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Programming questions are off-topic here. Do not ask questions about how to write code in C++. However, conceptual questions about computer science are more appropriate. See our help center for the scope of this site.

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Looking for a proof on why my algorithm in codeforces works

An easy-to-see algorithm that is potentially much simpler than yours but runs on the same idea is just to simply remove all "AB" pairs first, then remove all "BB" pairs. As you observed, we must have …
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