
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
Let $\{a_{n}\}$ be a sequence of positive numbers, and let $S_{n}$ denote the sum of the first $n$ terms. Given that $a_{1}=1$ and $(S_{n+1}+1)a_{n}=(S_{n}+1)a_{n+1}$ holds for all $n\in \mathbb{N}^{*}$, find:<br/>
$(1)$ The general formula for the sequence $\{a_{n}\}$;<br/>
$(2)$ Insert $k$ numbers between $a_{k}$ and $a_{k+1}$ to form an arithmetic sequence of $k+2$ numbers. Let the sum of the inserted $k$ numbers be denoted as $c_{k}$, where $k=1,2,\ldots,n$. Find the sum of the first $n$ terms of the sequence $\{c_{n}\}$.

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