
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]


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Given a sequence \\(\{a_{n}\}\\) with the sum of its first \\(n\\) terms being \\(S_{n}\\), \\(a_{1}=1\\), and \\(a_{n+1}=2S_{n}+1\\) for \\(n\in\mathbb{N}^{*}\\).
\\((1)\\) Find the general term formula for the sequence \\(\{a_{n}\}\\);
\\((2)\\) Let \\(c_{n}=\log_{3}a_{2n}\\) and \\(b_{n}= \frac{1}{c_{n}\cdot c_{n+1}}\\), denote the sum of the first \\(n\\) terms of the sequence \\(\{b_{n}\}\\) as \\(T_{n}\\), and find \\(T_{n}\\).

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