Task 877

Task 877: development task in emrekuruu/math-openenv, an RL environment on the Hugging Face Hub. Let $a,$ $b,$ $c$ be distinct complex numbers such that \ \frac{a}{1 - b} = \frac{b}{1 - c} = \frac{c}{1 - a} = k.\ Find the sum of all possible values of $k.$

The task

Let $a,$ $b,$ $c$ be distinct complex numbers such that \[\frac{a}{1 - b} = \frac{b}{1 - c} = \frac{c}{1 - a} = k.\]Find the sum of all possible values of $k.$