
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
Let  $ ABC$  be an equilateral triangle. Let  $ \Omega$  be its incircle (circle inscribed in the triangle) and let  $ \omega$  be a circle tangent externally to  $ \Omega$  as well as to sides  $ AB$  and  $ AC$ . Determine the ratio of the radius of  $ \Omega$  to the radius of  $ \omega$ .

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