
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 $\{a\_n\}$ with the sum of its first $n$ terms denoted as $S\_n$, and it satisfies the equation $a\_n - \frac{1}{2}S\_n - 1 = 0 (n \in \mathbb{N}^*)$.

1. Find the general term formula for the sequence $\{a\_n\}$.
2. Determine if there exists a real number $\lambda$ such that the sequence $\{S\_n + (n + 2^n)\lambda\}$ is an arithmetic sequence. If it exists, find the value of $\lambda$; otherwise, explain the reason.

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