
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 arithmetic sequence $\{a_{n}\}$ with the first term $a_{1}=-1$ and a common difference $d \gt 1$. Let $S_{n}(n\in N^{*})$ denote the sum of the first $n$ terms of $\{a_{n}\}$.

$(\color{red}{Ⅰ})$ If $S_{4}-2a_{2}a_{3}+6=0$, find $S_{n}$.

$(\color{red}{Ⅱ})$ If for each $n\in N^{*}$, there exists a real number $c_{n}$ such that $a_{n}+c_{n}$, $a_{n+1}+4c_{n}$, $a_{n+2}+15c_{n}$ form a geometric sequence, find the range of $d$.

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