
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
In an arithmetic sequence $\{a_n\}$ with a non-zero common difference and a geometric sequence $\{b_n\}$, it is known that $a_1=b_1=1$, $a_2=b_2$, and $a_6=b_3$.  
(1) Find the general formula for the $n$-th term $a_n$ of the arithmetic sequence $\{a_n\}$ and the general formula for the $n$-th term $b_n$ of the geometric sequence $\{b_n\}$.  
(2) Calculate the sum of the first $n$ terms $S_n$ of the sequence $\{a_n \cdot b_n\}$.

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