
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 triangle $ABC$, the sides opposite angles $A, B, C$ are denoted as $a, b, c$, respectively. It is given that the vector $\overrightarrow{m} = (\cos A, \cos B)$ and the vector $\overrightarrow{n} = (b - 2c, a)$, and that $\overrightarrow{m}$ is perpendicular to $\overrightarrow{n}$.  
(1) Determine the measure of angle $A$.  
(2) If $a = 3$, find the maximum value of the area of triangle $ABC$.

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