
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 \( \triangle ABC \), \( \sin A = \frac{4 \sqrt{5}}{9} \), with \( a \), \( b \), and \( c \) being the sides opposite to angles \( \angle A \), \( \angle B \), and \( \angle C \), respectively.

1. Find the value of \( \sin 2(B+C) + \sin^2 \frac{B+C}{2} \).

2. Given \( a = 4 \), determine the value of the area \( S \) of \( \triangle ABC \) when the dot product of vectors \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \) is maximized.

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