
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 dissected into nine non-overlapping squares, where the side lengths of these squares form specific relationships. The width and the height of the rectangle are relatively prime positive integers. Determine the perimeter of this rectangle. Given the relationships:
\[
a_1 + a_2 = a_4, \quad a_1 + a_4 = a_5, \quad a_2 + a_5 = a_6, \quad a_1 + a_6 = a_7, \quad a_4 + a_7 = a_8, \quad a_5 + a_8 = a_9
\]
where $a_1, \ldots, a_9$ are the side lengths of the squares.

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