
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
Find all integers  $ n \ge 3$  such that there are  $ n$  points in space, with no three on a line and no four on a circle, such that all the circles pass through three points between them are congruent.

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