Absolute Value System

Algebra Level 4

ab=2bc=3cd=4 \begin{aligned} |a - b | &=& 2 \\ |b - c | &=& 3 \\ |c - d | &=& 4 \\ \end{aligned}

Given that a,b,c,da,b,c,d are real numbers that satisfy the system of equations above, what is the sum of all distinct values of ad|a-d| ?

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