
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) Clearly, any plane polygonal chain of length 1 can be enclosed in a circle of radius 1: this is achieved by having the center \(O\) of the circle coincide with one of the endpoints of the polygonal chain. What is the radius of the smallest circle that can enclose any polygonal chain of length \(l\)?

b) Clearly, any closed plane polygonal chain with a perimeter of 1 can be enclosed in a circle of radius \(\leq \frac{1}{2}\): this is achieved by having the center \(O\) of the circle coincide with some point on the polygonal chain. (In this case, for any point \(M\) on the polygonal chain, the length of one of the two segments of the polygonal chain from \(O\) to \(M\) will be \(\leq \frac{1}{2}\), and therefore the distance \(OM\) will also be \(\leq \frac{1}{2}\).) What is the radius of the smallest circle that can enclose any closed polygonal chain of length 1?

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