
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 a sequence $\{a_{n}\}$ where each term is not equal to $0$, and the sum of the first $n$ terms is $S_{n}$, and $2^{n}a_{n+1}=n^{b}S_{n}$.<br/>$(Ⅰ)$ If $b=0$, find the sum $T_{n}$ of the first $n$ terms of the sequence $\left\{{\frac{{{a_{n+1}}}}{{{a_n}}}}\right\}$;<br/>$(Ⅱ)$ If $b=2$, find the maximum value of $\frac{{{S_{n+1}}}}{{{S_n}}$.

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