
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 $\overrightarrow{a} = (2\cos{x}, \sin{x})$ and $\overrightarrow{b} = (\cos{x}, 2\sqrt{3}\cos{x})$, define the function $f(x) = \overrightarrow{a} \cdot \overrightarrow{b} + 1$.  
(1) Find the intervals where the function $f(x)$ is monotonically increasing.  
(2) In triangle $\triangle ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite angles A, B, and C respectively, with $a=1$ and $f(A)=3$, find the maximum value of the area $S$ of $\triangle ABC$.

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