
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
Let $\{a_{n}\}$ be a sequence of positive terms with the sum of the first $n$ terms denoted as $S_{n}$, and $4{S_n}=a_n^2+2{a_n}-8$. <br/>$(1)$ Find the general formula for the sequence $\{a_{n}\}$; <br/>$(2)$ Can we select from $\{a_{n}\}$ a geometric sequence composed in the original order with $a_{1}$ as the first term, denoted as ${a_{{k_1}}},{a_{{k_2}}},⋯,{a_{{k_m}}},⋯,({{k_1}=1})$? If yes, find the group with the smallest common ratio and write down the general formula for this geometric sequence; if not, explain the reason.

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