
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
Definition: If the two real roots of a quadratic equation in $x$, $ax^2+bx+c=0 (a\neq 0)$, are $x_1$ and $x_2 (x_1<x_2)$, and the point $M(x_1,x_2)$ is obtained by taking $x_1$ and $x_2$ as the abscissa and ordinate respectively, then point $M$ is called the marvelous point of the quadratic equation.  
1. If the quadratic equation in $x$ is $x^{2}-2(m-1)x+m^{2}-2m=0$.  
2. Is there exist $b$, $c$, such that regardless of the value of $k$, the marvelous point $M$ of the quadratic equation in $x$, $x^2+bx+c=0$, always lies on the graph of the line $y=kx-2(k-2)$? If yes, please calculate the values of $b$ and $c$; if not, please explain the reason.

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