
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
Let \( n \) be an odd integer different from 1. How many permutations \( p \) of \(\{1,2, \ldots, n\}\) satisfy

\[ \sum_{i=1}^{n} |p(i) - i| = \frac{n^2 - 1}{2} \]

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