
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
Given that real numbers $x$ and $y$ are chosen as follows: two numbers are chosen uniformly at random from the interval $[0,1]$, with the following conditions: if the result of flipping a fair coin twice is two heads, the number is 0, two tails results in 1, one head followed by a tail results in a number chosen uniformly at random from the closed interval $[0, \frac{1}{2}]$, and one tail followed by a head results in a number chosen uniformly at random from the closed interval $[\frac{1}{2}, 1]$. Calculate the probability that $|x-y| > \frac{1}{2}$.

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