
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 \( AB \) be a segment of length 2 with midpoint \( M \). Consider the circle with center \( O \) and radius \( r \) that is externally tangent to the circles with diameters \( AM \) and \( BM \) and internally tangent to the circle with diameter \( AB \). Determine the value of \( r \).

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