
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 an arithmetic sequence $\{a_n\}$ with a non-zero common difference, the sum of its first $n$ terms is denoted as $S_n$. It is known that $S_3=9$ and the terms $a_1, a_3, a_7$ form a geometric sequence.

(1) Find the general term formula for the sequence $\{a_n\}$.

(2) If sequence $\{b_n\}$ satisfies $b_n=(a_n-1)2^n$, find the sum of the first $n$ terms of sequence $\{b_n\}$, denoted as $T_n$.

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