
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 $\{a_n\}$ satisfies $a_5=8$, $a_7=12$.
$(1)$ Find the general formula for the sequence $\{a_n\}$;
$(2)$ Suppose that the geometric sequence $\{b_n\}$ consists of positive terms, and its sum of the first $n$ terms is $T_n$. If $b_3=a_3$ and $T_2=3$, find $T_n$.

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