
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
Find the sum of all the different possible areas of 'amazing' rectangles, where an amazing rectangle's area is equal to three times its perimeter, and one side is always double the other side. The rectangles must have integer side lengths.

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