
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 integer greater than or equal to  $2$ .  $n$  people from each of  $n$  countries gather, a total of  $n^2$  people facing inward and surround the round table. For any two different people in the same country, their left neighbors belong to different countries. Find the maximum number of people whose left and right neighbors belong to the same country.

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