
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 circles \( K_{1} \) and \( K_{2} \) with centers at points \( O_{1} \) and \( O_{2} \) and equal radii \( r \) touch the circle \( K \) with radius \( R \) and center at point \( O \) internally. The angle \( O_{1} O O_{2} \) is \( 120^{\circ} \). Find the radius \( q \) of the circle that touches \( K_{1} \) and \( K_{2} \) externally and the circle \( K \) internally.

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