
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 $a_1=1$ and $a_5=5$, and a geometric sequence $\{b_n\}$ whose sum of the first $n$ terms is $S_n = 2 - \frac{1}{2^{n-1}}$ $(n \in \mathbb{N}^*)$.  
(1) Find the general term formulas for sequences $\{a_n\}$ and $\{b_n\}$.  
(2) If $c_n = a_nb_n$ $(n=1, 2, 3, \ldots)$ and $T_n$ is the sum of the first $n$ terms of the sequence $\{c_n\}$, find $T_n$.

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