
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
In the arithmetic sequence $\{a_n\}$, $d>0$. If $a_1 + a_4 + a_7 = 12$ and $a_1a_4a_7 = 28$, then there is a geometric sequence $\{b_n\}$ with $b_1 = 16$ and $a_2b_2 = 4$.  
(1) Find the general formula for $\{a_n\}$ and $\{b_n\}$.  
(2) Define $c_n = a_n \cdot b_n (n \in \mathbb{N}^*)$, and find the sum $T_n$ of the first $n$ terms of $\{c_n\}$.

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