
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
We have a rectangle \(ABCD\), as shown in the picture. The side lengths \(AB\) and \(BC\) are in the ratio 7:5. Inside the rectangle \(ABCD\) lie points \(X\) and \(Y\) such that triangles \(ABX\) and \(CDY\) are right-angled isosceles triangles with right angles at vertices \(X\) and \(Y\). The area common to both triangles is shaded gray and forms a square with an area of \(72 \, \mathrm{cm}^2\). Determine the side lengths \(AB\) and \(BC\).

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