
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 sequence $\{a\_n\}$ with its first $n$ terms sum $S\_n$, where $a_1=3$ and $2S_n = a_{n+1} + 2n$.

1. Find $a_2$.
2. Find the general formula for the $n$-th term $a_n$ of the sequence $\{a_n\}$.
3. Let $b_n = (2n - 1)(a_n - 1)$, find the sum $T_n$ of the first $n$ terms of the sequence $\{b_n\}$.

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