
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 \( n \geq 2 \), the integer sequence \( x_0, x_1, \ldots, x_n \) satisfies:
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 an "excellent sequence". Find the number of excellent sequences.

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