
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 quadratic polynomials \( f_{1}(x), f_{2}(x), \ldots, f_{100}(x) \) with identical coefficients for \( x^{2} \) and \( x \), but differing constant terms; each polynomial has two roots. For each polynomial \( f_{i}(x) \), one root is chosen and denoted by \( x_{i} \). What values can the sum \( f_{2}\left(x_{1}\right) + f_{3}\left(x_{2}\right) + \ldots + f_{100}\left(x_{99}\right) + f_{1}\left(x_{100}\right) \) take?

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