1033C · Permutation Game

Problem: After 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.

Input Format: The 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 $a_i = a_j$.

Output Format: Print $s$ — 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".

Note: In 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.