
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 to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given vectors $\overrightarrow{m} = (\sin \frac{A}{2}, \cos \frac{A}{2})$ and $\overrightarrow{n} = (\cos \frac{A}{2}, -\cos \frac{A}{2})$, and $2\overrightarrow{m} \cdot \overrightarrow{n} + |\overrightarrow{m}| = \frac{\sqrt{2}}{2}$, $\overrightarrow{AB} \cdot \overrightarrow{AC} = 1$.

(1) Find the measure of angle $A$.
(2) Find the area of $\triangle ABC$.

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