The Absolute's Minimum

Algebra Level 4

x1+2x1+3x1++300x1 \large |x - 1| + |2x-1| + |3x-1| + \cdots + |300x-1|

If the minimum value of the expression above is AB \dfrac AB, and it occurs when x=CDx = \dfrac CD, where (A,B)(A, B) and (C,D)(C, D) are each coprime pairs of positive integers, find A+B+C+DA+B+C+D.

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