to push on

from The Collaborative International Dictionary of English v.0.48
Push \Push\, v. i.
   1. To make a thrust; to shove; as, to push with the horns or
      with a sword. --Shak.
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   2. To make an advance, attack, or effort; to be energetic;
      as, a man must push in order to succeed.
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            At the time of the end shall the kind of the south
            push at him and the king of the north shall come
            against him.                          --Dan. xi. 40.
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            War seemed asleep for nine long years; at length
            Both sides resolved to push, we tried our strength.
                                                  --Dryden.
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   3. To burst pot, as a bud or shoot.
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   {To push on}, to drive or urge forward; to hasten.
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            The rider pushed on at a rapid pace.  --Sir W.
                                                  Scott.
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