1, 2, 3, _ ?

Algebra Level 4

\(x, y\) and \(z\) are complex numbers satisfying

\[ \begin{cases} x^1+y^1+z^1 & = 1\\ x^2 + y^2 + z^2 & = 2 \\ x^3 + y^3 + z^3 & = 3 \\ \end{cases} \]

The value of \( x^4 + y^4 + z^4 \) can be expressed as \( \frac {a}{b} \), where \( a\) and \(b\) are positive coprime integers. What is the value of \( a +b \)?

This problem is proposed by Harshit.

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