
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 three conditions where $S_{1}$, $S_{2}$, $S_{4}$ form a geometric sequence, $a_{4}=2a_{2}+2$, and $S_{8}=S_{4}+S_{7}-2$, choose any two of them to supplement the following question and complete the solution. Given that the sequence $\{a_{n}\}$ is an arithmetic sequence with a non-zero common difference, and the sum of its first $n$ terms is $S_{n}$, and satisfies ___, ___.  
$(1)$ Find the general formula for $\{a_{n}\}$;  
$(2)$ Find $\frac{1}{{{a_1}{a_2}}}+\frac{1}{{{a_2}{a_3}}}+\frac{1}{{{a_3}{a_4}}}+⋯+\frac{1}{{{a_n}{a_{n+1}}}$.

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