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leaf through (something)
To peruse a book, magazine, or other reading material by casually turning the pages, often without much attention or scrutiny. I leafed through a few fashion magazines while waiting at the doctor's office. I leafed through your proposal, and even at first glance I can tell you that it won't be approved by the board.
Turn pages, as in browsing or searching for something. For example, There she sat, leafing through the various catalogs. This expression employs leaf in the sense of "turn over the leaves of a book," a usage dating from the mid-1600s.
To go through some reading material quickly or superficially, turning from page to page, as in searching or browsing: On Sunday mornings, I leaf through the newspaper while eating breakfast.