
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, with $a_1=1$, and the sum of the first $n$ terms is $S_n$; the sequence $\{b_n\}$ is a geometric sequence, with $b_1=1$, and it is given that $b_2S_2=6$, $b_2+S_3=8$.  
$(1)$ Find the general formula for the sequences $\{a_n\}$ and $\{b_n\}$.  
$(2)$ Find the sum of the first $n$ terms $T_n$ of the sequence $\{a_n \cdot b_n\}$.

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