
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 sum of the first $n$ terms $S_n = 2n^2$, and in the sequence $\{b_n\}$, $b_1 = 1$, $b_{n+1} = 3b_n$ ($n \in \mathbb{N}^*$),  
(Ⅰ) Find the general formula for sequences $\{a_n\}$ and $\{b_n\}$.  
(Ⅱ) Let $c_n = a_n \cdot b_n$, 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