Amazing quintuples

5,7,17,5, 7, 17, and 1919 are four prime numbers such that the sum of any three of them is also prime: 5+7+17=295+7+19=315+17+19=417+17+19=43.\begin{aligned} 5+7+17&=29\\ 5+7+19&=31\\ 5+17+19&=41\\ 7+17+19&=43. \end{aligned} Are there 5 distinct positive primes such that the sum of any three of them is also prime?

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