
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]


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Given a sequence $\{a_n\}$ with a sum of the first $n$ terms denoted as $S_n$, and the point $(n, S_n)$ lies on the graph of the function $y=2^{x+1}-2$.

(I) Find the general term formula for the sequence $\{a_n\}$;
(II) Consider a sequence $\{b_n\}$ satisfying: $b_1=0$, $b_{n+1}+b_n=a_n$, find the sum formula of the first $n$ terms of the sequence $\{b_n\}$;
(III) Under the conditions of question (II), if for any $n\in\mathbb{N}^*$ the inequality $b_n < \lambda b_{n+1}$ always holds, determine the range of real values for $\lambda$.

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