More splitting

Computer Science Level pending

Consider the shape in the problem Splitting Them Up; that is, the irregular hexagon given by the points \((0,0)\), \((0,6)\), \((6,6)\), \((6,2)\), \((12,2)\), \((12,0)\). Let \(f(k)\) be equal to the sum of the slopes of the lines through \((0,0)\) which divide the figure into \(k\) equal areas (\(f(2) = \frac{2}{3}\), \(f(4) = \frac{7}{3}\)). If \(f(1000) = \frac{m}{n}\), where \(m\) and \(n\) are co-prime, and \(m ≡ q\space(mod\space n)\), find the sum of the digits of q.

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