
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
Given the sequence \(\{a_{n}\}\) that satisfies the condition \((n-1) a_{n+1}=(n+1)\left(a_{n}-1\right)\) with \(a_{2}=6\), let \(b_{n} = a_{n} + n\) for \(n \in \mathbf{N}_{+}\).

1. Find the first 4 terms of the sequence \(\{a_{n}\}\).
2. Determine the sequence \(b_{n}\).
3. Is there a non-zero constant \(p, q\) such that the sequence \(\left\{\frac{a_{n}}{p \cdot n+q}\right\}\) forms an arithmetic sequence? If it exists, find the relationship that \(p, q\) must satisfy; if it does not exist, provide an explanation.

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