
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 $\{a_n\}$ with the sum of its first $n$ terms denoted as $S_n$, if $a_n$ and $S_n$ satisfy $(t-1)S_n=t(a_n-2)$ (where $t$ is a constant, $t \neq 0$ and $t \neq 1$),
(Ⅰ) find the general formula for the sequence $\{a_n\}$;
(Ⅱ) let $b_n=(-a_n)\cdot \log_{3}(1-S_n)$, when $t= \frac{1}{3}$, 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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