Knowing that the system:
\[
\begin{align*}
x + y + z &= 3, \\
x^3 + y^3 + z^3 &= 15, \\
x^4 + y^4 + z^4 &= 35,
\end{align*}
\]
has a real solution \(x, y, z\) for which \(x^2 + y^2 + z^2 < 10\), find the value of \(x^5 + y^5 + z^5\) for that solution.

Let's think step by step and output the final answer within \boxed{}.
user