
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
Two balls, a red one and a blue one, are independently tossed into bins numbered with the positive integers. For the red ball, the probability that it is tossed into bin $k$ is $\frac{1}{2^k}$ for $k = 1, 2, 3, \ldots$. For the blue ball, the probability that it is tossed into bin $k$ is $\frac{1}{3^k}$ for $k = 1, 2, 3, \ldots$. What is the probability that the red ball is tossed into a higher-numbered bin than the blue ball?
A) $\frac{1}{10}$
B) $\frac{1}{7}$
C) $\frac{1}{5}$
D) $\frac{2}{5}$
E) $\frac{1}{2}$

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