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.