
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 diagram shows a rectangle dissected into nine non-overlapping squares. Given that the width and the height of the rectangle are relatively prime positive integers, find the perimeter of the rectangle.
[asy]draw((0,0)--(77,0)--(77,56)--(0,56)--(0,0));draw((30,0)--(30,30)--(0,30)); draw((30,29)--(77,29));draw((39,29)--(39,56));draw((18,30)--(18,56)); draw((33,30)--(33,37)--(18,37)); draw((30,30)--(30,33)--(33,33)); draw((30,33)--(39,33)); draw((33,37)--(39,37));[/asy]

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