
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 sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, and $S_n=2a_n-2$ ($n\in\mathbb{N}^*$).

$(1)$ Find the general formula for the sequence $\{a_n\}$.

$(2)$ If the sequence $\{b_n\}$ satisfies $\frac{1}{a_n} = \frac{b_1}{2+1} - \frac{b_2}{2^2+1} + \frac{b_3}{2^3+1} - \ldots + (-1)^{n+1} \frac{b_n}{2^n+1}$, find the general formula for the sequence $\{b_n\}$.

$(3)$ Under the condition of $(2)$, let $c_n=2^n+\lambda b_n$, is there a real number $\lambda$ that makes the sequence $\{c_n\}(n\in\mathbb{N}^*)$ a monotonically increasing sequence? If it exists, find the range of $\lambda$; if not, explain your reasoning.

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