HMMT 2024 A9

Suppose $a, b$ , and $c$ are complex numbers satisfying

$\begin{aligned} a^{2} & =b-c, \\ b^{2} & =c-a, \text { and } \\ c^{2} & =a-b . \end{aligned}$

Compute all possible values of $a+b+c$ .

Dataset text NuminaMath olympiads. Quoted with source.

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0, \pm i \sqrt{6}

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