
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 real numbers \( x \) and \( y \) are chosen independently and uniformly at random from the interval \( (0,1) \). What is the probability that \( \lfloor\log_3x\rfloor = \lfloor\log_4y\rfloor \) given that \( x < 0.5 \)?
**A)** \( \frac{1}{16} \)
**B)** \( \frac{1}{12} \)
**C)** \( \frac{1}{8} \)
**D)** \( \frac{1}{4} \)
**E)** \( \frac{1}{3} \)

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