SUMIF/SUMPRODUCT Multiple Criteria

Hi all,

I've seen similar posts on here but they have not quite answered my question. I have a range of data that I need to sum based on multiple criteria. Please see the attached excel file.

I'd like to be able to sum Column A with a criteria that is A,B,C,D (defined in E6:E9) and 1,2,3,4 (defined in F6:F9). I'd like to reference Columns E and F so that I could add or remove criteria in the future without changing the formula. I've seen this done with one criteria using something like =SUMPRODUCT(SUMIF(B1:B9,E6:E9,A1:A9)) to get 423, but I'm looking further to then see out of that, the sum of numbers that have 1,2,3, or 4 in column c.

Thank you in advance!