
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 its first $n$ terms denoted as $S\_n$. It is known that $S\_n = n a_{n+1} + n(n+1)$. In a geometric sequence $\{b\_n\}$, $b\_1 = a\_2$, $b\_2 = a\_5$, and $b\_3 = a\_6$.

(I) Find the general term formulas for the sequences $\{a\_n\}$ and $\{b\_n\}$.
(II) If $c\_n = a\_n b\_n$, find the sum of the first $n$ terms, $T\_n$, for the sequence $\{c\_n\}$.

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