}+\frac{1}{b}}}\]

Let \(a\), \(b\), \(c\), \(d\), \(e\), and \(f\) be positive real numbers such that \(abcdef=1\). Find the minimum value of expression above.

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