
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 rectangle with side lengths $5$ and $12$ is inscribed in a right triangle with sides of length $5$, $12$, and $13$ so that one vertex of the rectangle coincides with the right-angle vertex of the triangle. Another square with side length $y$ is inscribed in the same right triangle such that one side of the square lies along the hypotenuse of the triangle. What is the ratio of the longer side of the rectangle to the side length $y$ of the square?
A) $\frac{12}{5}$ 
B) $\frac{13}{5}$ 
C) $\frac{5}{13}$ 
D) $\frac{5}{12}$

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