That's an Ugly Looking Octagon

Geometry Level 4

As shown in the diagram above, \(ABCDEFGH\) is an inscribed octagon where \(AB=BC=CD=EF=1\) and \(DE=FG=GH=HA=2\).

If the area of the octagon can be expressed as \(a+b\sqrt c,\) where \(a, b, c\) are all integers and \(c\) is square-free, then find the value of \(a+b+c\).

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