Vertex \(E\) of equilateral \(\triangle{ABE}\) is in the interior of unit square \(ABCD\). Let \(R\) be the region consisting of all points inside \(ABCD\) and outside \(\triangle{ABE}\) whose distance from \(\overline{AD}\) is between \(\frac{1}{3}\) and \(\frac{2}{3}\). Find the area of \(R\). The original answer is in the form \(\frac{k-m\sqrt{3}}{n}\), please provide the value of k + m + n.

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