
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 $\{a_n\}$ whose sum of the first $n$ terms $S_n$ satisfies: $S_n = \frac{a}{a-1}(a_n - 1)$ (where $a$ is a constant, and $a \neq 0$, $a \neq 1$);  
(1) Find the general formula for $\{a_n\}$.  
(2) Let $b_n = \frac{2S_n}{a_n} + 1$, if the sequence $\{b_n\}$ is a geometric sequence, find the value of $a$.  
(3) If the sequence $\{b_n\}$ is the geometric sequence from (2), and the sequence $c_n = (n-1)b_n$, find the sum of the first $n$ terms of $\{c_n\}$, $T_n$.

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