P7389 "EZEC-6" Game

Alice and Bob play a game. Alice initially has $n$ stones, and Bob initially has $m$ stones. They play for several rounds. In round $i$, the loser gives the winner $i$ stones. The game stops when, in some round, the loser does not have enough stones. Given $n, m$, find the maximum number of rounds the game can be played (not including the ending round), and construct a win-loss sequence containing only `A` and `B` such that the length of the sequence equals your answer, and at any time the number of stones each person has is non-negative.

**This problem has multiple test cases**. The first line contains a positive integer $T$, representing the number of test cases. For each test case: One line contains $2$ non-negative integers $n, m$.

For each test case: The first line contains an integer, representing the maximum number of rounds the game can be played. Let your answer be $x$. The second line contains a string of length $x$, representing the win-loss sequence. If Alice wins round $i$, then the $i$-th character of your sequence is `A`. If Bob wins round $i$, then the $i$-th character is `B`.

**This problem uses bundled tests**. - Subtask 1 (6 points): $n+m\le15$. - Subtask 2 (8 points): $n=m$. - Subtask 3 (8 points): $\sum n+m\le10^3$. - Subtask 4 (8 points): $\sum n+m\le2\times10^4$. - Subtask 5 (25 points): $\sum n+m\le4\times10^5$. - Subtask 6 (45 points): No special constraints. For $100\%$ of the testdata, $1\le T\le10^3$, $0\le n,m\le2\times10^7$, $1\le n+m\le2\times10^7$, $\sum n+m\le2\times10^7$. If you correctly answer the maximum number of rounds, but your constructed win-loss sequence is invalid, you will get $20\%$ of the score for that test point (rounded down). Note that if you cannot construct a win-loss sequence, you still need to output a non-empty string. **The output size of this problem is large, so please use a fast output method.** Translated by ChatGPT 5