
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
Given $4\cos\left(\theta + \frac{\pi}{3}\right)\cos\left(\theta - \frac{\pi}{6}\right) = \sin 2\theta$, find the value of $\tan\left(2\theta - \frac{\pi}{6}\right)$. The options are:
A) $\frac{1}{6}$,
B) $\frac{\sqrt{3}}{9}$,
C) $-\frac{\sqrt{3}}{6}$,
D) $-\frac{\sqrt{3}}{3}$.

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