Total cost (hours * rate) across many months where rate changes by range of months

Hello and thank you in advance for your advice. I am attempting to create a single formula to calculate the total cost in Column Z (hours * rate) where the rate (B5:E6) varies by a data range (B2:E3) and hours (F5:Y6) cross many months. Depending on the date in F4:Y4, multiply by correct rate and sum all of them together.

I have been researching sumproduct and arrays like crazy but have not been able to figure it out.

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