
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  $\Gamma$  be a circle in the plane and  $S$  be a point on  $\Gamma$ . Mario and Luigi drive around the circle  $\Gamma$  with their go-karts. They both start at  $S$  at the same time. They both drive for exactly  $6$  minutes at constant speed counterclockwise around the circle. During these  $6$  minutes, Luigi makes exactly one lap around  $\Gamma$  while Mario, who is three times as fast, makes three laps. 

While Mario and Luigi drive their go-karts, Princess Daisy positions herself such that she is always exactly in the middle of the chord between them. When she reaches a point she has already visited, she marks it with a banana.

How many points in the plane, apart from  $S$ , are marked with a banana by the end of the  $6$  minutes.


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