
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 the sum of the first n terms being $S_n$, it satisfies the equation $S_n = 2a_n - n$.
(1) Find the general formula for the $n$-th term of the sequence $\{a_n\}$.
(2) Let $b_n = (2n+1)(a_n + 1)$, find the sum of the first n terms, $T_n$, of the sequence $\{b_n\}$.

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