A little long recursion

Algebra Level 4

If \(a\), \(b\) and \(c\) are complex numbers that satisfy: \[a^2+b^2+c^2=-3\] \[a^3+b^3+c^3=-46\] \[a^4+b^4+c^4=-123\] And it's known that the value of \(a+b+c\) is an integer.

Find \((a^{10}+b^{10}+c^{10}) \mod 1000\).

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