
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
Numbers between 0 and 1 are chosen as follows: A fair coin is flipped twice. If the first flip is heads and the second is tails, the number is 1. If the first flip is heads and the second is heads, the number is 0. If the first flip is tails, regardless of the second flip, the number is 0.5. Two independent numbers x and y are chosen using this method. Compute the probability that the absolute difference between x and y, |x-y|, is greater than 0.5.

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