Help summing values across multiple sheets with criteria

Hello folks,

I have a problem that has been plaguing me for far too long. Basically I have 6 worksheets in my book. 5 are titled "DEC" "JAN" "FEB" "MAR" "APR" and the last one is just "Sheet2". On the month sheets I have a list of about 120 numbers each 7 characters long. On Sheet 2 I have another list of the same type of numbers and also an amount beside them. What I am trying to do is compare the numbers in Sheet2 to the month sheets and if they match take the value beside it and sum it with all other matching values. However, some of the numbers in Sheet2 appear in more than 1 of the month sheets, so I need to avoid double counting. I can get the desired result with a VLOOKUP and then summing the values by hand, but this is extremely tedious and not at all future proof. I know there must be some combination of SUMPRODUCT/SUMIF statements that I can use to get what I am looking for. Thank you for taking the time to read my problem and helping!