
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 $\left\{ a_n \right\}$ are all positive, with $a_1=1$, and the sum of the first $n$ terms is $S_n$; the sequence $\left\{ b_n \right\}$ is a geometric sequence, with $b_1=1$, and $b_2S_2=6$, $b_2+S_3=8$.

$(1)$ Find the general formula for the sequences $\left\{ a_n \right\}$ and $\left\{ b_n \right\}$;

$(2)$ Find the sum of the first $n$ terms $S_n$ for the sequence $\left\{ a_n \cdot b_n \right\}$.

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