Industrious bees

Bees are known as some of the most industrious creatures in the world. In the beehive, only certain bees are responsible for guarding the collected honey. Our hive uses the following rules:

Each bee's working area depends on its grade:

• A first-grade bee works in exactly $1$ cell (hexagon);

• A second-grade bee works in $7$ cells (one cell and the $6$ surrounding cells);

• A third-grade bee works in $19$ cells (one cell, the $6$ cells around it, and $12$ more cells surrounding them);

• In general, a bee of grade $K$ oversees a figure consisting of hexagons of radius $K$.

Cells are numbered starting from the bottom-left cell, then continuing row by row in the direction along the bottom-left edge (see figure).

For inspection, the bee moves from cell $1$ to cell $N$, each time choosing a different path in order to visit as many distinct cells as possible. At every step, the bee can move in one of three directions: up, up-right, or down-right. In how many ways can a bee of grade $K$ get from cell $1$ to cell $N$?