
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 points A and B, take C on the perpendicular bisector of AB. Define the sequence \( C_1, C_2, C_3, \dots \) as follows. \( C_1 = C \). If \( C_n \) is not on AB, then \( C_{n+1} \) is the circumcenter of the triangle \( ABC_n \). If \( C_n \) lies on AB, then \( C_{n+1} \) is not defined and the sequence terminates. Find all points C such that the sequence is periodic from some point on.

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