
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 that the sequence $\{a_n\}$ is an arithmetic sequence with $a_1=1$ and $a_5=5$, and the sum of the first $n$ terms of the geometric sequence $\{b_n\}$ is $S_n=2- \frac{1}{2^{n-1}}$, where $(n\in\mathbb{N}^*)$.  
$(1)$ Find the general formula for the 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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