
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]


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Given a circle  $\gamma$  and two points  $A$  and  $B$  in its plane.  By  $B$  passes a variable secant that intersects  $\gamma$  at two points  $M$  and  $N$ . Determine the locus of the centers of the circles circumscribed to the triangle  $AMN$ .

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