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I have a feeling that the answer is either a "circular linked list" or a finite state machine, but the answer is probably a circular linked list.

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The answer is circular buffer.
Such structure could be implemented using array or linked list, but you cannot call explicit array-based circular buffer as linked-list.

FSA cannot be "executed" infinitely (type 3 machine only uses fixed input and reads it ones), unless your FSA definition covers Turing Machines.

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