
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 $\left\{ a_n \right\}$ with the sum of the first $n$ terms denoted as $S_n$, and $S_n=2a_n-2$ $(n\in \mathbb{N}^*)$.

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

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

(3) Under the condition of (2), let $c_n=2^n+\lambda b_n$, is there a real number $\lambda$ such that the sequence $\left\{ c_n \right\}(n\in \mathbb{N}^*)$ is 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}
user