
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 an arithmetic sequence $\{a_n\}$ and a geometric sequence $\{b_n\}$ with the sum of the first $n$ terms denoted as $S_n$ and $T_n$ ($n \in \mathbb{N}^*$) respectively. If $S_n = \frac{3}{2}n^2 + \frac{1}{2}n$, $b_1 = a_1$, and $b_2 = a_3$, then $a_n = \_\_\_\_\_\_$, $T_n = \_\_\_\_\_\_$.

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