
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
For real  $\theta_i$ ,  $i = 1, 2, \dots, 2011$ , where  $\theta_1 = \theta_{2012}$ , find the maximum value of the expression

\[
\sum_{i=1}^{2011} \sin^{2012} \theta_i \cos^{2012} \theta_{i+1}.
\]

*Proposed by Lewis Chen*

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