The dimensions of a rectangle $R$ are $a$ and $b$, with $a < b$. It is required to obtain a rectangle with dimensions $x$ and $y$, where $x < a$ and $y < a$, such that its perimeter is one-third that of $R$, and its area is one-third that of $R$. Find the number of such (different) rectangles.

Let's think step by step and output the final answer within \boxed{}.
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