
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 the sequence $\{a\_n\}$ that satisfies $a\_1=1$, $a\_{n+1}=2S\_n+1$, where $S\_n$ is the sum of the first $n$ terms of $\{a\_n\}$, $n∈N^*$.
(1) Find the general term formula of the sequence $\{a\_n\}$;
(2) Suppose $\{b\_n-a\_n\}$ is an arithmetic sequence with the first term 1 and common difference 2, find the sum of the first $n$ terms $T\_n$ of the sequence $\{b\_n\}$.

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