
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 $\overrightarrow{m} = (\sqrt{3}\sin x - \cos x, 1)$ and $\overrightarrow{n} = (\cos x, \frac{1}{2})$, let the function $f(x) = \overrightarrow{m} \cdot \overrightarrow{n}$.
(1) Find the interval where the function $f(x)$ is monotonically increasing.
(2) If $a$, $b$, $c$ are the lengths of the sides opposite to angles A, B, C in triangle $\triangle ABC$, with $a=2\sqrt{3}$, $c=4$, and $f(A) = 1$, find the area of triangle $\triangle ABC$.

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