A square piece of paper, \(4\) inches on a side, is folded in half vertically. Both layers are then cut in half parallel to the fold. Three new rectangles are formed, a large one and two small ones. Find the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

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