
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 that $\{a_n\}$ is an arithmetic sequence with a positive common difference and it satisfies $a_1 + a_3 = 8$ and $a_2 + a_4 = 12$,
1. Find the general formula for the sequence $\{a_n\}$.
2. If sequence $\{a_n\}$ and geometric sequence $\{b_n\}$ satisfy $b_1 = a_1 - 1$, $b_3 = a_3 + 3$ where $n$ is a positive integer, and the common ratio $q$ of $\{b_n\}$ is greater than $0$, 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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