
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 $\{a_n\}$, $a_1=1$, $a_{2n}=2a_n+1$ ($n\in\mathbb{N}^*$), and $S_n$ is the sum of the first $n$ terms of the sequence $\{a_n\}$.
(Ⅰ) Find $a_n$ and $S_n$;
(Ⅱ) Let the sequence $\{b_n\}$ satisfy $\frac{b_1}{a_1} + \frac{b_2}{a_2} + \ldots + \frac{b_n}{a_n} = 1 - \frac{1}{2^n}$ ($n\in\mathbb{N}^*$), find the sum of the first $n$ terms of $\{b_n\}$, $T_n$.

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