
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 $f(x)=2\cos x \cdot \sin\left(x+\frac{\pi}{6}\right)$, where $x\in \mathbb{R}$, and $\triangle ABC$ with internal angles $A$, $B$, $C$ opposite sides $a$, $b$, $c$ respectively. If the maximum value of $f(x)$ is $f(A)$, find:

$(1)$ The value of $A$;

$(2)$ When $a=2$ and $b=2\sqrt{3}$, find the area of $\triangle ABC$.

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