
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
In the arithmetic sequence $\{a\_n\}$, $a\_1=3$, and the sum of the first $n$ terms is $S\_n$. In the geometric sequence $\{b\_n\}$, all terms are positive, $b\_1=1$, and $b\_2+S\_2=12$. The common ratio $q$ of $\{b\_n\}$ is given by $q=\frac{S\_2}{b\_2}$.

1. Find the general term formulas for the sequences $\{a\_n\}$ and $\{b\_n\}$.
2. If the sequence $\{c\_n\}$ satisfies $c\_n=\frac{3}{2}b\_n(b\_n+1)$, find the sum $T\_n$ of the first $n$ terms of $\{c\_n\}$.

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