
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
In $\vartriangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively. Given $3\cos B=2\sin \left( \frac{\pi }{3}+A \right)\cdot \sin \left( \frac{\pi }{3}-A \right)+2{{\sin }^{2}}A$.

(1) Find the value of angle $B$;

(2) If $b=2\sqrt{3}$, find the maximum value of the perimeter of $\vartriangle ABC$.

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