
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 $n$ for which the quantity
\[\frac{\tan \frac{\pi}{3} + i}{\tan \frac{\pi}{3} - i}\]
is a sixth root of unity. Express it as $\cos \frac{2n\pi}{6} + i\sin \frac{2n\pi}{6}$ where $n$ is an integer between 0 and 5 inclusive.

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