
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Given a positive integer $n \geq 2$, let the integer sequence $x_{0}, x_{1}, \cdots, x_{n}$ satisfy:
(1) For any $0 \leq i \leq n$, $|x_{i}| \leq n$;
(2) For any $0 \leq i < j \leq n$, $x_{i} \neq x_{j}$;
(3) For any $0 \leq i < j < k \leq n$,
\[
\begin{aligned}
\max \left\{\left|x_{k} - x_{i}\right|, \left|x_{k} - x_{j}\right|\right\} = \frac{1}{2}\left(\left|x_{i} - x_{j}\right| + \left|x_{j} - x_{k}\right| + \left|x_{k} - x_{i}\right|\right).
\end{aligned}
\]

Then the sequence is called an “excellent sequence.” Find the number of excellent sequences.

Present the answer in LaTex format: \boxed{Your answer}
user