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$\frac{\color{#69047E}{b}+\color{#3D99F6}{c}}{\color{#D61F06}{a}} + \frac{\color{#3D99F6}{c}+\color{#D61F06}{a}}{\color{#69047E}{b}} + \frac{\color{#D61F06}{a}+\color{#69047E}{b}}{\color{#3D99F6}{c}} = \ ?$

If $\color{#D61F06}{a}+\color{#69047E}{b}+\color{#3D99F6}{c} = 6$ and $\dfrac{1}{\color{#D61F06}{a}} + \dfrac{1}{\color{#69047E}{b}} + \dfrac{1}{\color{#3D99F6}{c}} = 2,$ find the value of the expression above.