Let $x_{1}, x_{2}, \ldots, x_{n}$ be a sequence where each term is 0, 1, or -2. If

$$
\left\{\begin{array}{l}
x_{1}+x_{2}+\cdots+x_{n}=-5 \\
x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=19
\end{array}\right.
$$

determine $x_{1}^{5}+x_{2}^{5}+\cdots+x_{n}^{5}$.
Let $a$ be the number of terms equal to 1 and $b$ be the number of terms equal to -2. We can write:

$$
\left\{\begin{array} { l } 
{ a \cdot 1 + b \cdot ( - 2 ) = - 5 } \\
{ a \cdot 1 ^ { 2 } + b \cdot ( - 2 ) ^ { 2 } = 1 9 }
\end{array} \Longleftrightarrow \left\{\begin{array}{l}
a-2 b=-5 \\
a+4 b=19
\end{array}\right.\right.
$$

Solving the system, we get $a=3$ and $b=4$. Therefore,

$$
x_{1}^{5}+x_{2}^{5}+\cdots+x_{n}^{5}=a \cdot 1^{5}+b \cdot(-2)^{5}=3-4 \cdot 32=-125
$$