
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 $\left\{ a_n \right\}$ satisfies $a_3=2$, and the sum of the first $3$ terms $S_3=\frac{9}{2}$, the sum of the first $n$ terms of the sequence $\left\{ b_n \right\}$, $T_n$, satisfies $2T_n=3^n+3$.

$(1)$ Find the general formula for the sequence $\left\{ a_n \right\}$;

$(2)$ Find the general formula for the sequence $\left\{ b_n \right\}$.

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