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近づいてきて

came up to

→ 近付く
phrase Rare #41,773
came up to The て-form of 近づいてくる, used to connect an approach with what happens next. It often means that someone or something came closer to the speaker or another person.
JA: 犬が近づいてきて、私の手をなめた。
EN: A dog came up to me and licked my hand.

ルーシーは少年に近づいてきて、少年の頭の上に手を置いた。

Lucy approached the boy and placed her hand on his head.

知らない人が近づいてきて、そして私に病院へ行く道をたずねた。

A stranger approached me and asked for directions to the hospital.

車に向かって歩いていたら、男が近づいてきて、火を貸してくれと言うんだ

As I was walking toward my car, a man approached me and asked if he could borrow my lighter.

— ミスティック・リバー (2003)

見知らぬ人が近づいてきて私に道を尋ねた。

A stranger approached me and asked for directions.