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The answer is infinite number of lines
Any line segment, no matter how short it is, has an infinite number of points.
A line has an infinite amount of points.
A line segment consists of an infinite number of points. Depending on how you define "breaking it down", yes, it's possible. But of course not in a finite number of steps.
In Euclidean geometry each line contains a minimum of an infinite number of points. In projective geometry, though, a line may have as few as two points.