
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
Calculate the value of 
\[
\frac{\tan \frac{\pi}{7} + i}{\tan \frac{\pi}{7} - i}
\]
and determine if it is a fourteenth root of unity, i.e., does it equal $\cos \frac{2n \pi}{14} + i \sin \frac{2n \pi}{14}$ for some integer $n$ between 0 and 13 inclusive? If so, find the value of $n$.

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