
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
Triangle XYZ is a right triangle with a right angle at Y. Around this triangle, three squares are constructed such that one square lies on each side of the triangle. The sum of the areas of the three squares is 500 square centimeters. Determine the number of square centimeters in the area of the square constructed on XY.

[asy]
draw((0,0)--(15,0)--(0,8)--cycle);
dot((0,0));
dot((15,0));
dot((0,8));
draw((0,0)--(0,8)--(-8,8)--(-8,0)--cycle);  // Square on XY
draw((0,0)--(0,-15)--(15,-15)--(15,0));
draw((0,8)--(8,20)--(20,15)--(15,0)--cycle);
label("$X$",(0,8),NW);
label("$Y$",(0,0),SE);
label("$Z$",(15,0),E);
[/asy]

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