
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
The probability that the three balls fall into buckets labeled in an arithmetic sequence, where the probability of a ball being thrown into bucket \(i\) is \(2^{-i}\), is given by \(\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers. Find \(p+q\).

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