
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  $P$  and  $Q$  be two distinct points in the plane. Let us denote by  $m(PQ)$  the segment bisector of  $PQ$ . Let  $S$  be a finite subset of the plane, with more than one element, that satisfies the following properties:
(i) If  $P$  and  $Q$  are in  $S$ , then  $m(PQ)$  intersects  $S$ .
(ii) If  $P_1Q_1, P_2Q_2, P_3Q_3$  are three diferent segments such that its endpoints are points of  $S$ , then, there is non point in  $S$  such that it intersects the three lines  $m(P_1Q_1)$ ,  $m(P_2Q_2)$ , and  $m(P_3Q_3)$ .
Find the number of points that  $S$  may contain.

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