Expecting a positive sign?

Algebra Level 5

If \(a\) is a real number and the following system is given:

\[a-b+c=4 \\ a^2+b^2+c^2=-8 \\ a^3+b^3+c^3=8(1+\sqrt{7}i)\]

Where \(i^2=-1\), find \(a^4+b^4+c^4\).

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