
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, one blue and one orange, are independently tossed into bins numbered with positive integers. The probability that a ball is tossed into bin $k$ is $\frac{2^{-k}}{S}$ for $k = 1, 2, 3, \ldots$, where $S = \sum_{k=1}^{\infty} 2^{-k}$. Determine the probability that the blue ball is tossed into a higher-numbered bin than the orange ball.
A) $\frac{1}{4}$
B) $\frac{2}{7}$
C) $\frac{1}{3}$
D) $\frac{3}{8}$
E) $\frac{3}{7}$

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