
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
On a circle, 60 red points and one blue point are marked. Consider all possible polygons with vertices at the marked points. Which type of polygons is more common, those with the blue vertex or those without? And by how many?

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