
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 $S_{n}$, and $S_{n}=2a_{n}-1$; the sequence $\{b_{n}\}$ satisfies $b_{n-1}-b_{n}=b_{n}b_{n-1}$ $(n\geqslant 2$, $n\in N^{*})$, $b_{1}=1$.<br/>$(Ⅰ)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$;<br/>$(Ⅱ)$ Find the sum $T$ of the first $n$ terms of the sequence $\{\frac{{a}_{n}}{{b}_{n}}\}$.

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