
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Two concentric squares share the same center $O$, each with sides of length 2. The length of the line segment $\overline{AB}$ is $\frac{1}{3}$, and the area of the octagon $ABCDEFGH$ formed by the intersection of lines drawn from the vertices of the inner square to the midpoints of sides of the outer square is sought. Express this area as a fraction $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
[asy] //code taken from thread for problem
real alpha = 15;
pair W=dir(225), X=dir(315), Y=dir(45), Z=dir(135), O=origin;
pair w=dir(alpha)*W, x=dir(alpha)*X, y=dir(alpha)*Y, z=dir(alpha)*Z;
draw(W--X--Y--Z--cycle^^w--x--y--z--cycle);
pair A=intersectionpoint(Y--Z, y--z),  C=intersectionpoint(Y--X, y--x),  E=intersectionpoint(W--X, w--x),  G=intersectionpoint(W--Z, w--z),  B=intersectionpoint(Y--Z, y--x),  D=intersectionpoint(Y--X, w--x),  F=intersectionpoint(W--X, w--z),  H=intersectionpoint(W--Z, y--z);
dot(O);
label("$O$", O, SE);
label("$A$", A, dir(O--A));
label("$B$", B, dir(O--B));
label("$C$", C, dir(O--C));
label("$D$", D, dir(O--D));
label("$E$", E, dir(O--E));
label("$F$", F, dir(O--F));
label("$G$", G, dir(O--G));
label("$H$", H, dir(O--H));[/asy]

Present the answer in LaTex format: \boxed{Your answer}
user