
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, there is a set $F$ of points which consists 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 $180^{\circ}$, there are two points in the set $F$ that are separated by an angle $\alpha$.) What is the minimum total length that the 100 arcs forming the set $F$ can have? What would be the answer if instead of 100 arcs, there are $n$ arcs?

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