
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 that the sum of the first $n$ terms of the sequence $\{a_{n}\}$ is $S_{n}$, choose one of the following conditions:①$a_{n}=\frac{n}{n-1}a_{n-1}(n\geqslant 2)$ and $a_{1}=1$; ②$2S_{n}=n^{2}+n$; ③$a_{n}+a_{n+2}-2a_{n+1}=0$ and $a_{1}=1$, $a_{3}=3$. Supplement the selected condition in the question below and find the solution:<br/>$(1)$ Given that the sequence $\{a_{n}\}$ satisfies ____, find the general formula for $\{a_{n}\}$; <br/>$(2)$ Given a positive geometric sequence $\{b_{n}\}$ such that $b_{1}=a_{2}$ and $b_{2}+b_{3}=12$, find the sum of the first $n$ terms of the sequence $\{\frac{2}{\log_{2}{b}_{n} \cdot \log_{2}{b}_{n+1}}\}$ denoted as $T_{n}$.

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