
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 sequence $\{a\_n\}$ is an arithmetic sequence with a common difference $d$ not equal to $0$, where $a\_1=2$, and $S\_n$ denotes the sum of its first $n$ terms.

1. When $a\_3=6$, if $a\_1$, $a\_3$, $a\;\_{n\_1}$, $a\;\_{n\_2}…$, $a\;\_{n\_k}$ form a geometric sequence (where $3 < n\_1 < n\_2 < … < n\_k$), find the expression for $n\_k$.
2. Does there exist a suitable common difference $d$ such that the ratio of the sum of any first $3n$ terms to the sum of the subsequent $n$ terms in the sequence $\{a\_n\}$ is equal to a constant? Find $d$, or explain why it does not exist.

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