
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
Find all odd integers \( n > 1 \) for which there is at least one permutation \( a_1, a_2, \ldots, a_n \) of \( 1, 2, 3, \ldots, n \) such that the sums \( a_1 - a_2 + a_3 - \ldots + a_n \), \( a_2 - a_3 + a_4 - \ldots - a_n + a_1 \), \( a_3 - a_4 + \ldots + a_n - a_1 + a_2 \), \( \ldots \), \( a_n - a_1 + a_2 - \ldots + a_{n-1} \) are all positive.

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