
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 that $\{a_n\}$ is an arithmetic sequence, and $a_3=-6$, $a_6=0$.  
$(1)$ Find the general formula for $\{a_n\}$.  
$(2)$ If the geometric sequence $\{b_n\}$ satisfies $b_1=8$, $b_2=a_1+a_2+a_3$, find the formula for the sum of the first $n$ terms of $\{b_n\}$.

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