
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. If it lands heads, then a number is chosen uniformly at random from the interval $[0,1]$. If it lands tails, the coin is flipped again; based on the second flip, the number 0 is selected for heads and 0.5 is selected for tails. When two numbers, $x$ and $y$, are chosen independently via this method, what is the probability that $|x-y| \geq \frac{1}{2}$?
A) $\frac{1}{16}$
B) $\frac{1}{8}$
C) $\frac{1}{4}$
D) $\frac{1}{2}$
E) $\frac{3}{8}$

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