
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 coin is flipped three times. If the first two flips are heads, then the number is 0 if the third flip is heads, and 1 if the third flip is tails. If the first two flips result in exactly one head, the number is uniformly chosen from the closed interval $[0,\frac{1}{2}]$. Otherwise, the number is uniformly chosen from the closed interval $[\frac{1}{2}, 1]$. Two random numbers $x$ and $y$ are chosen independently using this method. What is the probability that $|x-y| > \tfrac{1}{4}$?
A) $\frac{261}{1024}$\qquad B) $\frac{423}{1024}$\qquad C) $\frac{1}{2}$\qquad D) $\frac{5}{8}$\qquad E) $\frac{2}{3}$

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