I am going to need some luck to justify that astounding 4N call... your partnership must have incredible requirements for splinters to warrant this auction!
I cross in hearts. If they break 3-2, I hook the diamond.. if it loses, and they ruff a spade, I wasn't making anyway. So if it loses, I will win the return and pull the last trump. Exactly how I play depends on the defence, but clearly I am cold if spades are 3-2 or diamonds 3-3, and I think I have some squeeze possibilities as well.
If the diamond hook wins, I pull the last trump, cross in spades and repeat the diamond hook (which may lose and revert to the previous situation absent one potentially crucial entry, but I am not going to spend time here worrying about it). If 2 hooks win in diamonds, then I am in great shape... at least 5 hearts, 3 diamonds, 3 spades and a club adds to 12.
If hearts are 4-1... I am going to ignore 4-1 offside, altho the contract is not down yet, the odds are against the 9 from J9xx offside and the lines that make are not complex... I need luck!
Assuming hearts are 4-1 onside: cross in trump (after all, trump could be 32) and hook the diamond. If this loses, especially if it loses on the second hook (and good opps will probably know enough to duck smoothly), I am in real trouble. I think I can handle a first round duck only if diamonds and spades are both breaking perfectly.
Assume the diamond K is onside: after the 1st winning hook, cross in spades and play another diamond.
If LHO pops the K, then cash a top heart in dummy and then play the
♦Q for LHO to ruff... if he doesn't ruff, we play the 4th diamond and ruff it ourselves.. we score 5 trumps, 2 diamonds, 4 spades and a club... we need spades to be 3-2, but we always did when trump were 4-1, I think.
If LHO doesn't pop the K, on the second diamond, we win the hook, cash the A and, if needed, ruff the 4th diamond and hook the heart, and make 13 tricks

But I wouldn't expect LHO to be 3=4=4=2 or 2=4=4=3 because no one bid clubs.
There are other permutations, as there usually are in hands from real life.
'one of the great markers of the advance of human kindness is the howls you will hear from the Men of God' Johann Hari
1♦-1♥
4♣-4NT
5♦-6♥