
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
Let's agree: if the graph of function $C_{1}$ is rotated $180$ degrees around a point and coincides with the graph of function $C_{2}$, then the functions $C_{1}$ and $C_{2}$ are said to be "centrally symmetric functions" with respect to this point, and this point is called the "symmetric center" of functions $C_{1}$ and $C_{2}$. There are various methods to find the "centrally symmetric function" of a function. For example, to find the centrally symmetric function of the function $C_{1}$: $y=x$ with respect to $\left(1,0\right)$, we can take points $\left(0,0\right)$ and $\left(1,1\right)$ on the function $C_{1}$. The points symmetric to $\left(1,0\right)$ with respect to $\left(0,0\right)$ and $\left(1,1\right)$ are $\left(2,0\right)$ and $\left(1,-1\right)$, respectively. In this way, we can obtain the centrally symmetric function of the function $y=x$ with respect to $\left(1,0\right)$ as $y=x-2$.

$(1)$ Find the centrally symmetric function of the function $y=x+1$ with respect to $\left(0,0\right)$.

$(2)$ Given the function $C_{1}$: $y=2x+b$, if the intersection points of the graph of the centrally symmetric function $C_{2}$ of function $C_{1}$ with respect to $\left(0,-b\right)$ and the graph of $y=-bx+2$ are integer points (points with both horizontal and vertical coordinates as integers), find the positive integer value of $b$.

$(3)$ Given the function $C_{1}: y=-ax^{2}+2bx-c$ (where $a$, $b$, $c$ are constants and $a\neq 0$), if the centrally symmetric function $C_{2}$ of function $C_{1}$ with respect to $\left(0,0\right)$ satisfies the following two conditions: ① $a+b+c=0$, ② $\left(2c+b-a\right)\left(2c+b+3a\right) \lt 0$, find the range of the length of the segment on the $x$-axis obtained by function $C_{2}$.

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