
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 the vertices \( A_{1}, A_{2}, \ldots, A_{n} \) of a regular \( n \)-gon, assign the numbers \( 1, 2, \ldots, n \) to the vertices in some order.

a) What is the maximum possible sum of the absolute values of the differences between neighboring numbers?

b) For how many different arrangements does this maximum sum occur?

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