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After executing a search, click on "Save Search" to save that query. This is helpful so that you don't have to build the same search query over and over. When you select "Save Search" you'll be saving this as a search filter and can choose whether or not to have this search filter display in your quicklist, and whether it should be publicly viewable.
You'll find that your saved search query queries using the "Load Search" pull down menu in the advanced search or in your quicklist to the left of the Library thumbnails where the other filters are (All, Drafts, Starred, Batches, Labels).
To manage these saved searches click on "Manage Saved Searches".