
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
Suppose that \( x \) and \( y \) are chosen randomly and uniformly from \( (0,1) \). What is the probability that \( \left\lfloor \sqrt{\frac{x}{y}} \right\rfloor \) is even? 

Hint: \( \sum_{n=1}^{\infty} \frac{1}{n^{2}} = \frac{\pi^{2}}{6} \).

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