
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 $f(x)=\sqrt{3}\sin 2x-2\sin^2 x$.  
$(1)$ Find the smallest positive period and the interval of monotonic increase of $f(x)$.  
$(2)$ If $x\in[-\frac{\pi}{3},\frac{\pi}{3}]$, find the minimum value of $f(x)$ and the corresponding values of $x$ when the minimum value is achieved.

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