
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 $f(x)=2\sin x\cos x+2\sqrt{3}\cos^{2}x-\sqrt{3}$, $x\in R$.
(1) Find the smallest positive period and the monotonically increasing interval of the function $f(x)$;
(2) In acute triangle $ABC$, if $f(A)=1$, $\overrightarrow{AB}\cdot\overrightarrow{AC}=\sqrt{2}$, find the area of $\triangle ABC$.

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