
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
On a line lie three points \( A \), \( B \), and \( C \) such that \( AB = BC = 3 \). Three circles with radius \( R \) have centers at points \( A \), \( B \), and \( C \).

Find the radius of a fourth circle that touches all three given circles if (a) \( R = 1 \); (b) \( R = 2 \); (c) \( R = 5 \).

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