Points \(K, L, M,\) and \(N\) lie in the plane of the square \(ABCD\) such that \(AKB\), \(BLC\), \(CMD\), and \(DNA\) are equilateral triangles. If \(ABCD\) has an area of \(16\), find the area of \(KLMN\). The original answer is in the form \(k + m\sqrt{3}\). Please give the value of \(k + m\).

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