Example 7 Let $x_{1}, x_{2}, \cdots, x_{2006}$ be integers, and satisfy the following conditions:
$$
\begin{array}{l}
-1 \leqslant x_{n} \leqslant 2 \quad (n=1,2, \cdots, 2006), \\
x_{1}+x_{2}+\cdots+x_{2006}=200, \\
x_{1}^{2}+x_{2}^{2}+\cdots+x_{2006}^{2}=2006 .
\end{array}
$$

Find the minimum and maximum values of $x_{1}^{3}+x_{2}^{3}+\cdots+x_{2000}^{3}$. (2006, Zhejiang Province Junior High School Mathematics Competition)
Given the problem, let there be $a$ numbers of $-1$, $b$ numbers of $0$, $c$ numbers of $1$, and $d$ numbers of $2$. Then,
$$
\begin{array}{c}
a+b+c+d=2006, \\
-a+c+2 d=200, \\
a+c+4 d=2006 .
\end{array}
$$

Thus, $b=3 d \geqslant 0$,
$$
\begin{array}{l}
c=1103-3 d \geqslant 0, \\
a=903-d \geqslant 0 .
\end{array}
$$

Therefore, $0 \leqslant d \leqslant 367$.
Hence, $x_{1}^{3}+x_{2}^{3}+\cdots+x_{2006}^{3}$
$$
=-a+c+8 d=200+6 d
$$

The maximum value is 2402, and the minimum value is 200.