
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 a diagram, there exists a larger circle and a smaller circle centered within it. The gray area visible between the circles is equal to twice the area of the smaller circle. Determine the ratio of the radius of the smaller circle to the radius of the larger circle and the fraction of the total area of the larger circle that is occupied by the smaller circle. Express your answers as common fractions.

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