
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
Textbook knowledge recap:<br/>$(Ⅰ)$ Induction (page 70 of the eighth-grade textbook): The coordinates of the point $\left(x,y\right)$ symmetric about the x-axis are $\left(x,-y\right)$; the coordinates of the point $\left(x,y\right)$ symmetric about the y-axis are $\left(-x,y\right)$;<br/>$(Ⅱ)$ Induction (page 68 of the ninth-grade textbook): When two points are symmetric about the origin, their coordinates have opposite signs, that is, the symmetric point of point $P\left(x,y\right)$ about the origin is ${P'}\left(-x,-y\right)$. After studying the chapters on "Rotation" and "Quadratic Functions," Xiaoying considered the symmetry of function graphs from the perspective of point symmetry and found that the symmetry properties of points can also be applied to linear and quadratic function graphs.<br/>$(1)$ Based on the above knowledge, find the analytical expression of the line that is symmetric about the y-axis with the known line $y=2x+3$;<br/>Solution: Since the coordinates of the point $\left(x,y\right)$ symmetric about the y-axis are $\left(-x,y\right)$;<br/>that is, the point $\left(x,y\right)$ on the line $y=2x+3$ symmetric about the y-axis lies on the line with the point $\left(-x,y\right)$,<br/>Therefore, $y=2\left(-x\right)+3=-2x+3$.<br/>Therefore, the analytical expression of the line symmetric about the y-axis with the known line $y=2x+3$ is $y=-2x+3$. Understand the above problem-solving process and complete the blanks:<br/>The analytical expression of the line symmetric about the x-axis with the known line $y=2x+3$ is ______;<br/>$(2)$ Given that the graph of the quadratic function $y=ax^{2}+bx+c$ is symmetric about the origin with the parabola $y=-2x^{2}+8x-7$, find the values of $a$, $b$, and $c$;<br/>$(3)$ Determine the graphs of the following pairs of functions:<br/>①$y=2x+1$ and $y=-2x-1$;<br/>②$y=x^{2}+1$ and $y=-x^{2}$;<br/>③$y=x^{2}+1$ and $y=-x^{2}-1$;<br/>④$y=\left(x-1\right)^{2}$ and $y=\left(x+1\right)^{2}$. One of them that is definitely symmetric about the origin is ______$(fill in the number)$.

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