
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
Given are  $n$  points with different abscissas in the plane. Through every pair points is drawn a parabola - a graph of a square trinomial with  leading coefficient equal to  $1$ . A parabola is called  $good$  if there are no other marked points on it, except for the two through which it is drawn, and there are no marked points above it (i.e. inside it).  What is the greatest number of  $good$  parabolas?

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