
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 1. The length of  $\overline{AB}$  is  $43/99$  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(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