
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
Let \\(\{a_n\}\\) be a positive geometric sequence with \\(a_4=81\\), and the arithmetic mean of \\(a_2\\) and \\(a_3\\) is \\(\dfrac{3}{2}(a_1+a_2)\\).
\\((1)\\) Find the general formula for the sequence \\(\{a_n\}\\).
\\((2)\\) If \\(b_n=\log_3a_{2n-1}\\), the sum of the first \\(n\\) terms of the sequence \\(\{b_n\}\\) is \\(S_n\\), and the sequence \\(\{c_n\}\\) satisfies \\(c_n= \dfrac{1}{4S_n-1}\\), \\(T_n\\) is the sum of the first \\(n\\) terms of the sequence \\(\{c_n\}\\), find \\(T_n\\).

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