
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 $b_n=1+3^n a_n (n\in \mathbb{N}^*)$, and $3b_1=10a_1$, where $\{a_n\}$ is an arithmetic sequence with a positive common difference, and $a_2$, $a_7$, $b_2-1$ form a geometric sequence.  
(I) Find the general formula for the sequence $\{a_n\}$.  
(II) Find the sum of the first $n$ terms $S_n$ of the sequence $\{b_n\}$.

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