Shuffle the plane

We prove that any permutation $p$ of the plane can be obtained as a composition of a fixed number (209) of simple transformations of the form $(x,y)\to (x,y+f(x))$ and $(x,y)\to (x+g(y),y)$, where $f$ and $g$ are arbitrary $\mathbb{R}\to\mathbb{R}$ functions.

As a corollary we get that the full symmetric group acting on a set of continuum cardinal is a product of finitely many (209) copies of two isomorphic Abelian subgroups.