Advanced Number Theory · Superchallenge problem

A dart hits a uniformly random point in the square with vertices $(0,0)$, $(2,0)$, $(2,2)$, and $(0,2)$. The probability that the dart lands closer to the vertex $(0,0)$ than to BOTH of the vertices $(2,0)$ and $(2,2)$ is $\tfrac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m + n$.