
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 interior angles \( A, B, \) and \( C \) are opposite the sides \( a, b, \) and \( c \) respectively. Given that the area of \( \triangle ABC \) is \( \frac{a^2}{3 \sin A} \):
1. Find \( \sin B \sin C \).
2. If \( 6 \cos B \cos C = 1 \) and \( a = 3 \), find the perimeter of \( \triangle ABC \).

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