
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 $f(x)=2\sin{x}\cos{x}+\sqrt{3}(\sin^{2}x-\cos^{2}x)$.  
$(1)$ Find the minimum positive period and the interval of monotonic increase of the function $f(x)$;  
$(2)$ In acute triangle $\triangle ABC$, where the angles $A$, $B$, $C$ are opposite to sides $a$, $b$, $c$ respectively, if $a=2$ and $f(A)=\sqrt{3}$, find the maximum perimeter of triangle $\triangle ABC$.

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