
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 $f(x)=2\sqrt{3}\sin x\cos x-\cos (\pi +2x)$.

(1) Find the interval(s) where $f(x)$ is monotonically increasing.

(2) In $\Delta ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $f(C)=1,c=\sqrt{3},a+b=2\sqrt{3}$, find the area of $\Delta ABC$.

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