
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  $B$  points in the plane and  $C$  a point in the perpendiclar bisector of  $AB$ . It is constructed a sequence of points  $C_1,C_2,\dots, C_n,\dots$  in the following way:  $C_1=C$  and for  $n\geq1$ , if  $C_n$  does not belongs to  $AB$ , then  $C_{n+1}$  is the circumcentre of the triangle  $\triangle{ABC_n}$ .

Find all the points  $C$  such that the sequence  $C_1,C_2,\dots$  is defined for all  $n$  and turns eventually periodic.

Note: A sequence  $C_1,C_2, \dots$  is called eventually periodic if there exist positive integers  $k$  and  $p$  such that  $C_{n+p}=c_n$  for all  $n\geq{k}$ .

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