
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 with a common difference greater than $0$, and it satisfies $a_3a_6=55$, $a_2+a_7=16$.  
$(1)$ Find the general formula for the sequence $\{a_n\}$.  
$(2)$ If the sequence $\{b_n\}$ satisfies $\frac{b_1}{2}+ \frac{b_2}{2^2}+ \frac{b_3}{2^3}+…+ \frac{b_n}{2^n}=a_n$ $(n\in\mathbb{N}^*)$, find the sum of the first $n$ terms of the sequence $\{b_n\}$.

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