
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 $p$ and $q$, where $p > q$, design a new rectangle with dimensions $x$ and $y$, such that $x < p$, $y < q$, the perimeter is one-half that of the original rectangle, and the area is one-fourth that of the original rectangle, and calculate the number of distinct rectangles that can fulfill these criteria.

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