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All answers having "Cores" as clue:

4 Character Answers:

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HUBS

PITS



5 Character Answers:

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GISTS

PITHS

ROOTS



6 Character Answers:

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HEARTS

INNERS

NUCLEI



7 Character Answers:

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CENTERS



8 Character Answers:

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ESSENCES



9 Character Answers:

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INTERIORS