
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 vectors $\overrightarrow{m}=(\sin x, -1)$ and $\overrightarrow{n}=(\sqrt{3}\cos x, -\frac{1}{2})$, let $f(x)=(\overrightarrow{m}+\overrightarrow{n})\cdot \overrightarrow{m}$.
(1) Find the analytic expression for $f(x)$ and its intervals of monotonic increase;
(2) Given that $a$, $b$, and $c$ are the sides opposite angles $A$, $B$, and $C$ in triangle $\triangle ABC$, respectively, and $A$ is an acute angle with $a=2\sqrt{3}$ and $c=4$. If $f(A)$ is the maximum value of $f(x)$ on the interval $[0, \frac{\pi}{2}]$, find $A$, $b$, and the area $S$ of $\triangle ABC$.

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