
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 regular octagon ABCDEFGH, points P, Q, R, and S are chosen on sides BC, DE, FG, and HA respectively so that lines AB, PS, RQ, and GH are parallel and equally spaced. What is the ratio of the area of octagon APBQCRDS to the area of octagon ABCDEFGH?

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