
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 $\{a_n\}$, $a_1=1$, and the sum of the first $n$ terms $S_n$ satisfies the condition $\frac{S_{2n}}{S_n}=4$ for $n=1,2,\ldots$  
$(1)$ Find the formula for the general term $a_n$ and $S_n$;  
$(2)$ Let $b_n=a_n \cdot 2^{n-1}$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $T_n$.

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