Prime Subset Sums

Let $S = {2, 3, 5, \dots, 4999}$ be the set of prime numbers less than $5000$ .

Find the number of subsets of $S$ , the sum of whose elements is a prime number.
Enter the rightmost $16$ digits as your answer.

Write-up coming later

The complete problem is available here. An approach, code, and answer will be added later.