
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]


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Let the sum of the first $n$ terms of a geometric sequence $\{a_n\}$ be $S_n$, given that $a_{n+1}=2S_n+2$ ($n\in\mathbb{N}^*$).

(1) Find the general formula for the sequence $\{a_n\}$.

(2) Insert $n$ numbers between $a_n$ and $a_{n+1}$ to form an arithmetic sequence of $n+2$ numbers with a common difference of $d_n$.

   ① Is there a set of three terms $d_m$, $d_k$, $d_p$ (where $m$, $k$, $p$ form an arithmetic sequence) in the sequence $\{d_n\}$ that form a geometric sequence? If so, find such terms; if not, explain why.

   ② Let $T_n = \frac{1}{d_1} + \frac{1}{d_2} + \frac{1}{d_3} + \ldots + \frac{1}{d_n}$ ($n\in\mathbb{N}^*$), find the values of $n$ for which $T_n \leq \frac{3}{4}$.

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