
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
A sequence \(\{a_n\}\) satisfies the equation \(\frac{a_{n+1} + a_n - 1}{a_{n+1} - a_n + 1} = n \left(n \in \mathbf{N}^{*}\right)\), and \(a_2 = 6\).

1. Find the general formula for the sequence \(\{a_n\}\).
2. Let \(b_n = \frac{a_n}{n+c} \left(n \in \mathbf{N}^{*}\right)\), where \(c\) is a nonzero constant. If the sequence \(\{b_n\}\) is an arithmetic sequence, define \(c_n = \frac{b_n}{2^n}\) and \(S_n = c_1 + c_2 + \cdots + c_n\). Find \(S_n\).

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