
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 \( O_{1} \) and \( O_{2} \) respectively, have radii of 2 units and \( r \), and are tangent to each other externally. The Thales circle of the segment \( O_{1} O_{2} \) intersects circle \( k_{1} \) at points \( A \) and \( B \), and circle \( k_{2} \) at points \( C \) and \( D \). The reflection of \( O_{1} \) over segment \( AB \) is \( O_{1}' \), and the reflection of \( O_{2} \) over segment \( CD \) is \( O_{2}' \). The distance \( O_{1}'O_{2}' \) is 1 unit.

What is the value of \( r \)?

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