
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 $\{a_n\}$ whose sum of the first $n$ terms is $S_n$ and it is known that $S_n = n(n+1)$ where $n \in \mathbb{N}^*$.  
(i) Find the general term formula for the sequence $\{a_n\}$.  
(ii) If a sequence $\{b_n\}$ satisfies $$a_n = \frac{b_1}{3^1+1} + \frac{b_2}{3^2+1} + \frac{b_3}{3^3+1} + \ldots + \frac{b_n}{3^n+1},$$ find the general term formula for the sequence $\{b_n\}$.  
(iii) Let $$c_n = \frac{a_n \cdot b_n}{4}$$ for $n \in \mathbb{N}^*$, find the sum of the first $n$ terms of the sequence $\{c_n\}$, denoted as $T_n$.

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