Where's Number One?

Algebra Level 5

\[ \begin{cases} {a^2 + b^2 + c^2= 26} \\ {a^3 + b^3 + c^3 = -129} \\ {a^4 + b^4 + c^4 = 650} \\ \end{cases} \]

Given that \(a,b,c\) are complex numbers that satisfy the system of equations above, and that their sum is an integer. Find the last three digits of \(a^6 + b^6 + c^6 \).

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