The fractional distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ is defined as \[ \sqrt{ \left\| x_1 - x_2 \right\|^2 + \left\| y_1 - y_2 \right\|^2},\]where $\left\| x \right\|$ denotes the distance between $x$ and its nearest integer. Find the largest real $r$ such that there exists four points on the plane whose pairwise fractional distance are all at least $r$.