Any way you slice it

Geometry Level 5

Suppose that two distinct lines passing through the origin cut through the region \(R\) in the first quadrant bounded by the \(x\)-axis and the circle \((x - 1)^{2} + y^{2} = 1\) such that the lines divide \(R\) into \(3\) subregions of equal area.

If the sum of the slopes of these two lines is \(S\), find \(\lceil 1000*S \rceil\).

Note: Numerical methods can be used in the process of solving this question.

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