Let $a$ and $b$ be real numbers such that the quadratic equations $x^2 + ax + b = 0$ and $ax^2 + bx + 1 = 0$ have a root in common.  Enter all possible values of $a + b,$ separated by commas.
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $a$ and $b$ be real numbers such that the quadratic equations $x^2 + ax + b = 0$ and $ax^2 + bx + 1 = 0$ have a root in common.  Enter all possible values of $a + b,$ separated by commas.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
-1,2
Let $a$ and $b$ be real numbers such that the quadratic equations $x^2 + ax + b = 0$ and $ax^2 + bx + 1 = 0$ have a root in common.  Enter all possible values of $a + b,$ separated by commas.
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