<|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
 
Different numbers \(a\), \(b\), and \(c\) are such that the equations \(x^{2}+a x+1=0\) and \(x^{2}+b x+c=0\) have a common real root. In addition, the equations \(x^{2}+x+a=0\) and \(x^{2}+c x+b=0\) also have a common real root. Find the sum \(a+b+c\). 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
-3

Different numbers \(a\), \(b\), and \(c\) are such that the equations \(x^{2}+a x+1=0\) and \(x^{2}+b x+c=0\) have a common real root. In addition, the equations \(x^{2}+x+a=0\) and \(x^{2}+c x+b=0\) also have a common real root. Find the sum \(a+b+c\).
test