
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 in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is $0$ if the second flip is heads, and $1$ if the second flip is tails. However, if the first coin flip is tails, then the number is chosen uniformly at random from two disjoint intervals $[0, \frac{1}{3}]$ or $[\frac{2}{3}, 1]$. Two random numbers $x$ and $y$ are chosen independently using this method. What is the probability that $|x-y| < \tfrac{1}{3}$?
A) $\frac{1}{24}$
B) $\frac{1}{16}$
C) $\frac{1}{8}$
D) $\frac{1}{4}$
E) $\frac{1}{3}$

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