There is a curve such that the middle point of the normal from any point on the curve to the \(x\)-axis lies on the parabola \(2y^2=x\). The equation of the curve is:

\[ y^2 = bx + c + de^{nx} \]

for some constants \(b,c,d,n\), where e is Euler's number.

What is the value of \( b + c + d + n \)?

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