
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]


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Given a sequence $\{a\_n\}$ whose first $n$ terms sum $S\_n$ satisfies: $S\_n= \frac {a}{a-1}(a\_n-1)$ ($a$ is a constant, and $a\neq 0$, $a\neq 1$);
(1) Find the general term 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 in (2), and the sequence $c\_n=(n-1)b\_n$, find the first $n$ terms sum $T\_n$ of the sequence $\{c\_n\}$.

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