
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
A set \( F \) of points is placed on a circle, consisting of 100 arcs. For any rotation \( R \) of the circle, the set \( R(F) \) has at least one common point with the set \( F \). (In other words, for any angle \( \alpha \) from \( 0^\circ \) to \( 180^\circ \), the set \( F \) contains two points separated by an angle \( \alpha \).) What is the minimum sum of the lengths of the 100 arcs forming the set \( F \)? What will be the answer if there are not 100 arcs, but \( n \)?

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