
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 $f(x)=\sin (2x+\frac{\pi }{6})+\sin (2x-\frac{\pi }{6})+\cos 2x+1$

     $(1)$ Find the symmetry center of the function $f(x)$ and the interval where the function is monotonically increasing;

     $(2)$ Given in $\Delta ABC$, the sides opposite angles $A, B, C$ are $a, b, c$ respectively, if $f(A)=3, B=\frac{\pi }{4}, a=\sqrt{3}$, find $AB$．

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