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Black is attempting to get to a stalemate situation. He therefore needs to try to either capture the pawn or to get trapped in a8 by the K and p. His moves are therefore aimed at a situation where White's king is on c7 and pawn on b7.

White obviously wants to avoid this stalemate and promote the pawn. To do this needs to manoeuvre the black king into the position at move 8. If black now moves to a8, the pawn advances and eventually forces the black king to move away and therefore allows promotion of the pawn.