
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 $y=kx^{2}-2kx+2k-1$.  
$(Ⅰ)$ If the solution set of the inequality $y\geqslant 4k-2$ with respect to $x$ is denoted as $R$, find the range of real numbers for $k$;  
$(Ⅱ)$ The equation $y=0$ has two distinct real roots $x_{1}$ and $x_{2}$,  
1. Does there exist a real number $k$ such that ${x}_{1}^{2}+{x}_{2}^{2}=3x_{1}x_{2}-4$ holds? If yes, find the value of $k$; if not, explain the reason;  
2. If $x_{1}$ and $x_{2}$ are both greater than zero, find the range of $k$.

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