
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 $\{a_n\}$, the sum of the first $n$ terms is $S_n$, and $S_n = 2a_n - 2$ for any positive integer $n$.  
(1) Find the general formula for $\{a_n\}$.  
(2) Let $b_{n+1} = 2b_n - 2^{n+1}$ with $b_1 = 8$. If $T_n$ is the sum of the first $n$ terms of the sequence $\{b_n\}$, find the positive integer $k$ such that for any positive integer $n$, $T_k \geq T_n$ always holds.  
(3) Let $c_n = \frac{a_{n+1}}{(1 + a_n)(1 + a_{n+1})}$. If $R_n$ is the sum of the first $n$ terms of the sequence $\{c_n\}$, and for any positive integer $n$, $R_n < \lambda$ always holds, find the minimum value of $\lambda$.

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