
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 $S_n = n^2$ ($n \in \mathbb{N}^*$), and a geometric sequence $\{b_n\}$ satisfies $b_1 = a_1$, $2b_3 = b_4$.  
(1) Find the general formula for sequences $\{a_n\}$ and $\{b_n\}$.  
(2) If $c_n = a_n \cdot b_n$ ($n \in \mathbb{N}^*$), find the sum of the first $n$ terms of sequence $\{c_n\}$, $T_n$.

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