
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 $f(x)=2\sqrt{3}\sin x \cos x - 2\cos^{2}x$ where $x\in \mathbb{R}$.  
$(1)$ Find the range of the function $f(x)$.  
$(2)$ In triangle $ABC$, with sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively, if $f(A)=-2$ and $a=\sqrt{3}$, find the maximum value of the area $S$ of triangle $ABC$.

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