
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 $\{a\_n\}$ is a monotonically increasing arithmetic sequence with first term $a\_1=3$, and the sum of the first $n$ terms is $S\_n$. The geometric sequence $\{b\_n\}$ has first term $b\_1=1$, and it is given that $a\_2b\_2=12$, $S\_3+b\_2=20$.
(1) Find the general term formulas for $\{a\_n\}$ and $\{b\_n\}$;
(2) Find the sum of the first $n$ terms, $T\_n$, for the sequence $\{a\_n b\_n\}$.

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