
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
Let the function $f(x) = \sin x \cos x - \sqrt{3} \cos^2 x + \frac{\sqrt{3}}{2}$.

(1) Find the smallest positive period of $f(x)$ and the symmetry center of its graph;  
(2) Find the intervals where $f(x)$ is monotonically increasing;  
(3) Find the maximum and minimum values of $f(x)$ in the interval $\left[0, \frac{\pi}{2}\right]$.

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