Mixmatch

Algebra Level 1

If a,b,\color{#3D99F6}{a},\color{#D61F06}{b}, and c\color{#20A900}{c} are distinct reals such that a+1b=b+1c=c+1a,\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 abc.|\color{#3D99F6}{a}\color{#D61F06}{b}\color{#20A900}{c}|.

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