
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 how many integers $n$ with $1 \le n \le 3000$ is the product
\[
  \prod_{k=0}^{n-1} \left( \left( 1 + e^{2 \pi i k / n} \right)^{2n} + 1 \right) 
\]equal to zero?

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