
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
On square  $ABCD,$  points  $E,F,G,$  and  $H$  lie on sides  $\overline{AB},\overline{BC},\overline{CD},$  and  $\overline{DA},$  respectively, so that  $\overline{EG} \perp \overline{FH}$  and  $EG=FH = 34.$  Segments  $\overline{EG}$  and  $\overline{FH}$  intersect at a point  $P,$  and the areas of the quadrilaterals  $AEPH, BFPE, CGPF,$  and  $DHPG$  are in the ratio  $269:275:405:411.$  Find the area of square  $ABCD$ . 


[asy]
size(200);
defaultpen(linewidth(0.8)+fontsize(10.6));
pair A = (0,sqrt(850));
pair B = (0,0);
pair C = (sqrt(850),0);
pair D = (sqrt(850),sqrt(850));
draw(A--B--C--D--cycle);
dotfactor = 3;
dot(" $A$ ",A,dir(135));
dot(" $B$ ",B,dir(215));
dot(" $C$ ",C,dir(305));
dot(" $D$ ",D,dir(45));
pair H = ((2sqrt(850)-sqrt(120))/6,sqrt(850));
pair F = ((2sqrt(850)+sqrt(306)+7)/6,0);
dot(" $H$ ",H,dir(90));
dot(" $F$ ",F,dir(270));
draw(H--F);
pair E = (0,(sqrt(850)-6)/2);
pair G = (sqrt(850),(sqrt(850)+sqrt(100))/2);
dot(" $E$ ",E,dir(180));
dot(" $G$ ",G,dir(0));
draw(E--G);
pair P = extension(H,F,E,G);
dot(" $P$ ",P,dir(60));
label(" $w$ ", (H+E)/2,fontsize(15));
label(" $x$ ", (E+F)/2,fontsize(15));
label(" $y$ ", (G+F)/2,fontsize(15));
label(" $z$ ", (H+G)/2,fontsize(15));
label(" $w:x:y:z=269:275:405:411$ ",(sqrt(850)/2,-4.5),fontsize(11));
[/asy]

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