
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 triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $b=2\sqrt{3}$, $c=2$, and $b\sin C-2c\sin B\cos A=0$.
$(1)$ Find the area $S$ of triangle $\triangle ABC$;
$(2)$ Let $f(x)=4\cos x(\sin x\cos A+\cos x\sin A)$, where $x\in[0,\frac{\pi}{3}]$. Find the maximum value of the function $f(x)$ and determine the corresponding value of $x$.

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