Triangle \(ABC\) is equilateral with \(AB=1\). Points \(E\) and \(G\) are on \(\overline{AC}\) and points \(D\) and \(F\) are on \(\overline{AB}\) such that both \(\overline{DE}\) and \(\overline{FG}\) are parallel to \(\overline{BC}\). Furthermore, triangle \(ADE\) and trapezoids \(DFGE\) and \(FBCG\) all have the same perimeter. Find the value of \(DE+FG\). 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{}.
user