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This Python 3 program constructs all possible length 10 chains, and then checks the additional conditions. It runs in 84ms.

Solution:The third prime in the chain is 73. The sixth prime in the chain is 19. The ninth prime in the chain is 17.The entire chain is: (13, 37, 73, 31, 11, 19, 97, 71, 17, 79).