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 \(AD\) is between \(\frac{1}{3}\) and \(\frac{2}{3}\). The area of \(R\) is given in the form \(\frac{k-m\sqrt{n}}{p}\), where \(n\) is a positive integer. Find the value of \(k + m + n + p\).

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