
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
Let \( a = \frac{\pi}{4032} \). Determine the smallest positive integer \( n \) so that
\[
2[\cos(a)\sin(a) + \cos(8a)\sin(4a) + \cos(27a)\sin(9a) + \cdots + \cos(n^3a)\sin(n^2a)]
\]
is an integer.

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