# A Four-Function Functional Equation

Algebra Level 5

$\large f(x)+g(y)=h(x)k(y)$

Let the real-valued functions $$f, g, h, k$$ be defined on $$\left[0, 1\right],$$ such that they satisfy the previous functional equation for all possible values of $$x$$ and $$y$$ in $$\left[0, 1\right],$$ $$f(0)=h(0)=1,$$ and $$k$$ is not a constant function.

If $$M$$ represents the largest possible value of $$f(1)$$ and $$m$$ the smallest value of $$f(1),$$ find $$M-m.$$ Enter 666 if either $$M$$ or $$m$$ does not exist.

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