
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
When $x$ takes the values $\frac{1}{2020}$, $\frac{1}{2019}$, $\frac{1}{2018}$, $\frac{1}{2017}$, $\ldots $, $\frac{1}{2}$, $0$, $1$, $2$, $\ldots $, $2017$, $2018$, $2019$, $2020$, calculate the value of the algebraic expression $\frac{x^{2}-1}{3x^{2}+3}$, add up the results obtained, and the sum is ______.

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