
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
The terms of the arithmetic sequence $\{a_n\}$ are all positive numbers, with $a_1=1$ and the sum of the first $n$ terms denoted as $S_n$. The sequence $\{b_n\}$ is a geometric sequence with $b_1=1$ and it is known that $b_2S_2=6$, and $b_2+S_3=8$.  
(1) Find the general formulas for the sequences $\{a_n\}$ and $\{b_n\}$.  
(2) Find the sum of the first $n$ terms of the product sequence $\{a_n \cdot b_n\}$, denoted as $T_n$.

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