
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 the expansion of $(1+2x)^n$, the sum of the binomial coefficients of all terms is $a_n$, and the coefficient of the second term is $b_n$.  
(1) Find $a_n$ and $b_n$;  
(2) Calculate the sum of the first $n$ terms of the sequence $\{a_nb_n\}$, denoted as $S_n$.

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