
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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