A particle starts at $(0,0)$. Every second it moves with equal probability to one of the eight lattice points (points with integer coordinates) closest to its current position, independently of its previous moves. In other words, the probability is $1/8$ that the particle will move from $(x,y)$ to each of $(x,y+1)$, $(x+1,y+1)$, $(x+1,y)$, $(x+1,y-1)$, $(x,y-1)$, $(x-1,y-1)$, $(x-1,y)$, or $(x-1,y+1)$. A rectangular region is defined by vertices at $(3,1)$, $(-3,1)$, $(-3,-1)$, $(3,-1)$. The particle will eventually hit the boundary of this region for the first time, either at one of the 4 corners of the rectangle or at one of the 14 lattice points in the interior of one of the sides of the rectangle. The probability that it will hit at a corner rather than at an interior point of a side is $m/n$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?