
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 the function $f(x)=\log_{k}x$ ($k$ is a constant, $k>0$ and $k\neq 1$).<br/>$(1)$ Choose one of the following conditions _____ to make the sequence $\{a_{n}\}$ a geometric sequence, and explain the reason;<br/>① The sequence $\{f(a_{n})\}$ is a geometric sequence with the first term being $2$ and the common ratio being $2$;<br/>② The sequence $\{f(a_{n})\}$ is an arithmetic sequence with the first term being $4$ and the common difference being $2$;<br/>③ The sequence $\{f(a_{n})\}$ is a sequence of the sum of the first $n$ terms of an arithmetic sequence with the first term being $2$ and the common difference being $2$.<br/>$(2)$ Let ${a}_{n}{b}_{n}=\frac{{2}^{n+1}}{4{n}^{2}-1}$, when $k=\sqrt{2}$, find the sum of the first $n$ terms of the sequence $\{b_{n}\}$, denoted as $T_{n}$.

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