
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 $\{a_n\}$ whose sum of the first $n$ terms $S_n = 2^{n+1} - 2$ ($n \in \mathbb{N}^*$),  
(Ⅰ) Find the general formula for the $n$-th term $a_n$ of the sequence;  
(Ⅱ) If $b_n = a_n \log_{2} a_n$ ($n \in \mathbb{N}^*$), 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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