The function \(F(x, y, z)\) is defined in the following manner:

\(F(x, y, z) = \nabla^2 F(x, y, z) \text{, where } \nabla^2 \text{ is the Laplacian.}\)

The solution to this can be written in the following form:

\(F(x, y, z) = a(u) + a'(u) \text{, where } u = x+y+z.\)

The function \(a(u)\) obeys the nonlinear differential equation \(a(2u) = 2 a(u) \cdot a'(u)\). The average of the possible values of \(a(0)\) can be written as \(-\frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers. What is the value of \(a+b\)?

It may help to read about the Laplacian here.

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