
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 $\{a_{n}\}$, given $a_{1}=1$, $\frac{a_{n+1}}{n+1}-\frac{a_n}{n}=\frac{1}{n(n+1)}$, where $n\in{{N}^*}$.<br/>$(1)$ Find the general formula for the sequence $\{a_{n}\}$;<br/>$(2)$ Let $b_{n}=(-1)^{n-1}\frac{4n}{a_{n}a_{n+1}}$, 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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