
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 that the sequence $\{a\_n\}$ satisfies $a\_1=1$, $a_{n+1}=3a\_n$, and the sum of the first $n$ terms of the sequence $\{b\_n\}$ is $S\_n=n^2+2n+1$.
(I) Find the general term formulas for the sequences $\{a\_n\}$ and $\{b\_n\}$;
(II) If $c\_n=a\_nb\_n$, find the sum of the first $n$ terms of the sequence $\{c\_n\}$, denoted as $T\_n$.

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