
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
The circle with radius \( r \) is contained within the region bounded by a circle with radius \( R \). Given that the area of the region bounded by the larger circle is three times the area of the region that lies outside the smaller circle but inside the larger one, determine the ratio \( R:r \).

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