
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 $\overrightarrow{m}=(2\cos x+2\sqrt{3}\sin x, 1)$, $\overrightarrow{n}=(\cos x, -y)$, and $\overrightarrow{m} \perp \overrightarrow{n}$,

(I) Express $y$ as a function of $x$, $f(x)$, and find the interval where $f(x)$ is monotonically increasing.

(II) Given $a, b, c$ are the lengths of the sides opposite to angles $A, B, C$ of $\triangle ABC$, respectively. If $f(\frac{A}{2})=3$ and $a=3, b+c=4$, find the area of $\triangle ABC$.

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