<|im_start|>system
You are a helpful assistant. You first think about the reasoning process in the mind and then provide the user with the answer. You should try to truely solve the question by yourself without cheating.  <|im_end|>
<|im_start|>user
 
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? Show your work in <think> </think> tags. And return the final answer in <answer> </answer> tags.<|im_end|>
<|im_start|>assistant
Let me solve this step by step.
<think>
user
0

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?
test