
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 a sequence $\{a_{n}\}$ that satisfies $a_{1}+2a_{2}+3a_{3}+\ldots +na_{n}=(n-1)\cdot 2^{n+1}+2(n\in \mathbb{N}^{*})$.  
$(1)$ Find the general formula for the sequence $\{a_{n}\}$.  
$(2)$ Let $b_{n}=\frac{a_{n}}{(a_{n}+1)(a_{n+1}+1)}$. Find the sum of the first $n$ terms of the sequence $\{b_{n}\}$ denoted as $S_{n}$.

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