Giant expression

With AM-GM-HM, we have $\frac{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}{3}\geq\sqrt[3]{\frac{1}{a}\cdot\frac{1}{b}\cdot\frac{1}{c}}$ $=\sqrt[3]{\frac{\color{#3D99F6}{1}}{abc}}$ $=\sqrt[3]{\frac{\color{#3D99F6}{abcdef}}{abc}}$ $=\sqrt[3]{def}\geq\frac{3}{\frac{1}{d}+\frac{1}{e}+\frac{1}{f}}\cdots(1)$ Similarly, we get $\frac{\frac{1}{b}+\frac{1}{c}+\frac{1}{d}}{3}\geq\frac{3}{\frac{1}{e}+\frac{1}{f}+\frac{1}{a}}\cdots(2)$ $\frac{\frac{1}{c}+\frac{1}{d}+\frac{1}{e}}{3}\geq\frac{3}{\frac{1}{f}+\frac{1}{a}+\frac{1}{b}}\cdots(3)$ $\frac{\frac{1}{d}+\frac{1}{e}+\frac{1}{f}}{3}\geq\frac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}\cdots(4)$ $\frac{\frac{1}{e}+\frac{1}{f}+\frac{1}{a}}{3}\geq\frac{3}{\frac{1}{b}+\frac{1}{c}+\frac{1}{d}}\cdots(5)$ $\frac{\frac{1}{f}+\frac{1}{a}+\frac{1}{b}}{3}\geq\frac{3}{\frac{1}{c}+\frac{1}{d}+\frac{1}{e}}\cdots(6)$ By adding $(1)+(2)+(3)+(4)+(5)+(6)$ , we get $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}+\frac{1}{f}\geq\frac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}+\frac{3}{\frac{1}{b}+\frac{1}{c}+\frac{1}{d}}+\frac{3}{\frac{1}{c}+\frac{1}{d}+\frac{1}{e}}+\frac{3}{\frac{1}{d}+\frac{1}{e}+\frac{1}{f}}+\frac{3}{\frac{1}{e}+\frac{1}{f}+\frac{1}{a}}+\frac{3}{\frac{1}{f}+\frac{1}{a}+\frac{1}{b}}$ $\frac{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}+\frac{1}{f}}{\frac{1}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}+\frac{1}{\frac{1}{b}+\frac{1}{c}+\frac{1}{d}}+\frac{1}{\frac{1}{c}+\frac{1}{d}+\frac{1}{e}}+\frac{1}{\frac{1}{d}+\frac{1}{e}+\frac{1}{f}}+\frac{1}{\frac{1}{e}+\frac{1}{f}+\frac{1}{a}}+\frac{1}{\frac{1}{f}+\frac{1}{a}+\frac{1}{b}}}\geq\boxed{3}$