BALTIC-WAY 2024 Problem 10

A frog is located on a unit square of an infinite grid oriented according to the cardinal directions. The frog makes moves consisting of jumping either one or two squares in the direction it is facing, and then turning according to the following rules: (i) If the frog jumps one square, it then turns $90^{\circ}$ to the right; (ii) If the frog jumps two squares, it then turns $90^{\circ}$ to the left.

Is it possible for the frog to reach the square exactly 2024 squares north of the initial square after some finite number of moves if it is initially facing: (a) North; (b) East?