
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 rectangle subdivided into nine non-overlapping squares with distinct side lengths such that two of the side lengths sum to a third. Further, assume that the width and the height of the rectangle are relatively prime positive integers, like before, but the largest square's side is now equal to half the rectangle's width. Find the perimeter of the rectangle.

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