
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 immediate neighbors to the left and right at the table. A kind wizard wants to seat the gnomes at one round table so that every pair of adjacent gnomes is friends. He has the ability to make $2n$ pairs of gnomes (these gnomes can be from the same table or from different tables) friends, but after that, an evil wizard will make $n$ pairs of these $2n$ pairs unfriends. For which values of $n$ can the kind wizard achieve his goal, no matter how the evil wizard acts?

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