\[\large \begin{cases} a+b+c=0 \\ a^{3}+b^{3}+c^{3}=3 \\ a^{5}+b^{5}+c^{5}=10 \end{cases} \]

If \(a\), \(b\) and \(c\) are real numbers satisfying the system of equations above, find \( a^4 + b^4 + c^4 \).

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