
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 $\{a_{n}\}$ satisfying $a_{n}=2a_{n-1}+2^{n}-1$ for $n\geq 2$, and $a_{1}=5$,  
(1) If there exists a real number $\lambda$ such that the sequence $\left( \frac{a_{n}+\lambda}{2^{n}}\right)$ forms an arithmetic progression, find the value of $\lambda$;  
(2) Under the condition of (1), find the sum of the first $n$ terms of the sequence $\{a_{n}\}$, denoted as $S_{n}$.

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