UKMT - Hamilton Mathematical Olympiad - 2026 - 3

Starting at $(0, 0)$, a frog makes $99$ jumps of distance $\sqrt{5}$ between points on the lines $y = -1$, $y = 0$ and $y = 1$, increasing its $x$-coordinate by $1$ or $2$ each time.

How many possibilities are there for the final position of the frog?