Is there a single word that means "to add an amount to a running total"?

single-word-requests

I'm trying to determine what word would best be used to indicate an instruction to, when someone comes across a number of things in a given situation, add the number there (maybe written, or it may require counting a number in that situation) to a running total of similar things.

It would be similar in usage to count or tally, but would specifically imply that there is more than one to add to the running total
Something more similar in meaning to accumulate or aggregate except we're just tracking the numbers of things (like rocks or plants), not actually collecting them in a bag or box, so I'd prefer to avoid those words, which have that potential meaning
Add might be close, but there's a reason the phrase "add to the running total" isn't just "add"
Whereas count and tally and even sum, I think, imply "determine a number and report it", I want something that is more like "determine the number of these and then add that to this total"
So, it would fit in this sentence in place of accumulate: "If you come across some hair pins, accumulate them in the 'sharp' count and the 'metal' count, but if you find buttons, they should be accumulated in the 'round' count - if they're round - and the 'flat' count."
The more precise meaning of that sentence above is: "If you come across some hair pins, count them and add that number to the 'sharp' count and the 'metal' count, but if you find buttons, they should be counted and that total should be added to the both the 'round' count and the 'flat' count."