
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 sum of the first $n$ terms of a geometric sequence $\left\{ a_n \right\}$ be $S_n$. It is known that $S_3 = a_4 - a_1$, and $a_1, a_3 + 1, a_4$ form an arithmetic sequence.

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

(Ⅱ) If the sequence $\left\{ b_n \right\}$ satisfies $a_n \cdot b_n = n \cdot 2^{n+1} + 1$, find the sum of the first $n$ terms of the sequence $\left\{ b_n \right\}$, denoted as $T_n$.

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