
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 $S_n = \frac{n(n+1)}{2}$, and a sequence $\{b_n\}$ satisfies $b_n = a_n + a_{n+1} (n \in \mathbb{N}^*)$.

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

(Ⅱ) If $c_n = 2^{a_n} \cdot (b_n - 1) (n \in \mathbb{N}^*)$, find the sum of the first $n$ terms of the sequence $\{c_n\}$, denoted as $T_n$.

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