
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 are randomly and independently tossed into bins numbered with the positive integers such that for each ball, the probability that it is tossed into bin $i$ is $2^{-i}$ for $i=1,2,3,...$ One ball is fixed in bin 1 while the other two can fall into any bin numbered 2 or higher. Calculate the probability that the two free balls end up exactly 3 bins apart from each other and from the ball in bin 1.

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