
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 center of symmetry of $f(x)$;
$(2)$ Find the smallest positive period and the interval of monotonic increase of $f(x)$;
$(3)$ If $x∈[-\frac{π}{2},0]$, find the minimum value of $f(x)$ and the corresponding values of $x$ when the minimum value is reached.

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