
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]


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Let $\{a_{n}\}$ be a sequence of positive terms with the sum of the first $n$ terms denoted as $S_{n}$, and ${a}_{n}+1=\sqrt{4{S}_{n}+9}$.  
$(1)$ Find the general formula for the sequence $\{a_{n}\}$;  
$(2)$ Can we select a geometric sequence ${a}_{{k}_{1}},{a}_{{k}_{2}},⋯,{a}_{{k}_{m}},⋯$ from $\{a_{n}\}$ with $a_{1}$ as the first term and in the original order? If yes, identify the set with the smallest common ratio, write down the general formula for this geometric sequence, and calculate the sum $T_{n}$ of the first $n$ terms of the sequence $\{k_{n}\}$. If not, explain the reason.

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