Let $n$ be a positive integer. Consider a square $S$ of side $2n$ units with sides parallel to the coordinate axes. Divide $S$ into $4 n ^ { 2 }$ unit squares by drawing $2n - 1$ horizontal and $2n - 1$ vertical lines one unit apart. A circle of diameter $2n - 1$ is drawn with its centre at the intersection of the two diagonals of the square $S$. How many of these unit squares contain a portion of the circumference of the circle?\\
(A) $4 n - 2$\\
(B) $4 n$\\
(C) $8 n - 4$\\
(D) $8 n - 2$