
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
Three balls, red, green, and blue, are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability of being tossed into bin k is $2^{-k}$ for $k = 1, 2, 3, \ldots$. Find the probability that the red ball is tossed into a higher-numbered bin than both the green and blue balls.

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