
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
Determine the integer \( n \) such that the quantity
\[
\frac{\tan \frac{\pi}{6} + i}{\tan \frac{\pi}{6} - i}
\]
is a twelfth root of unity, i.e., it is equal to \(\cos \frac{2n \pi}{12} + i \sin \frac{2n \pi}{12}\) for some integer \( n \) between 0 and 11 inclusive.

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