
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 right triangle with sides of length 6, 8, and 10 has a square inscribed in it with side length x and a square inscribed in it with side length y, such that one vertex of the first square coincides with the right-angle vertex of the triangle and a side of the second 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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