
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\}\) that satisfies \\(3S_n = (n+2)a_n (n \in \mathbb{N}^*)\\), where \\(S_n\\) is the sum of the first \\(n\\) terms of the sequence \\(\{a_n\}\), and \\(a_1 = 2\\).

\\((1)\\) Find the general formula for the sequence \\(\{a_n\}\).

\\((2)\\) Let \\(T_n\\) be the sum of the first \\(n\\) terms of the sequence \\(\left\{\frac{1}{{a_n}}\right\}\). Is there an infinite set \\(M\\) such that for all \\(n \in M\\), \\(|T_n - 1| < \frac{1}{10}\\) holds? If such a set exists, please find one; if not, explain why.

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