
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 the complex number  $z = \cos\tfrac{1}{1000} + i \sin\tfrac{1}{1000}.$  Find the smallest positive integer  $n$  so that  $z^n$  has an imaginary part which exceeds  $\tfrac{1}{2}.$ 

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