The real numbers $x$, $y$, and $z$ satisfy the system of equations:
\[
\begin{align*}
  x^2 + 27 &= -8y + 10z, \\
  y^2 + 196 &= 18z + 13x, \\
  z^2 + 119 &= -3x + 30y.
\end{align*}
\]
Find $x + 3y + 5z$.

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