Alex starts at the origin O of a hexagonal lattice. Every second, he moves to one of the six vertices adjacent to the vertex he is currently at. If he ends up at X after 2018 moves, then let p be the probability that the shortest walk from O to X (where a valid move is from a vertex to an adjacent vertex) has length 2018. Then p can be expressed as \frac{a^{m}-b}{c^{n}}, where a, b, and c are positive integers less than 10; a and c are not perfect squares; and m and n are positive integers less than 10000. Find a+b+c+m+n.