
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
Consider a regular polygon with  $2^n$  sides, for  $n \ge 2$ , inscribed in a circle of radius  $1$ . Denote the area of this polygon by  $A_n$ . Compute  $\prod_{i=2}^{\infty}\frac{A_i}{A_{i+1}}$ 

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