2018 AMC 10B Problem 17

In rectangle PQRS, PQ=8 and QR=6. Points A and B lie on \overline{PQ}, points C and D lie on \overline{QR}, points E and F lie on \overline{RS}, and points G and H lie on \overline{SP} so that AP=BQ<4 and the convex octagon ABCDEFGH is equilateral. The length of a side of this octagon can be expressed in the form k+m\sqrt{n}, where k, m, and n are integers and n is not divisible by the square of any prime. What is k+m+n?

\textbf{(A) }1 \qquad \textbf{(B) }7 \qquad \textbf{(C) }21 \qquad \textbf{(D) }92 \qquad \textbf{(E) }106 \qquad

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.