
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
The sequence $\{a_{n}\}$ satisfies $a_{1}=2$, $a_{n+1}=\lambda a_{n}+2^{n}$, where $n\in \mathbb{N}^{*}$ and $\lambda$ is a constant.  
$(1)$ Does there exist a real number $\lambda$ such that the sequence $\{a_{n}\}$ becomes a geometric sequence? If so, find all possible values of $\lambda$ and the corresponding general formula; if not, explain the reason.  
$(2)$ When $\lambda =2$, let $b_{n}=\frac{a_{n}}{2^{n}}$, find the sum of the first $n$ terms of the sequence $\{b_{n}\}$.

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