
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 an arithmetic sequence $\{a_{n}\}$ with a common difference greater than $0$, satisfying $\frac{1}{{{a_1}{a_2}}}+\frac{1}{{{a_2}{a_3}}}+…+\frac{1}{{{a_n}{a_{n+1}}}}=\frac{n}{{2n+4}}$ for $n\in \mathbb{N^*}$.<br/>$(1)$ Find the general formula for the sequence $\{a_{n}\}$;<br/>$(2)$ If $b_{n}=\left(-1\right)^{n}a_{n}a_{n+1}$, find the sum $S_{20}$ of the first $20$ terms of the sequence $\{b_{n}\}$.

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