Careful play at trick1  will help South to solve the problem.

Win the heart lead with Dummy's Queen, playing the 7 hearts from hand instead of the 4.
Draw Trumps in 3 Rounds ending in Dummy
Lead the 8 Hearts which East must cover to protect his potential Heart trick
Win this trick with the King
Play a Diamond to the Ace and now play the 6 hearts from Dummy
East is caught in a dilemma with his J 3 hearts
If he covers Dummy's 6 with the Jack, South wil win with Ace and play his four to Dummy's five
Then a club towards his King of clubs for the twelveth trick
If East does not cover Dummy's 6 hearts,South  will play his carefully preserved 4
Then a club towards his King  for the eventual twelveth trick.

This hand  demonstrates the usual futility of ruffing with a winning Trump. 

Various answers have been suggested 
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If East wins  the Trick cheaply with the Jack , the Heart suit is blocked and the defence loses a vital  tempo and with it the certainty of defeating the Contract

So win the first trick wtht the Ace and return the Jack and let your Partner force out the King while he still has an Entry

(South's Heart should not be the Queen, because if he had that card, then West would have the 10 9 8 7 and would have led the 10)

After winning the the first trich with the A Clubs, South leads dummy's JS and wins this trick with  AS.

Now a Diamond to Dummy, Club ruff and cash A-J diamonds.

Now South cashes A-K Hearts and exits with a Spade to West's K

West must now give Declarer a ruff and a sluff on next trick.

Six Spades made,

South must resist the temptaion to let the diamond  run to the QJ 4 in the South hand

Otherwise with good defence he will be cut off from the Clubs in his own hand

a

With Eleven tricks cold, the twelvth has to come from Hearts
Win  first trick with K Clubs
Draw trumps in three rounds
Cash A Hearts
Play a small Heart towards Dummy
West will win with the J  and continue Clubs (best)
Win this trick with A Clubs
Cash remaining 3 Trumps
Cash K Hearts
Play remaing Club towards the Q in Dummy
West is destroyed

If South ruffs the KH in Dummy she will eventually find herself blocked from the Diamond suit. So, she should let KH win. Contract is now safe

After ruffing the opening KS, cash the AH and then the JH, followed by the 10H if West does not cover. Then start cashing the Minor suits. West may take his QH whenever he likes for his only trick for the Defence.

If the KH is cashed befors the AH, West will force Declarer with another Spade when he wins the QH and will eventually score a second trump trick for the Defence

South should lead Clubs at once before drawing Trumps. West will win AC and continue Diamonds.Dummy wins AD and continues Clubs. East ruffs the third Club and South overruffs. South gets back to Dummy with a Heart ruff and continues Clubs. Whether or not East ruffs with Ace of Trumps, South discards her losing Diamond.

After winning the 2nd trick in Dummy with KS, play a small club towards your own hand. West will win and return a Trump (best). Win in Dummy with the QH and play another small club. Ruff this trick  small and play JS. Ruff the Jack in Dummy and play a 3rd small club. Ruff small in hand and draw 2 rounds of Trumps, leaving a Trump outstanding in West's Hand. Now play on Diamonds. West may ruff at her discretion, but the Contract is secure with  KS and a Spade ruff in Dummy, 6 Trump Tricks including 2 Club Ruffs and 2 Diamonds in Dummy.

East will overtake the QH opening lead. With 6 Hearts between the N/S hands the Rule of 7 applies. So duck the first heart and take the Heart continuation. With 8 playing tricks and no way of getting at the QS in the North hand without letting West in, the best play is a throw-in. So, after winning the Heart continuaton at trick 2, cash A K Spades and A K Q J diamonds, then exit with a Heart. West will cash his 3 Heart winners, but must give you the last 2 tricks by either playing a spade to the Queen in Dummy or a Club to the Club tenace in your own hand.