
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 that the sum of the first $n$ terms of the sequence $\{a\_n\}$ is $S\_n$, and $S\_n = 2a\_n - 2 (n \in \mathbb{N}^*)$.

1. Find the general term formula for $\{a\_n\}$.
2. Let $b_{n+1} = 2b\_n - 2^{n+1}$, $b\_1 = 8$, and $T\_n$ is the sum of the first $n$ terms of the sequence $\{b\_n\}$. Find the positive integer $k$ such that $T\_k \geq T\_n$ holds for any $n \in \mathbb{N}^*$.
3. Let $c\_n = \frac{a_{n+1}}{(1 + a\_n)(1 + a_{n+1})}$, and $R\_n$ is the sum of the first $n$ terms of the sequence $\{c\_n\}$. If $R\_n < \lambda$ holds for any $n \in \mathbb{N}^*$, find the minimum value of $\lambda$.

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