
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 $\{a\_n\}$, where $a\_1=1$, and the sum of the first $n$ terms $S\_n$ satisfies the condition $\frac{S_{2n}}{S_{n}}=4, n=1,2,…$,
(I) Find the general term formula for the sequence $\{a\_n\}$ and $S\_n$;
(II) Let $b\_n=a\_n \cdot 2^{n-1}$, find the sum of the first $n$ terms $T\_n$ for the sequence $\{b\_n\}$.

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