
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 the probability of ball ending up in bin \(i\) is \(3^{-i}\) for \(i=1,2,3,...\). More than one ball is allowed in each bin. The probability that the balls end up in an arithmetic progression in distinct bins (example, bins \(2,5,8\)) is \(\frac{p}{q},\) where \(p\) and \(q\) are relatively prime positive integers. What is \(p+q?\)
**A)** 331
**B)** 341
**C)** 351
**D)** 361

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