
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 R with dimensions a and b, where a < b, find the dimensions x and y for a new rectangle such that x < b, y < b, the perimeter of the new rectangle is half that of R, and its area is one-fourth that of R. Determine the number of such rectangles.

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