
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]


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Given a positive integer \( n \geq 2 \), an integer sequence \( x_{0}, x_{1}, \ldots, x_{n} \) is defined as follows:
1. For any \( 0 \leq i \leq n \), \( \left|x_{i}\right| \leq n \);
2. For any \( 0 \leq i < j \leq n \), \( x_{i} \neq x_{j} \);
3. For any \( 0 \leq i < j < k \leq n \), 
\[ 
\max \left\{\left|x_{k} - x_{i}\right|, \left|x_{k} - x_{j}\right|\right\} = \frac{1}{2} \left(\left|x_{i} - x_{j}\right| + \left|x_{j} - x_{k}\right| + \left|x_{k} - x_{i}\right|\right),
\]

such a sequence is called a "superior sequence". Find the number of superior sequences.

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