
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 $x\in R$, let $\overrightarrow{m}=(2\cos x,\sin x+\cos x)$ and $\overrightarrow{n}=(\sqrt{3}\sin x,\sin x-\cos x)$. Define the function $f(x)=\overrightarrow{m}\cdot\overrightarrow{n}$.
(1) Find the range of $x$ when the function $f(x)$ reaches its minimum value.
(2) Let $a$, $b$, and $c$ be the sides opposite to angles $A$, $B$, and $C$ in $\triangle ABC$, respectively. If $f(C)=2$ and $c=\sqrt{3}$, find the maximum value of the area $S$ of $\triangle ABC$.

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