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Default sum a range based on criteria in rowdata and column name

Suppose I have some data that looks something like:
A B C D E F G
H
1 Category Site Sep1 Sep2 Sep3 Oct1 Oct2 Oct3
2 Train GB 10 5 4 12
14
3 Meeting STL 10 10
10
4 Train STL 5 5 5
5 Meeting GB 14 4 10
15

I need a formula that can ask a question like: How many hours of meeting
activity are happening in GB during the month of October?

Assuming the real data includes a column for every weekday over a span of a
year or more, and needs to be summed by both week and year, or any date
range. The variables may be stored in a cell on another worksheet such as
which Site and which activity and which month the answer is needed for. I've
tried playing around with sumif, sumproduct, and a few other things - the
problem I keep running up against is the fact that the range I need to sum is
more than one column, I won't necessarily know how many columns I need to sum
(for month anyways, week is always 5,) etc.

Anyone have any ideas?
 
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