Cubic equation in three variables

Algebra Level 3

\[\begin{eqnarray} x + y + z &=& 4 \\ x^{2} + y^{2} + z^{2} &=& 14 \\ x^{3} + y^{3} + z^{3} &=& 34 \\ \end{eqnarray} \]

There are many different sets of solutions to the above system of equations.

Let \(S\) be the sum of all possible values of \(x\). Find \(S\).

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