
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 square is inscribed in a circle of unit radius. A circle is then inscribed in this square, and an octagon is inscribed in this circle, followed by another circle inscribed in this octagon, and so on. Generally, in the $n$-th circle, a regular $2^{n+1}$-gon is inscribed, and within this, the $(n+1)$-th circle is inscribed. Let $R_{n}$ be the radius of the $n$-th circle. What is $\lim _{n \rightarrow \infty} R_{n}$ equal to?

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