Problem 1002

The equilateral triangle $ABC$ has sides of integer length $N$. The triangle is completely divided (by drawing lines parallel to the sides of the triangle) into equilateral triangular cells of side length $1$.

A continuous route is chosen, starting inside the cell with vertex $A$ and always crossing from one cell to another through an edge shared by the two cells. No cell is visited more than once. Find, with proof, the greatest number of cells which can be visited.