
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
Read the following material and answer the questions:<br/>When the analytical expression of a parabola contains letter coefficients, the coordinates of the vertex of the parabola will change as the values of the letters change.<br/>For example: Given the parabola $y=x^{2}-2mx+m^{2}+2m-1$,①<br/>From ①, we have $y=\left(x-m\right)^{2}+2m-1$,②<br/>Therefore, the coordinates of the vertex of the parabola $y=x^{2}-2mx+m^{2}+2m-1$ are $\left(m,2m-1\right)$, i.e., $\left\{\begin{array}{l}{x=m③}\\{y=2m-1④}\end{array}\right.$<br/>When the value of $m$ changes, the values of $x$ and $y$ also change. Substituting ③ into ④, we get $y=2x-1$.⑤<br/>It can be seen that regardless of the value of $m$, the vertical coordinate $y$ and the horizontal coordinate $x$ of the vertex of the parabola satisfy $y=2x-1$.<br/>$(1)$ In the above process, the mathematical method used to obtain ② from ① is ______ (Fill in "$A$" or "$B$"), and the mathematical method used to obtain ⑤ from ③④ is ______ (Fill in "$A$" or "$B$").<br/>$A$. Elimination method<br/>$B$. Completing the square method<br/>$(2)$ Based on the methods provided in the above material, determine the functional relationship between the vertical coordinate $y$ and the horizontal coordinate $x$ of the vertex of the parabola $y=x^{2}-2mx+2m^{2}-3m+1$.<br/>Solution: $y=x^{2}-2mx+2m^{2}-3m+1=(x-$______)$^{2}+m^{2}-3m+1$,<br/>Therefore, the coordinates of the vertex of this parabola are $(m,$______), i.e., $\left\{\begin{array}{l}{x=()①}\\{y=()②}\end{array}\right.$<br/>When the value of $m$ changes, the values of $x$ and $y$ also change. Substituting ① into ②, we get $y=x^{2}-3x+1$.<br/>It can be seen that regardless of the value of $m$, the vertical coordinate $y$ and the horizontal coordinate $x$ of the vertex of the parabola satisfy______.

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