A point \(P\) is randomly selected from the rectangular region with vertices \((0,0),(2,0),(2,1),(0,1)\). What is the probability, expressed as a fraction \(\frac{k}{m}\), that \(P\) is closer to the origin than it is to the point \((3,1)\)? Please provide the value of \(k + m\).

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