
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 two circles, points \( A \) and \( B \) are taken on them such that they are equidistant from the midpoint of the segment connecting their centers. Find the locus of the midpoints of the segments \( AB \).

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