In this question, $x_1,x_2,x_3,\dots$ is an arithmetic progression, all of whose terms are integers.

Let $n$ be a positive integer. If the median of the first $n$ terms of the sequence is an integer, which of the following three statements must be true?

I The median of the first $n+2$ terms is an integer.

II The median of the first $2n$ terms is an integer.

III The median of $x_2,x_4,x_6,\dots ,x_{2n}$ is an integer.