
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 arithmetic sequence $\{a\_n\}$ and the geometric sequence $\{b\_n\}$ with the sum of the first $n$ terms being $S\_n$ and $T\_n$ respectively, where $S\_n = \frac{3}{2}n^2 + \frac{1}{2}n$, $b\_1 = a\_1$, and $b\_2 = a\_3$. Find the expressions for $a\_n$ and $T\_n$.

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