
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 arithmetic sequence $\{a_n\}$ has a first term $a_1 = -2$, and the geometric sequence $\{b_n\}$ has a common ratio $q$, and it's given that $a_2 = b_1$, $a_3 = b_2 + 1$, and $a_1b_2 + 5b_2 = b_3$.
$(1)$ Find the general term formula for the sequences $\{a_n\}$ and $\{b_n\}$;
$(2)$ Find the sum of the first $n$ terms $S_n$ of the sequence $\{a_nb_n\}$.

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