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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