A little long recursion

Algebra Level 4

If aa, bb and cc are complex numbers that satisfy: a2+b2+c2=3a^2+b^2+c^2=-3 a3+b3+c3=46a^3+b^3+c^3=-46 a4+b4+c4=123a^4+b^4+c^4=-123 And it's known that the value of a+b+ca+b+c is an integer.

Find (a10+b10+c10)mod1000(a^{10}+b^{10}+c^{10}) \mod 1000.

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