
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 two sequences $\{a_n\}$ and $\{b_n\}$, satisfying: $a_1=3$. For $n\geq2$, $a_{n-1}+a_n=4n$; for any positive integer $n$, $b_1+2b_2+\ldots+2^{n-1}b_n=na_n$.  
Let the sum of the first $n$ terms of the sequence $\{b_n\}$ be $S_n$.  
(1) Calculate $a_2$, $a_3$. And find the general formula for the sequence $\{a_n\}$.  
(2) Find the set of positive integers $n$ that satisfy $13<S_n<14$.

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