Solve the system of nonlinear equations:
\[
\begin{cases}
x^2 + y^2 + z^2 = 14 \\
x + y + z = 4 \\
xyz = 2
\end{cases}
\]
Find the value of \( x^3 + y^3 + z^3 \). Let's think step by step and output the final answer within \boxed{}.
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