
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 rectangle \( R \) has side lengths \( a \) and \( b \), where \( a < b \). To obtain a rectangle with side lengths \( x \) and \( y \) (where \( x < a \) and \( y < a \)) such that its perimeter is \(\frac{1}{3}\) the perimeter of \( R \) and its area is \(\frac{1}{3}\) the area of \( R \), find the number of different rectangles.

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