
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
Given an octagonal frame and an equilateral triangular plate with side lengths equal to the sides of the octagon. The plate is aligned with one of its sides to the inner side of the frame. The plate is rolled around the inner side of the frame. How much area does the plate sweep until it reaches the side of the frame directly opposite its starting position, and how much area does it sweep until it returns to its original position?

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