
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
Given that the function $f(x)$ defined on $\mathbb{R}$ is symmetric about the point $(1, 1)$, and $g(x) = (x-1)^3 + 1$. If the intersection points of the graph of $f(x)$ and $g(x)$ are $(x_1, y_1), (x_2, y_2), ..., (x_{2019}, y_{2019})$, find the value of $\sum\limits_{i=1}^{2019} (x_i + y_i) = \_\_\_\_\_\_.$

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