
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  $\Gamma$  a fixed circunference. Find all finite sets  $S$  of points in  $\Gamma$  such that:
For each point  $P\in \Gamma$ , there exists a partition of  $S$  in sets  $A$  and  $B$ ( $A\cup B=S$ ,  $A\cap B=\phi$ ) such that  $\sum_{X\in A}PX = \sum_{Y\in B}PY$ .

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