
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
Behind each of the two round tables sit $n$ gnomes. Each gnome is friends only with the gnomes immediately to their left and right at the table. A good wizard wants to seat the gnomes around one round table so that each pair of neighboring gnomes are friends. He has the power to make $2n$ pairs of gnomes become friends (the gnomes in a pair can be from the same table or different tables), but after this, an evil wizard will cause $n$ pairs of these $2n$ pairs to fall out of friendship. For which values of $n$ can the good wizard achieve his goal, regardless of the actions of the evil wizard?

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