Question 144
The points $(0,0)$, $(1,2)$, $(2,1)$, $(2,2)$ in the plane are colored red while the points $(1,0)$, $(2,0)$, $(0,1)$, $(0,2)$ are colored blue. Four segments are drawn such that each one connects a red point to a blue point and each colored point is the endpoint of some segment. The smallest possible sum of the lengths of the segments can be expressed as $a+\sqrt{b}$, where $a, b$ are positive integers. Compute $100 a+b$.