
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 a monotonically increasing geometric sequence $\{a_n\}$ satisfies $a_2+a_3+a_4=28$, and $a_3+2$ is the arithmetic mean of $a_2$ and $a_4$.  
(I) Find the general formula for the sequence $\{a_n\}$.  
(II) If sequence $\{b\}$ satisfies $\dfrac{1}{a_n} = \dfrac{b_1}{2+1} - \dfrac{b_2}{2^2+1} + \dfrac{b_3}{2^3+1} - \ldots + (-1)^{n+1} \dfrac{b_n}{2^n+1}$, find the general formula for the sequence $\{b_n\}$.  
(III) Under the condition of (II), let $c_n=2^n+\lambda b_n$. Is there a real number $\lambda$ that makes the sequence $\{c_n\} (n\in\mathbb{N}^*)$ monotonically increasing? If it exists, find the range of $\lambda$; if not, explain why.

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