
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
In acute triangle $\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given vectors $\overrightarrow{m} = (2, c)$ and $\overrightarrow{n} = (\frac{b}{2}\cos C - \sin A, \cos B)$, with $b = \sqrt{3}$ and $\overrightarrow{m} \perp \overrightarrow{n}$.
(1) Find angle $B$;
(2) Find the maximum area of $\triangle ABC$ and the lengths of the other two sides, $a$ and $c$, when the area is maximum.

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