
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 are selected based on a series of coin flips. A fair coin is flipped. If it lands heads, the coin is flipped again: if this second flip is heads, then the number is chosen uniformly from the closed interval $[1,2]$, and if it is tails, the number is 1. On the other hand, if the first coin flip is tails, then the number is chosen uniformly at random from the closed interval $[0,1]$. Two random numbers $x$ and $y$ are independently selected in this manner. What is the probability that $|x-y| > 0.5$?
A) $\frac{1}{8}$
B) $\frac{3}{16}$
C) $\frac{5}{16}$
D) $\frac{7}{16}$
E) $\frac{1}{4}$

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