
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 not equal to $0$, and $a_{1}=2$, $a_{1}$, $a_{3}$, $a_{7}$ form a geometric sequence.<br/>$(1)$ Find the general formula for the sequence $\{a_{n}\}$;<br/>$(2)$ If the sequence $\{b_{n}\}$ satisfies ${b}_{n}=\frac{1}{{a}_{n}•({a}_{n}-1)}$, and $S_{n}$ denotes the sum of the first $n$ terms of the sequence $\{b_{n}\}$, find $S_{n}$.

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