Rectangle \(ABCD\) has \(AB=5\) and \(BC=4\). Point \(E\) lies on \(\overline{AB}\) so that \(EB=1\), point \(G\) lies on \(\overline{BC}\) so that \(CG=1\), and point \(F\) lies on \(\overline{CD}\) so that \(DF=2\). Segments \(\overline{AG}\) and \(\overline{AC}\) intersect \(\overline{EF}\) at \(Q\) and \(P\), respectively. The value of \(\frac{PQ}{EF}\) is in the form \(\frac{k}{m}\). Please find the value of \(k + m\).

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