
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 $\{a\_n\}$, $a\_1=1$, $a\_2n=2a\_n+1\,(n∈N^∗)$, and $S\_n$ is the sum of the first $n$ terms of the sequence.
(I) Find $a\_n$ and $S\_n$;
(II) Suppose the sequence $\{b\_n\}$ satisfies $\frac{b\_1}{a\_1}+\frac{b\_2}{a\_2}+⋯+\frac{b\_n}{a\_n}=1−\frac{1}{2^n}\,(n∈N^∗)$, find the sum of the first $n$ terms of $\{b\_n\}$, denoted as $T\_n$.

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