
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
For a triangle $ABC$ with internal angles $A$, $B$, $C$ and respective opposite side lengths $a$, $b$, $c$, let $\mathbf{m} = (\cos A, \cos C)$ and $\mathbf{n} = (c - 2b, a)$, and suppose that $\mathbf{m}$ is perpendicular to $\mathbf{n}$.

(1) Determine the measure of angle $A$;

(2) If angle $B = 60^\circ$ and the length of the median $AM$ from vertex $A$ to side $BC$ is $\sqrt{3}$, find the area of triangle $ABC$.

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