
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 $x_{2} (x_{1}\leqslant x_{2})$, and the point $M(x_{1}, x_{2})$ is plotted with $x_{1}$ and $x_{2}$ as the coordinates, then point $M$ is called the derived point of the quadratic equation.<br/>$(1)$ If the equation is $x^{2}=3x$, write down the coordinates of the derived point $M$ of this quadratic equation;<br/>$(2)$ For the quadratic equation in $x$, $x^{2}-(2m-1)x+m^{2}-m=0$, find the value of $m$ when its derived point $M$ is closest to the origin.

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