Three Unique Primes

What is the largest integer less than 100 which cannot be written as the sum of three unique prime numbers? Less than 1000? Less than 1 million?

For example,
10 = 5+3+2, all unique.
11 = 7+2+2 = 5+3+3, but neither sum has all three primes unique.

Extension: The problem above actually has been investigated thoroughly. Try also finding the largest integer in each range which can be written as exactly one such sum.

Source: Original.

Solution
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