
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 $f(x)=-2\sin^2x+2\sqrt{3}\sin x\cos x+1$

$(1)$ Find the smallest positive period and the center of symmetry of $f(x)$;

$(2)$ If $x\in \left[-\frac{\pi}{6},\frac{\pi}{3}\right]$, find the maximum and minimum values of $f(x)$.

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