
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

At each of the two round tables, there are \( n \) gnomes. Each gnome is friends only with their neighbors on the left and right at their table. A kind wizard wants to seat all the gnomes at one round table such that every two neighboring gnomes are friends with each other. He has the ability to foster friendship among \( 2n \) pairs of gnomes (the gnomes in a pair can be from the same table or different tables), but after this, an evil wizard will cause \( n \) of these \( 2n \) pairs to quarrel. For which values of \( n \) can the kind wizard achieve his goal, no matter what the actions of the evil wizard?

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