QGIS is an open-source geographical information system (read: mapping program) built on, in true open-source style, piles and piles of older software and feature-sets meaning sometimes you'll be trying to find the tool for computing polygon intersections1 and you will instead find the menu option for fax previews.
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I found four different ways of doing this, one of which involves invoking a second different GIS inside of QGIS, and none of which actually did the thing I wanted.