
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. Calculate \( \dfrac{x}{y} \).

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