
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 $f(x) = \cos(\pi + x)\cos\left(\frac{3}{2}\pi - x\right) - \sqrt{3}\cos^2x + \frac{\sqrt{3}}{2}$.
(I) Find the smallest positive period and the maximum value of $f(x)$;
(II) Find the interval where $f(x)$ is monotonically increasing on $\left[\frac{\pi}{6}, \frac{2}{3}\pi\right]$.

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