
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 the first $n$ terms denoted as $S_{n}$, $a_{1}=3$, $\frac{{S}_{n+1}}{{S}_{n}}=\frac{{3}^{n+1}-1}{{3}^{n}-1}$, $n\in N^{*}$.  
$(1)$ Find $S_{2}$, $S_{3}$, and the general formula for $\{a_{n}\}$;  
$(2)$ Let $b_n=\frac{a_{n+1}}{(a_n-1)(a_{n+1}-1)}$, the sum of the first $n$ terms of the sequence $\{b_{n}\}$ is denoted as $T_{n}$. If $T_{n}\leqslant \lambda (a_{n}-1)$ holds for all $n\in N^{*}$, find the minimum value of $\lambda$.

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