
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 \( b = \frac{\pi}{2010} \). Find the smallest positive integer \( m \) such that
\[
2\left[\cos(b) \sin(b) + \cos(4b) \sin(2b) + \cos(9b) \sin(3b) + \cdots + \cos(m^2b) \sin(mb)\right]
\]
is an integer.

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