Condition on x-coordinate of a point such that three distinct normals can be drawn to a parabola

The set of points on the axis of the parabola $2{(x−1)^2+(y−1)^2}=(x+y)^2$ from which three distinct normals can be drawn is the set of points (h,k) lying on the axis of the parabola such that h>3/2?

How to prove this claim?I am not being able to proceed much.Just did some axis rotation to get the form $X^2=4(Y-1)$ (after rotating axes by 45 degrees).

Next step: find the equation of the normal to the parabola at $(u,1+u^2/4)$. Where does this intersect the axis? What other normals go through the same point on the axis?