
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
Let the sequence $\{a_n\}$ have a sum of the first $n$ terms denoted by $S_n$, where $S_n=2a_n-2$.

(Ⅰ) Find the general formula for the sequence $\{a_n\}$;

(Ⅱ) The sequence $\{b_n-a_n\}$ is an arithmetic sequence with the first term being $1$ and the common difference being $3$. Find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted by $T_n$.

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