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With ArcGIS for Desktop Advanced, you could Convert all depression lines vertices to points (with Feature Vertices to Points) Convert all vertices back to lines (with Points to Line, with the Close Line option checked) Revert the direction of all lines (with Flip Line).
If GISGe's solution works, I'd go for it. Nevertheless, if you still need a solution to remove the holes from polygons, you can just run the union tool and unselect the "gaps allowed" checkbox.
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