A white bug sits in one corner square of a $1000$ × $n$ chessboard, where $n$ is an odd positive integer and $n > 2020$. In the two nearest corner squares there are two black chess bishops. On each move, the bug either steps into a square adjacent by side or moves as a chess knight. The bug wishes to reach the opposite corner square by never visiting a square occupied or attacked by a bishop, and visiting every other square exactly once. Show that the number of ways for the bug to attain its goal does not depend on $n$.