Sum data based on multiple criteria across multiple columns

I apologize if there's a similar question in the forum already but I wasn't able to find it if there is.

I have a spreasheet for my business where I list my clients and all the payments I receive from them.

Column A - Date

Column E - Payment 1 Date
Column F - Payment 1 Amount
Column G - Cash, Check, or Credit

Column H - Payment 2 Date
Column I - Payment 2 Amount
Column J - Cash, Check, or Credit

I'm trying to create a summary tab where it will automatically calculate the total that I've received in Cash, Check, and Credit by month. I'd like it to look like....

summary.png

I cannot seem to figure out the right formula to use.