Let $O$ be the origin, and let $OABC$ be a rectangle, where $A$ and $C$ lie on the parabola $y = x^2.$  Then vertex $B$ must lie on a fixed parabola.  Enter the equation of the fixed parabola in the form "$y = px^2 + qx + r$".
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $O$ be the origin, and let $OABC$ be a rectangle, where $A$ and $C$ lie on the parabola $y = x^2.$  Then vertex $B$ must lie on a fixed parabola.  Enter the equation of the fixed parabola in the form "$y = px^2 + qx + r$".
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
y = x^2 + 2
Let $O$ be the origin, and let $OABC$ be a rectangle, where $A$ and $C$ lie on the parabola $y = x^2.$  Then vertex $B$ must lie on a fixed parabola.  Enter the equation of the fixed parabola in the form "$y = px^2 + qx + r$".
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