
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
Suppose that
\[
        \prod_{n=1}^{\infty}\left(\frac{1+i\cot\left(\frac{n\pi}{2n+1}\right)}{1-i\cot\left(\frac{n\pi}{2n+1}\right)}\right)^{\frac{1}{n}} = \left(\frac{p}{q}\right)^{i \pi},
 \]
    where  $p$  and  $q$  are relatively prime positive integers. Find  $p+q$ .
    
*Note: for a complex number  $z = re^{i \theta}$  for reals  $r > 0, 0 \le \theta < 2\pi$ , we define  $z^{n} = r^{n} e^{i \theta n}$  for all positive reals  $n$ .*

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