Unit squares on a circle

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Let $n$ be a positive integer. Consider a square $S$ of side $2n$ units with sides parallel to the coordinate axes. Divide $S$ into $4n^2$ 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?

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