C. Permutation Gametime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputAfter a long day, Alice and Bob decided to play a little game. The game board consists of n cells in a straight line, numbered from 1 to n, where each cell contains a number a_i between 1 and n. Furthermore, no two cells contain the same number. A token is placed in one of the cells. They take alternating turns of moving the token around the board, with Alice moving first. The current player can move from cell i to cell j only if the following two conditions are satisfied:   the number in the new cell j must be strictly larger than the number in the old cell i (i.e. a_j > a_i), and  the distance that the token travels during this turn must be a multiple of the number in the old cell (i.e. |i-j|\bmod a_i = 0). Whoever is unable to make a move, loses. For each possible starting position, determine who wins if they both play optimally. It can be shown that the game is always finite, i.e. there always is a winning strategy for one of the players.InputThe first line contains a single integer n (1 \le n \le 10^5) — the number of numbers.The second line contains n integers a_1, a_2, \ldots, a_n (1 \le a_i \le n). Furthermore, there are no pair of indices i \neq j such that a_i = a_j.OutputPrint s — a string of n characters, where the i-th character represents the outcome of the game if the token is initially placed in the cell i. If Alice wins, then s_i has to be equal to "A"; otherwise, s_i has to be equal to "B". ExamplesInput83 6 5 4 2 7 1 8OutputBAAAABABInput153 11 2 5 10 9 7 13 15 8 4 12 6 1 14OutputABAAAABBBAABAABNoteIn the first sample, if Bob puts the token on the number (not position):   1: Alice can move to any number. She can win by picking 7, from which Bob has no move.  2: Alice can move to 3 and 5. Upon moving to 5, Bob can win by moving to 8. If she chooses 3 instead, she wins, as Bob has only a move to 4, from which Alice can move to 8.  3: Alice can only move to 4, after which Bob wins by moving to 8.  4, 5, or 6: Alice wins by moving to 8.  7, 8: Alice has no move, and hence she loses immediately.  
Input83 6 5 4 2 7 1 8
OutputBAAAABAB
['brute force', 'dp', 'games', '*1600']