
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 \(\{x_n\}\) with the sum of the first \(n\) terms \(S_n = 2a_n - 2\) (\(n \in \mathbf{N}^*\)).

1. Find the general term \(a_n\) of the sequence \(\{a_n\}\).
2. Let \(b_n = \frac{1}{a_n} - \frac{1}{n(n+1)}\), and \(T_n\) be 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 \(n \in \mathbf{N}^*\).
3. Let \(c_n = \frac{a_{n+1}}{\left(1+a_n\right)\left(1+a_{n+1}\right)}\), and \(R_n\) be the sum of the first \(n\) terms of the sequence \(\{c_n\}\). If \(R_n < \lambda\) holds for any \(n \in \mathbf{N}^*\), find the minimum value of \(\lambda\).

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