
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 two sequences \\(\{a_n\}\) and \\(\{b_n\}\) with the first term of each being \\(1\\) and \\(b_n \neq 0\\) for \\(n \in \mathbb{N}^*\), satisfying \\(a_nb_{n+1}-a_{n+1}b_n+2b_{n+1}b_n=0\\),

  \\((1)\\) Let \\(c_n = \dfrac{a_n}{b_n}\\), find the general formula for the sequence \\(\{c_n\}\);

  \\((2)\\) If \\(b_n = 3^{n-1}\\), find the sum of the first \\(n\\) terms of the sequence \\(\{a_n\}\), denoted as \\(S_n\\).

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