
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 that the sequence $\{a_n\}$ is an arithmetic sequence with the first term being $1$, and the sequence $\{b_n\}$ is a geometric sequence with the common ratio $q (q \neq 1)$, it is also given that $a_2=b_1$, $a_3-a_1=b_2-b_1$, and $a_2b_2=b_3$.
$(1)$ Find the general term formulas for $\{a_n\}$ and $\{b_n\}$;
$(2)$ If $S_n=a_1b_n+a_2b_{n-1}+\dots+a_{n-1}b_2+a_nb_1$, find $S_n$.

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