
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 right triangle has legs of lengths \(a\) and \(b\), with \(b > a\), and the hypotenuse \(c\). The ratio of leg \(a\) to leg \(b\) is equal to half of the ratio of leg \(b\) to the hypotenuse \(c\). Additionally, the perimeter of the triangle is \(a + b + c = 12\). Find \( \left(\frac{a}{b}\right)^2 \).

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