
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 an arithmetic sequence $\{a_n\}$ with the first term $a_1 = 1$, and it satisfies $a_{n+1} = a_n + 2$. Let $S_n$ denote the sum of the first $n$ terms of $\{a_n\}$.
1. Find the $n$-th term $a_n$ and the sum $S_n$.
2. Suppose $\{b_n\}$ is a geometric sequence with the first term $b_1 = 2$ and common ratio $q$, and $q$ satisfies the quadratic equation $q^2 - (a_4 + 1)q + S_4 = 0$. Find the general term formula for $\{b_n\}$ and the sum of the first $n$ terms, $T_n$.

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