
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 real number between 0 and 1, inclusive, is chosen by the following updated method: A fair coin is flipped. If it lands heads, then a fair six-sided die is rolled. If the die roll is 1 or 2, the number chosen is 0; if 3, 4, or 5, the number is 1; if 6, the number is chosen uniformly at random from the interval $[0,1]$. If the first coin flip is tails, then the number is chosen uniformly at random from the interval $[0,1]$. Two random numbers, $x$ and $y$, are chosen independently by this method. What is the probability that $|x-y| > \tfrac{1}{2}$?
A) $\frac{3}{16}$
B) $\frac{9}{32}$
C) $\frac{17}{48}$
D) $\frac{18}{48}$
E) $\frac{1}{2}$

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