
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
In the sequence $\{a_{n}\}$, it is known that $a_{1}=1$, $a_{n+1}=\frac{2(n+1)}{n}{a}_{n}$, where $n\in N^{*}$.  
$(1)$ Find the general formula for the sequence $\{a_{n}\}$.  
$(2)$ If the general formula for the sequence $\{b_{n}\}$ is $b_{n}=\frac{{a}_{n}}{n}$, and $k$ numbers are inserted between $b_{k}$ and $b_{k+1}$ to form an arithmetic sequence of $k+2$ numbers, with the sum of the inserted $k$ numbers 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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