This may be the only time in your bridge career you see a hand like this. I can't see any scientific way of reaching 6♣. Gerber doesn't help you because you can't stop in 5♣ if you get a disappointing answer. Nor does Blackwood for the same reason.

Disappointing answers are anything less than two Aces and even two Aces may not be enough if they don't include the ♦ A.

However you will never see a hand like this again and well done to those who chose valour over discretion and bid to 6♣ .

I'm not sure I'd have found this auction at the table.

The jump rebid promises 17 points so South knows they have a combined 33 count at least and that 6NT is on unless missing Ace and King in same suit. Blackwood (4NT) reassures South on that score. Now 5NT asks for Kings and when the response shows one King is missing South probably signs off in 6NT (better at pairs than 6♠ - that extra 10 points is vital).

North knows from the 5NT that they have all four Aces and if South can make 6NT without knowing about his seventh Spade may feel like risking the Grand Slam

Personally I would bid 7♠ as North playing Teams where the size of the score difference matters but not at Pairs where 6NT is likely to be a good score