
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 chosen between 0 and 2, inclusive, using the following procedure. A fair coin is flipped. If it lands heads, then the coin is flipped again and the chosen number is 0 if the second flip is heads, and 2 if the second flip is tails. If the first coin flip lands tails, then a number is chosen uniformly at random from the closed interval $[0,2]$. Two random numbers $x$ and $y$ are chosen independently with this method. What is the probability that $|x-y| > 1$?
A) $\frac{7}{16}$
B) $\frac{1}{2}$
C) $\frac{5}{8}$  ⟵ Correct
D) $\frac{3}{4}$

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