The Absolute's Minimum

Algebra Level 5

\[ \large |x - 1| + |2x-1| + |3x-1| + \cdots + |300x-1| \]

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

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