
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
We know that the graph of the function $y=f(x)$ is symmetric about the origin if and only if the function $y=f(x)$ is an odd function. A student found that this can be generalized as follows: the graph of the function $y=f(x)$ is symmetric about the point $P(a,b)$ if and only if the function $y=f(x+a)-b$ is an odd function. 

(1) Let $f(x)=x^{3}-3x^{2}+1$.
- (i) Find the center of symmetry of the graph of the function $f(x)$ and calculate the value of $f(-2019)+f(-2020)+f(-2021)+f(2021)+f(2022)+f(2023)$.
- (ii) If the graph of the function $y=f(x)$ intersects the graph of $g(x)=\frac{1}{x-1}-1$ at points $A$ and $B$, and the coordinates of point $C$ are $(0,2)$, find the value of $|\overrightarrow{CA}+\overrightarrow{CB}|$.

(2) By analogy with the above generalization, write a generalization of the statement: "The graph of the function $y=f(x)$ is symmetric about the $y$-axis if and only if the function $y=f(x)$ is an even function."

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