
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
Two squares are inscribed in two right triangles, both with sides $5$, $12$, and $13$. In one triangle, a square of side length $x$ is inscribed so that one vertex coincides with the right-angle vertex. In the other triangle, a square of side length $y$ is inscribed such that its side lies along the hypotenuse. Determine the ratio of the side lengths $x$ to $y$.

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