About Frogs

Once $N$ white and $N$ black frogs decided to play a game. They found 2N + 1 tussocks and numbered them from $0$ to $2N$. Then the frogs occupied the tussocks in such a way that the white frogs sat on the tussocks with numbers from $0$ to N - 1, the black frogs sat on the tussocks with numbers from N + 1 to $2N$, and the tussock $N$ was empty. The goal is to swap the white and black frogs; that is, at the end of the game, the first $N$ tussocks must be occupied by the black frogs, and the last $N$ tussocks must be occupied by the white frogs.

In this game, the following moves are allowed: frogs may jump only to empty tussocks. A black frog may jump from a tussock numbered i > 0 to the tussock i - 1, or it may jump from a tussock j > 1 to the tussock j - 2 if there is a white frog on the tussock j - 1. Similarly, a white frog may jump from a tussock i < 2N to the tussock i + 1, or it may jump from a tussock j < 2N - 1 to the tussock j + 2 if there is a black frog on the tussock j + 1.

Usually in games, white and black make moves by turns, but here, white and black frogs have the same goal, so they may make moves in any order, and the total number of moves of white frogs may differ from the total number of moves of black frogs. If after one million moves the frogs still have not swapped, they become bored of the game and jump into the water.

Given $N$, determine if the frogs can achieve their goal.