
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
Real numbers $x$ and $y$ are chosen independently and uniformly at random from the interval $(0, \frac{1}{2})$. What is the probability that $\lfloor\log_3{x}\rfloor=\lfloor\log_3{y}\rfloor$?
A) $\frac{1}{36}$
B) $\frac{25}{324}$
C) $\frac{1}{81}$
D) $\frac{4}{81}$
E) $\frac{1}{16}$

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