
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 legs $5$ and $12$, and a hypotenuse of $13$, has a square inscribed such that one vertex of the square coincides with the right-angle vertex of the triangle. Let the side length of the square be $x$. Another right triangle, also with legs $5$ and $12$, but oriented differently, has a square inscribed such that one side of the square lies along one leg of $12$. The side length of this square is $y$. Calculate $\dfrac{x}{y}$.
A) $\frac{12}{17}$ 
B) $\frac{30}{17}$ 
C) $\frac{17}{30}$ 
D) $\frac{17}{12}$

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