
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 the first three terms of the sequence $\{a_n\}$ are equal to the first three terms of the sequence $\{b_n\}$, and $a_1+2a_2+2^2a_3+…+2^{n-1}a_n=8n$ holds for any $n\in\mathbb{N}^*$, the sequence $\{b_{n+1}-b_n\}$ is an arithmetic sequence.  
(1) Find the general formula for the sequences $\{a_n\}$ and $\{b_n\}$.  
(2) Does there exist $k\in\mathbb{N}^*$ such that $b_k-a_k\in(0,1)$? Please explain your reasoning.

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