
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 a triangle \( ABC \). Four circles of equal radius \(\rho\) are constructed such that one of them touches the other three circles, and each of these three circles touches two sides of the triangle. Find \(\rho\), if the radii of the inscribed and circumscribed circles of the triangle are \(r\) and \(R\) respectively.

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