
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
We consider \( n \) distinct points in a plane. For each segment formed by a pair of these points, we color the midpoint of the segment blue. How many blue points are needed, at minimum?

If two midpoints coincide, they count as a single point.

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