
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
Let the sequence $\{a_n\}$ have a sum of the first $n$ terms denoted by $S_n$, with $a_1=1$, and for any positive integer $n$, the point $(a_{n+1}, S_n)$ lies on the line $2x+y-2=0$.
(Ⅰ) Find the general formula for the sequence $\{a_n\}$;
(Ⅱ) Does there exist a real number $\lambda$ such that the sequence $\{S_n+\lambda n+ \frac{\lambda}{2^n}\}$ forms an arithmetic sequence? If it exists, find the value of $\lambda$; if not, explain why.

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