Suppose that $a, b, c$ are complex numbers satisfying the system of equations:  
\[a^2 = b - c,\]  
\[b^2 = c - a,\]  
\[c^2 = a - b.\]  
Compute all possible values of $a + b + c$.

Let's think step by step and output the final answer within \boxed{}.
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