
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
Given a rectangle with dimensions $a$ and $b$ ($a < b$), determine the number of rectangles with dimensions $x$ and $y$ such that $x < \frac{a}{2}$ and $y < \frac{a}{2}$, and the perimeter of the new rectangle is half the perimeter of the original rectangle and its area is half the area of the original rectangle.

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