Let \(n\) be a positive integer. Consider a square \(S\) of side \(2n\) units. Divide \(S\) into \(4n^2\) unit squares by drawing \(2n-1\) horizontal & \(2n-1\) vertical lines one unit apart. A circle of diameter \(2n-1\) is drawn with it's centre at the intersection of the two diagonals of the square \(S\). Prove that the number of unit squares which contain a portion of the circumference of the circle is \(8n-2\).