If $x$, $y$, $z$ are real numbers satisfying:
\[
\begin{align*}
(x + 1)(y + 1)(z + 1) &= 3, \\
(x + 2)(y + 2)(z + 2) &= -2, \\
(x + 3)(y + 3)(z + 3) &= -1,
\end{align*}
\]
find the value of
\[ (x + 20)(y + 20)(z + 20). \]

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