
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 between 0 and 1, inclusive, are chosen as follows: A fair six-sided die is rolled. If it rolls a 1 or 2, the number is chosen uniformly from the interval \([0, \frac{1}{2}]\). If it rolls a 3 or 4, the number is chosen uniformly from the interval \([\frac{1}{2}, 1]\). If it rolls a 5, the number is fixed at 0. Finally, if it rolls a 6, the number is fixed at 1. Two random numbers \(x\) and \(y\) are selected independently via this method. What is the probability that \(|x-y| > \frac{1}{3}\)?
A) \(\frac{1}{12}\)
B) \(\frac{1}{9}\)
C) \(\frac{2}{9}\)
D) \(\frac{1}{4}\)
E) \(\frac{1}{3}\)

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