Prime Summations

It is possible to write ten as the sum of primes in exactly five different ways:

\begin{aligned} 7 + 3 \\ 5 + 5 \\ 5 + 3 + 2 \\ 3 + 3 + 2 + 2 \\ 2 + 2 + 2 + 2 + 2 \end{aligned}

What is the first value which can be written as the sum of primes in over five thousand different ways?

Write-up coming later

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