
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 the sequence ${a_n}$, which is an arithmetic progression with a positive common difference, it also satisfies $a_3·a_6=55$ and $a_2+a_7=16$.  
(1) Find the general formula for the sequence ${a_n}$.  
(2) Let $S_n$ be the sum of the first $n$ terms of the sequence ${a_n}$. Does there exist a positive integer $n$ such that $S_n>30n+400$? If so, find the smallest value of $n$; if not, explain why.  
(3) If the sequence ${a_n}$ and the sequence ${b_n}$ satisfy the equation $a_n = \frac{b_1}{2} + \frac{b_2}{2^2} + \frac{b_3}{2^3} + \dots + \frac{b_n}{2^n}$ (where $n$ is a positive integer), find the sum of the first $n$ terms $S_n$ of the sequence ${b_n}$.

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