
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 the function $f(x) = 2\sqrt{3}\sin(\pi+x)\cos(-3\pi-x) - 2\sin\left(\frac{\pi}{2}-x\right)\cos(\pi-x)$.
1. Find the interval of monotonic increase for the function $f(x)$.
2. If $f\left(\frac{\alpha}{2} - \frac{\pi}{12}\right) = \frac{3}{2}$, and $\alpha$ is an angle in the second quadrant, find the value of $\cos\left(2\alpha + \frac{\pi}{3}\right)$.

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