
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
There are two round tables, each seating \( n \) dwarfs, where each dwarf is friends only with their immediate neighbors to the left and right. A kind wizard wants to seat all the dwarfs at one round table such that each pair of neighboring dwarfs are friends. The wizard can create new friendships between \( 2n \) pairs of dwarfs (these dwarfs can be from the same table or from different tables), but afterward, an evil wizard will break friendship between \( n \) of these \( 2n \) pairs. For which values of \( n \) can the kind wizard achieve his goal, no matter how the evil wizard acts?

(Mikhail Svyatlovsky)

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