All three paths take five moves to capture on a7, but if the Black KIng can get to c7 he'll be able to stalemate the White King in the corner.
The Key squares for the White a6-pawn are b7 and b8. Black must get his King to c7 and take control of the a-pawns key squares in order to stalemate White's King in the corner.
Now that we have the facts the answer is clear. The White King must head to the center.
Therefore, play continues;
1.Ke6 (1. Ke7 Kc3 2. Kd7 Kd4 3. Kc7 Ke5 4. Kb7 Kd6 5. Kxa7 Kc7 6. Ka8 Kc87. Ka7 Kc7 8. Ka8 Kc8 9. a7 Kc7) 1... Kc3 2.Kd5! (Body Check, keeping Black out of d4) 2... Kd3 (2... Kb4 3. Kc6 Ka5 4. Kb7 Kb5 5. Kxa7 Kc6 (5... Ka5 6. Kb7) 6. Kb8 ) 3. Kc6 Kd4 4. Kb7 Kd5 5. Kxa7 Kc6 6. Kb8 and White wins as he controls c7 and occupies b8 one of the a-pawns Key squares.