
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 \( \triangle ABC \), the sides opposite to angles \( A, B, C \) are \( a, b, c \) respectively, and \( \cos A = \frac{1}{3} \).
1. Find the value of \( \sin^2 \frac{B+C}{2} + \cos 2A \).
2. If \( a = \sqrt{3} \), find the maximum value of \( bc \).

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