Where's Number One?

Algebra Level 5

{a2+b2+c2=26a3+b3+c3=129a4+b4+c4=650 \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,ca,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 a6+b6+c6a^6 + b^6 + c^6 .

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