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All answers having "Assign" as clue:

3 Character Answers:

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LAY

PUT



4 Character Answers:

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CEDE

TASK



5 Character Answers:

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ALLOT

PINON

REFER



7 Character Answers:

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APPOINT

ASCRIBE



8 Character Answers:

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ALLOCATE

DELEGATE



9 Character Answers:

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ATTRIBUTE