HMIC 2020 Problem 2

Some bishops and knights are placed on an infinite chessboard, where each square has side length 1 unit. Suppose that the following conditions hold:

• For each bishop, there exists a knight on the same diagonal as that bishop (there may be another piece between the bishop and the knight).
• For each knight, there exists a bishop that is exactly $\sqrt{5}$ units away from it.
• If any piece is removed from the board, then at least one of the two conditions above is no longer satisfied.

If $n$ is the total number of pieces on the board, find all possible values of $n$.