How can 7NT be played for this hand?
There is a pentagonal squeeze on this hand. One line of play:
Win the club lead in dummy, cross to hand with the spade ace, take the diamond finesse, and play on diamonds, pitching spades. Eventually, west has to unguard a round suit to keep the spade control, and now you have a double squeeze, with west controlling spades and one round suit, and east controlling both round suits. Cash your hand's winner in the round suit controlled only by east, then return to dummy with the heart king. Now we've gotten to a standard double squeeze position; cash the last diamond, forcing east to unblock one suit, pitching the suit that east keeps. West will be forced to let go the last guard to one of your suits, and your thirteenth trick has materialized.
I've put together one possible play sequence.
(Thanks to Forget I was ever here for pointing out an error in my initial answer, and Mycroft for pointing out places that I had confused hand and dummy).

Here is the 6 card end position after winning ♣ Ace; coming to hand with ♠ Ace; and running ♢ after a successful finesse of ♢ Q:
         ♠ J
         ♡ K 7
         ♢ 3 2
         ♣ 7

 ♠ K            ♠ T
 ♡ Q 9 5        ♡ J T 6
 ♢ -            ♢ -
 ♣ J T          ♣ 9 8
         ♠ 5
         ♡ A 8 2
         ♢ -
         ♣ K 6

On the ♢ 3 East and South discard their last ♠ and West must unguard a round suit; establishing the double squeeze. Best defense is for West to pitch a
♡, as if South has ♡ J it's either doubleton, or West was always going to be victim of a successful ♠-♡ simple squeeze. (If West was dealt ♢ J or T instead of ♢ Q he should pitch a ♣ instead. Since South cannot see, or likely read, any of these cards, this option selection should be opaque to Declarer.)
On play of ♢ 2 East now discards ♣ 8 and South must guess the situation. If West is believed to be guarding ♣ (best above) then South must discard ♡ 2 and cash ♡ A and ♡ K in order to squeeze West in both black suits.
If West is believed to be guarding ♡ then South must discard ♣ 6 and cash ♣ K and ♡ K in order to complete the major suit squeeze on West.
Declarer's best bet for reading the situation is that East, whose ♣ attitude to the opening lead is known to all once Dummy comes down, probably gave an honest count signal in ♣'s at trick one. If East is astute enough to have seen the double squeeze coming that early, tip your hat to him as the better man/woman on this day.