
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
A rectangle is dissected into nine non-overlapping squares. The widths (side lengths) of the squares are such that when arranged correctly, they perfectly fill the rectangle without any gaps or overlaps. The side lengths of the squares follow these relationships:
1. The sum of the side lengths of the first and second smallest squares equals the side length of the third smallest square.
2. The sum of the side lengths of the first and third smallest squares equals the side length of the fourth smallest square.
3. The sum of the side lengths of the third and fourth smallest squares equals the side length of the fifth smallest square.
4. The sum of the side lengths of the fourth and fifth smallest squares equals the side length of the sixth smallest square.
5. The sum of the side lengths of the second, third, and fifth smallest squares equals the side length of the seventh smallest square.
6. The sum of the side lengths of the second and seventh smallest squares equals the side length of the eighth smallest square.
7. The sum of the side lengths of the first, fourth, and sixth smallest squares equals the side length of the ninth smallest square.
8. The sum of the side lengths of the sixth and ninth smallest squares equals the sum of the seventh and eighth smallest squares.

Given that the width and the height of the rectangle are relatively prime positive integers, find the perimeter of the rectangle.

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