
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
The sequence $\{a_{n}\}$ and $\{b_{n}\}$ are both sequences of positive numbers. The sum of the first $n$ terms of the sequence $\{b_{n}\}$ is $S_{n}$, where $b_{1}=1$, $\frac{{S}_{n+1}}{{S}_{n}}=(1+\frac{1}{n})^{2}$, and $b_{n+1}=a_{n}b_{n}$.  
$(1)$ Find the general formula for the sequence $\{b_{n}\}$;  
$(2)$ Let ${c}_{n}=(2-{a}_{n})\cdot({a}_{n+1}-1)\cdot{2}^{n}$, find the sum of the first $n$ terms of the sequence $\{c_{n}\}$ denoted by $T_{n}$.

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