Observe net \(ABCDEFGHIJ\).

Figure \(BEGJ\) is a unit square, with angle \(ABJ=90^\circ\) and \(BJ=AB=BC\). Also, \(DE=EF=FG=GH\) and \(CE=AJ=JI\), with the ratio of \(GH\) to \(IJ\) is \(1:2\). The net is to form a closed polyhedron no holes or overlaps. If the volume is written as a fraction in lowest terms, find the sum of the numerator and the denominator.

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