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She takes pride in the many achievements she has accomplished.
Literal
She [topic-は] her own accomplished many achievements [object-を] pride [as-に] is-doing.
自分の達成した数多くの業績 ('the many achievements she has herself accomplished') is a relative clause: past-tense 達成した stacks directly before 業績 with no relative pronoun, just attributive position. 数多くの ('many, numerous') is a slightly more formal/written variant of たくさんの. 誇りにする applies the [X]を[Y]にする pattern — 'take X as one's pride.'