KöMaL 2016 B4808

A point $(a;b)$ of the coordinate plane, where $a$ and $b$ are positive integers, is coloured red. Then if a point $(x;y)$ is red then point $(x+1;y+1)$, as well as point $\left(\frac x2;\frac y2\right)$ are also coloured red, provided that $x$ and $y$ are both even. If $(x;y)$ and $(y;z)$ have both been coloured, then we will also colour the point $(x;z)$ red. What is the distance from the origin to the closest point coloured red?