
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 $f(x)=\cos{x}\cos{\left(x-\frac{\pi}{6}\right)}+\sqrt{3}\sin^2{x}-\frac{3\sqrt{3}}{4}$.  
$(1)$ Find the smallest positive period and the interval of monotonic increase of $f(x)$;  
$(2)$ When $x\in\left[\frac{\pi}{12},\frac{\pi}{2}\right]$, find the maximum and minimum values of the function $f(x)$ and determine the corresponding $x$.

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