
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
Elmo is drawing with colored chalk on a sidewalk outside. He first marks a set  $S$  of  $n>1$  collinear points. Then, for every unordered pair of points  $\{X,Y\}$  in  $S$ , Elmo draws the circle with diameter  $XY$  so that each pair of circles which intersect at two distinct points are drawn in different colors. Count von Count then wishes to count the number of colors Elmo used. In terms of  $n$ , what is the minimum number of colors Elmo could have used?

*Michael Ren*

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