
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
The two squares shown share the same center $O$ and have sides of length 2. The length of $\overline{AB}$ is $55/99$ and the area of octagon $ABCDEFGH$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
[asy]
real alpha = 25;
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(scale(2)*(W--X--Y--Z--cycle)^^scale(2)*(w--x--y--z--cycle));
pair A=intersectionpoint(scale(2)*(Y--Z), scale(2)*(y--z)),  
    C=intersectionpoint(scale(2)*(Y--X), scale(2)*(y--x)),  
    E=intersectionpoint(scale(2)*(W--X), scale(2)*(w--x)),  
    G=intersectionpoint(scale(2)*(W--Z), scale(2)*(w--z)),  
    B=intersectionpoint(scale(2)*(Y--Z), scale(2)*(y--x)),  
    D=intersectionpoint(scale(2)*(Y--X), scale(2)*(w--x)),  
    F=intersectionpoint(scale(2)*(W--X), scale(2)*(w--z)),  
    H=intersectionpoint(scale(2)*(W--Z), scale(2)*(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