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Number theory Difficulty 5.0 AIME, harder Prove it United Kingdom

The number 35 can be written as the sum of four distinct primes in two different ways: $35 = 2 + 3 + 13 + 17$ and $35 = 2 + 3 + 11 + 19$ . What is the smallest number that can be written as the sum of four distinct primes in two different ways?

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