Question 3 Given a positive integer $n \geqslant 2$. If the integer sequence $x_{0}, x_{1}, \cdots, x_{n}$ satisfies:
(1) For any $0 \leqslant i \leqslant n$, we have $\left|x_{i}\right| \leqslant n$;
(2) For any $0 \leqslant i<j \leqslant n$, we have $x_{i} \neq x_{j}$;
(3) For any $0 \leqslant i<j<k \leqslant n$, we have
$$
\begin{array}{l}
\max \left\{\left|x_{k}-x_{i}\right|,\left|x_{k}-x_{j}\right|\right\} \\
=\frac{1}{2}\left(\left|x_{i}-x_{j}\right|+\left|x_{j}-x_{k}\right|+\left|x_{k}-x_{i}\right|\right) .
\end{array}
$$

then the sequence is called a "good sequence". Find the number of good sequences.
For a good sequence, for $0 \leqslant i<j<k \leqslant n$, we have
$x_{k}=\min \left\{x_{i}, x_{j}, x_{k}\right\}$ or $x_{k}=\max \left\{x_{i}, x_{j}, x_{k}\right\}$.
Thus, $x_{k}=\min _{1 \leqslant i \leqslant k}\left\{x_{i}\right\}$ or $x_{k}=\max _{1 \leqslant i \leqslant k}\left\{x_{i}\right\}$.
By the problem statement, we know that there are $\mathrm{C}_{2 n+1}^{n+1}$ ways to choose the values of the $n+1$ terms of the good sequence, arranged in non-decreasing order as
$$
a_{0}, a_{1}, \cdots, a_{n} \text {. }
$$

When $x_{0}=a_{k}(k=0,1, \cdots, n)$, the $k$ numbers smaller than $x_{0}$ are arranged in non-increasing order in $\left\{x_{1}, x_{2}, \cdots, x_{n}\right\}$, and the $n-k$ numbers larger than $x_{0}$ are arranged in non-decreasing order in $\left\{x_{1}, x_{2}, \cdots, x_{n}\right\}$. Therefore, there are $\mathrm{C}_{n}^{k}$ such good sequences.
In summary, the total number of good sequences is
$$
\mathrm{C}_{2 n+1}^{n+1} \sum_{k=0}^{n} \mathrm{C}_{n}^{k}=2^{n} \mathrm{C}_{2 n+1}^{n+1} .
$$