
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
Given a right triangle with sides of length $5$, $12$, and $13$, and a square with side length $x$ inscribed in it so that one vertex of the square coincides with the right-angle vertex of the triangle, and another square with side length $y$ inscribed in a different right triangle with sides of length $5$, $12$, and $13$ so that one side of the square lies on the hypotenuse of the triangle, find the value of $\frac{x}{y}$.

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