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Rolf Trautner | Mate in 5 | Main-Post, 1967-1968

A lucid problem showing a well-known theme but with a cheeky little twist. White to play & mate in 5. Make the key move.
1. Ra8 Kg2 - How should White proceed?
2. Qg8+ Kf1 - How should White continue?
3. Qb8 Kg2 - White to play. Keep going.
4. Qg3+ Kxg3 - White to play. Final move?
But what if 4... Kf1; How do you finish off?