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third-person singular simple present presses on, present participle pressing on, simple past and past participle pressed on
(idiomatic) to persist, continue to advance quotations examples
And now she has the real dilemma: press on to King’s Landing, where I have no doubt she could get as close to killing Cersei as anyone else, or double back to Winterfell to reunite with her family.
2017 July 23, Brandon Nowalk, “The great game begins with a bang on Game Of Thrones (newbies)”, in The Onion AV Club