Mixmatch

Algebra Level 1

If $\color{#3D99F6}{a},\color{#D61F06}{b},$ and $\color{#20A900}{c}$ are distinct reals such that $\color{#3D99F6}{a}+\dfrac {1}{\color{#D61F06}{b}}=\color{#D61F06}{b}+\dfrac {1}{\color{#20A900}{c}}=\color{#20A900}{c}+\dfrac {1}{\color{#3D99F6}{a}},$ then find the value of $|\color{#3D99F6}{a}\color{#D61F06}{b}\color{#20A900}{c}|.$