
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
The apothem of a regular pentagon having its area numerically equal to its perimeter is compared with the apothem of a rectangle where the length is twice the width and the area is numerically equal to its perimeter. Determine the ratio of the apothem of the pentagon to that of the rectangle.

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