
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
We extend a given rectangle inscribed in a circle into an octagon, making the four vertices of the rectangle the vertices of the octagon, and the other four vertices the intersection points of the perpendicular bisectors of the rectangle's sides and the circle. How should we choose the rectangle to minimize the ratio of the areas of the octagon and the rectangle?

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