
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  $A$  and  $A' $  be fixed points on two equal circles in the plane and let  $AB$  and  $A' B'$   be arcs of these circles of the same length  $x$ . Find the locus of the midpoint of segment  $BB'$  when  $x$  varies:
(a) if the arcs have the same direction,
(b) if the arcs have opposite directions.

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