
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 $f(x)=\sqrt{3}\sin(2x-\frac{\pi}{6})+2\sin^{2}(x-\frac{\pi}{12})$ where $(x\in R)$.

(1) Find the smallest positive period of the function $f(x)$;

(2) Find the interval(s) where $f(x)$ is monotonically increasing;

(3) If $x\in [0,2\pi]$, find the zeros of the function $f(x)$.

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