
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 an infinite geometric sequence $\{a_{n}\}$ with the sum of the first $n$ terms being $S_{n}$, and $a_{n+1}=2S_{n}+2$ for $n\in \mathbb{N}^{*}$.  
$(1)$ 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 with a common difference of $d_{n}$. If the sequence $\{c_{n}\}$ satisfies $c_{n}=\frac{d_{n}}{na_{n}}$, 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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