
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
Given an acute triangle \( \triangle ABC \) with angles \( A, B, C \) opposite to sides \( a, b, c \) respectively, and the condition \( \sin A = 2a \sin B \). The circumradius of \( \triangle ABC \) is \( \frac{\sqrt{3}}{6} \). Determine the range of values for the perimeter of \( \triangle ABC \).

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