
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 $S\_n$ satisfying $2S\_n + a_n = n^2 + 2n + 2$, where $n \in \mathbb{N}^*$. Another sequence $\{b\_n\}$ is defined as $b\_n = a_n - n$.
(I) Find the general formula for the sequence $\{b\_n\}$;
(II) Find the first $n$ terms sum $T\_n$ for the sequence $\{n b\_n\}$.

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