
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 $\triangle ABC$, $\sin A = \sin^2 B + \sin\left( \frac{\pi}{4} + B \right)\sin\left( \frac{\pi}{4} - B \right)$.  
(1) Find the value of angle $A$;  
(2) If $\overrightarrow{AB} \cdot \overrightarrow{AC} = 12$, find the area of $\triangle ABC$.

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