
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
A red ball and a blue ball are independently thrown into bins numbered with the positive integers. The probability that a ball lands in bin $k$ is given by $3^{-k}$. Additionally, there is an extra rule: any ball that lands in an even-numbered bin has this probability halved (making it $\frac{1}{2} \times 3^{-k}$ for even $k$). What is the probability that the red ball is tossed into a higher-numbered bin than the blue ball?
A) $\frac{1}{8}$
B) $\frac{1}{4}$
C) $\frac{1}{2}$
D) $\frac{3}{4}$
E) $\frac{1}{3}$

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