
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 a sequence $\{a_n\}$ with the sum of the first $n$ terms denoted by $S_n$, where $S_n=2a_n-2$,  
(Ⅰ) Find the general term formula of the sequence $\{a_n\}$;  
(Ⅱ) Let $b_n=\log_2 a_n$ and $c_n=\frac{1}{b_n b_{n+1}}$, with the sum of the first $n$ terms of the sequence $\{c_n\}$ denoted by $T_n$, find $T_n$;  
(Ⅲ) Let $d_n=n a_n$ and the sum of the first $n$ terms of the sequence $\{d_n\}$ denoted by $G_n$, find $G_n$.

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