
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. Width and height of the rectangle are relatively prime integers. The sides of the smallest three squares are known as \( a_1, a_2, a_3 \) respectively, where \( a_1 + a_2 = a_3 \) and \( a_2 + a_3 = a_4 \). Then the next four squares are arranged such that \( a_4 + a_3 = a_5 \) and \( a_5 + a_4 = a_6 \), and the last setup includes \( a_6 + a_5 = a_7 \) and \( a_7 + a_6 = a_8 \). The largest square, \( a_9 \) is such that \( a_1 + a_4 + a_7 = a_9 \). Given \( a_1 = 1 \) and \( a_2 = 3 \), find the perimeter of the rectangle.

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