
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 that $\tan\left(\alpha + \frac{\pi}{3}\right)=2$, find the value of $\frac{\sin\left(\alpha + \frac{4\pi}{3}\right) + \cos\left(\frac{2\pi}{3} - \alpha\right)}{\cos\left(\frac{\pi}{6} - \alpha\right) - \sin\left(\alpha + \frac{5\pi}{6}\right)}$.

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