
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
Regarding the quadratic polynomials \( f_{1}(x) = a x^{2} + b x + c_{1} \), \( f_{2}(x) = a x^{2} + b x + c_{2} \), \(\ldots\), \( f_{2020}(x) = a x^{2} + b x + c_{2020} \), it is known that each of them has two roots.

Denote by \( x_{i} \) one of the roots of \( f_{i}(x) \), where \( i = 1,2, \ldots, 2020 \). Find the value

$$
f_{2}\left(x_{1}\right) + f_{3}\left(x_{2}\right) + \cdots + f_{2020}\left(x_{2019}\right) + f_{1}\left(x_{2020}\right).
$$

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