
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  $n\ge 4$  be a positive integer and let  $M$  be a set of  $n$  points in the plane, where no three points are collinear and not all of the  $n$  points being concyclic. Find all real functions  $f:M\to\mathbb{R}$  such that for any circle  $\mathcal{C}$  containing at least three points from  $M$ , the following equality holds:
\[\sum_{P\in\mathcal{C}\cap M} f(P)=0\]
*Dorel Mihet*

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