optimize

After a little thought, one can see that the sum of minimum and maximum is twice the distance from the point to the centre. And the distance to the centre is $\sqrt{9+16+144}=13$ , so, the answer is $\boxed{26}$ .

The sphere has it's center at the origin of coordinates. The unit vector along the line joining the center of the sphere and the point $(3,4,12)$ is $\hat n=\dfrac{1}{13}(3\hat i+4\hat j+12\hat k)$ . So the line intersects the sphere at the point $\left (\dfrac{3}{13},\dfrac{4}{13},\dfrac{12}{13}\right)$ . So the minimum distance of the given point from the sphere is $\sqrt {\left (3-\dfrac{3}{13}\right )^2+\left (4-\dfrac{4}{13}\right )^2+\left (12-\dfrac{12}{13}\right )^2}=12$ and the maximum distance is $12+2=14$ . Hence the required sum is $12+14=\boxed {26}$