In the figure below, $\triangle DOG$, $\triangle ION$, and $\triangle IDO$ are congruent and isosceles, each with perimeter $55$. The quadrilaterals $FLIN,$ $DIES,$ and $DRAG$ are all squares. The perimeter of the $11$-sided polygon $DRAGONFLIES$ is $127$. What is the area of $DRAG$? [asy] size(5cm); pair cis(real magni, real argu) { return (magni*cos(argu),magni*sin(argu)); } real t=asin(9/46); pair o=(0,0); pair g=cis(23,pi/2-3*t); pair d=cis(23,pi/2-t); pair i=cis(23,pi/2 t); pair n=cis(23,pi/2 3*t); pair a=g cis(9,pi/2-2*t); pair r=d cis(9,pi/2-2*t); pair s=d (0,9); pair e=i (0,9); pair l=i cis(9,pi/2 2*t); pair f=n cis(9,pi/2 2*t); draw(d--r--a--g--o--n--f--l--i--e--s--d,black 1); draw(g--d--i--n,black 1); draw(o--i,black 1); draw(o--d,black 1); draw(0.6*n-cis(1,3*t)--0.6*n cis(1,3*t)); draw(0.6*i-cis(1,t)--0.6*i cis(1,t)); draw(0.6*d-cis(1,-t)--0.6*d cis(1,-t)); draw(0.6*g-cis(1,-3*t)--0.6*g cis(1,-3*t)); dot(d); dot(r); dot(a); dot(g); dot(o); dot(n); dot(f); dot(l); dot(i); dot(e); dot(s); label(scale(0.8)*"$D$",d,SSE); label(scale(0.8)*"$R$",r,SSE); label(scale(0.8)*"$A$",a,SSE); label(scale(0.8)*"$G$",g,SSE); label(scale(0.8)*"$O$",o,S); label(scale(0.8)*"$N$",n,SSW); label(scale(0.8)*"$F$",f,SSW); label(scale(0.8)*"$L$",l,SSW); label(scale(0.8)*"$I$",i,SSW); label(scale(0.8)*"$E$",e,SE); label(scale(0.8)*"$S$",s,SW); [/asy]
