
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
A polygon  $A_1A_2A_3\dots A_n$  is called *beautiful* if there exist indices  $i$ ,  $j$ , and  $k$  such that  $\measuredangle A_iA_jA_k = 144^\circ$ . Compute the number of integers  $3 \le n \le 2012$  for which a regular  $n$ -gon is beautiful.

*Proposed by Aaron Lin*

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