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to run up (a hill, stairs etc.); to dash up; to rush up
「駆け上る」 has another reading with different meanings.
階段を一気に駆け上がった。
I ran up the stairs in one breath.
丘を駆け上がって景色を眺めた。
I ran up the hill and looked at the scenery.
遅刻しそうになって、階段を駆け上がる羽目になった。
Running late, I ended up racing up the stairs.