
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 a geometric sequence $\{a_n\}$ with a common ratio $q > 1$, the sum of the first $n$ terms is $S_n$, $S_3 = 7$, and $a_1 + 3$, $3a_2$, $a_3 + 4$ form an arithmetic sequence. The sum of the first $n$ terms of sequence $\{b_n\}$ is $T_n$, where $6T_n = (3n+1)b_n + 2$, and $n \in \mathbb{N}^*$.  
(Ⅰ) Find the general formula for the sequence $\{a_n\}$.  
(Ⅱ) Find the general formula for the sequence $\{b_n\}$.

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