
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
Let the sequence $\{a_n\}$ have a sum of the first $n$ terms denoted by $S_n$, with $a_1=1$, and $a_{n+1}=\lambda S_n+1$ $(n\in\mathbb{N}^*, \lambda\neq -1)$. Also, $a_1$, $2a_2$, and $a_3+3$ are the first three terms of the arithmetic sequence $\{b_n\}$.

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

$(2)$ Find the sum of the first $n$ terms of the sequence $\{a_nb_n\}$.

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