
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 $\overrightarrow{a}=(2\cos x, \sin x - \cos x)$, $\overrightarrow{b}=(\sqrt{3}\sin x, \sin x + \cos x)$, let the function $f(x)=\overrightarrow{a}\cdot\overrightarrow{b}$.
$(1)$ Find the expression for $f(x)$ and the set of values of $x$ when $f(x)$ reaches its maximum value;
$(2)$ Suppose $\triangle ABC$ has internal angles $A, B, C$ with corresponding sides $a, b, c$ respectively. If $a+b=2\sqrt{3}, c=\sqrt{6}, f(C)=2$, find the area of $\triangle ABC$.

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