
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 positive integer \( i \) is chosen at random such that the probability of a positive integer \( k \) being chosen is \(\frac{3}{2}\) times the probability of \( k+1 \) being chosen. What is the probability that the \( i^{\text{th}} \) digit after the decimal point of the decimal expansion of \(\frac{1}{7}\) is a 2?

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