
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 an arithmetic sequence \\(\{a_n\}\) with the first term \\(1\\), the term \\(a_8\\) is the geometric mean of \\(a_5\\) and \\(a_{13}\\).
\\((1)\\) Find the general formula of the sequence \\(\{a_n\}\);  
\\((2)\\) If the sequence \\(\{a_n\}\) is monotonic and another sequence \\(\{b_n\}\) is defined by \\(b_n= \dfrac{a_{2n+1}}{3^n}\), find the sum of the first \\(n\\) terms of \\(\{b_n\}\), denoted by \\(T_n\\).

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