
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 satisfies $S_n = \frac{1}{2}n^2 + \frac{1}{2}n$. The sequence $\{b_n\}$ satisfies $b_1 = 1$, $2b_n - b_{n-1} = 0$ ($n \geq 2$, $n \in \mathbb{N}^*$).  
(1) Find the general formula for sequences $\{a_n\}$ and $\{b_n\}$.  
(2) Let $c_n = a_n b_n$, find the sum of the first $n$ terms of sequence $\{c_n\}$, $T_n$.

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