
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 in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again; if the second flip is heads, the chosen number is 0, and if tails, the chosen number is 0.5. 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 chosen in this manner. What is the probability that $|x-y| > \tfrac{1}{3}$?
A) $\frac{1}{9}$
B) $\frac{2}{9}$
C) $\frac{5}{18}$
D) $\frac{5}{9}$
E) $\frac{23}{36}$

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