
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 the sequence \(\left\{a_{n}\right\}\) with the sum of the first \(n\) terms 
\[ 
S_n = 2a_n - 2 \quad (n \in \mathbf{Z}_{+}).
\]
(1) Find the general term formula \(a_n\);

(2) Let \(b_n = \frac{1}{a_n} - \frac{1}{n(n+1)}, T_n\) be the sum of the first \(n\) terms of the sequence \(\left\{b_n\right\}\). Find the positive integer \(k\) such that for any \(n \in \mathbf{Z}_{+}\), \(T_k \geqslant T_n\);

(3) Let \(c_n = \frac{a_{n+1}}{(1+a_n)(1+a_{n+1})}, R_n\) be the sum of the first \(n\) terms of the sequence \(\left\{c_n\right\}\). If for any \(n \in \mathbf{Z}_{+}\), \(R_n < \lambda\), find the minimum value of \(\lambda\).

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