
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 the acute triangle \( \triangle ABC \), \( \sin A = \frac{4 \sqrt{5}}{9} \), and \( a, b, c \) represent the sides opposite to angles \( A, B, \) and \( C \), respectively.
1. Find the value of \( \sin 2(B + C) + \sin^{2} \frac{B+C}{2} \).
2. If \( a = 4 \), find the area \( S_{\triangle ABC} \) of the triangle when \( \overrightarrow{AB} \cdot \overrightarrow{AC} \) reaches its maximum value.

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