
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 new method of choosing real numbers between 0 and 1, inclusive, is as follows: Flip a fair coin three times. If all three flips are heads, the chosen number is 0. If all three flips are tails, the chosen number is 1. If the sequence of flips is not all heads or all tails, then the number is chosen uniformly at random from the interval $[0,1]$. Two random numbers, $x$ and $y$, are chosen independently using this method. What is the probability that $|x-y| > \tfrac{1}{3}$?
A) $\frac{1}{32}$
B) $\frac{3}{32}$
C) $\frac{5}{32}$
D) $\frac{7}{32}$
E) $\frac{9}{32}$

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