
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 the sequence $\{a\_n\}$ that satisfies $a\_n = 2a_{n-1} + 1$ ($n \in \mathbb{N}^*, n \geq 2$), and $a\_1 = 1$, $b\_n = a\_n + 1$.
(I) Find the general term formula for the sequence $\{a\_n\}$;
(II) Find the sum of the first $n$ terms, $T\_n$, for the sequence $\{nb\_n\}$.

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