A cricket wants to move across a $2n \times 2n$ board that is entirely covered by dominoes, with no overlap. He jumps along the vertical lines of the board, always going from the midpoint of the vertical segment of a $1 \times 1$ square to another midpoint of the vertical segment, according to the rules: $(i)$ When the domino is horizontal, the cricket jumps to the opposite vertical segment (such as from $P_2$ to $P_3$); $(ii)$ When the domino is vertical downwards in relation to its position, the cricket jumps diagonally downwards (such as from $P_1$ to $P_2$); $(iii)$ When the domino is vertically upwards relative to its position, the cricket jumps diagonally upwards (such as from $P_3$ to $P_4$). The image illustrates a possible covering and path on the $4 \times 4$ board. Considering that the starting point is on the first vertical line and the finishing point is on the last vertical line, prove that, regardless of the covering of the board and the height at which the cricket starts its path, the path ends at the same initial height.

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