# Cinch it

Calculus Level 5

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 $$a+b$$?