
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
A square with side length \( x \) is inscribed in a right triangle with sides of length \( 5 \), \( 12 \), and \( 13 \) so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length \( y \) is inscribed in another right triangle with sides of length \( 5 \), \( 12 \), and \( 13 \) so that one side of the square lies on the hypotenuse of the triangle. What is \( \frac{x}{y} \)?
A) \( \frac{17}{60} \)
B) \( \frac{5}{13} \)
C) \( 1 \)
D) \( \frac{17}{156} \)
E) \( \frac{13}{5} \)

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