For a real number a, let \lfloor a\rfloor denote the greatest integer less than or equal to a. Let \mathcal{R} denote the region in the coordinate plane consisting of points (x, y) such that
\lfloor x\rfloor^{2}+\lfloor y\rfloor^{2}=25
The region \mathcal{R} is completely contained in a disk of radius r (a disk is the union of a circle and its interior). The minimum value of r can be written as \frac{\sqrt{m}}{n}, where m and n are integers and m is not divisible by the square of any prime. Find m+n.