
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 $f(x)=4\cos x\sin (x-\frac{\pi}{6})$, $x\in R$.
(I) Find the smallest positive period and the monotonically increasing interval of $f(x)$;
(II) In $\triangle ABC$, $BC=4$, $\sin C=2\sin B$. If the maximum value of $f(x)$ is $f(A)$, find the area of $\triangle ABC$.

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