
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
\[
\frac{2 \cos ^{2} 2 \alpha + \sqrt{3} \sin 4 \alpha - 1}{2 \sin ^{2} 2 \alpha + \sqrt{3} \sin 4 \alpha - 1} = \frac{\sin \left(4 \alpha + \frac{\pi}{6}\right)}{\sin \left(4 \alpha - \frac{\pi}{6}\right)}
\]

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