
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 $f(x)=\sqrt{3}\sin\left(x+\frac{\pi}{4}\right)\sin\left(x-\frac{\pi}{4}\right)+\sin x\cos x$.  
$(Ⅰ)$ Find the smallest positive period and the center of symmetry of $f(x)$;  
$(Ⅱ)$ Given an acute triangle $\triangle ABC$ with sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively, if $f(A)=\frac{\sqrt{3}}{2}$ and $a=4$, find the maximum perimeter of $\triangle ABC$.

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