
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
With the same lawn and the same allowable mobot orientations as in the previous problem, let us call a formation "happy" if it is invariant under 120° rotations. (A rotation applies both to the positions of the mobots and to their orientations.) Find the number of happy formations for a given \( n \).

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