human
Can you provide a Python program to determine who wins if they both play optimally given the following:
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|mod 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

The first line contains a single integer n (1 ≤ n ≤ 10^5) — the number of numbers.

The second line contains n integers a_1, a_2, …, a_n (1 ≤ a_i ≤ n). Furthermore, there are no pair of indices i ≠ j such that a_i = a_j.

Output

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". 

Examples

Input

8
3 6 5 4 2 7 1 8

Output

BAAAABAB

Input

15
3 11 2 5 10 9 7 13 15 8 4 12 6 1 14

Output

ABAAAABBBAABAAB

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:...
The above is tricky. Write me a correct solution in PYTHON3.
assistant
#Here is Python program to determine who wins if they both play optimally.

import sys

next(sys.stdin)

positions = {}
pos2x = {}
for i, x in enumerate(next(sys.stdin).split()):
    x = int(x)
    positions[x] = i
    pos2x[i] = x

answers = ['' for _ in range(len(positions))]

for x in range(len(positions), 0, -1):

    position = positions[x]

    def can_go_to_looser():

        next_position = position + x
        while next_position < len(positions):

            if pos2x[next_position] > x and answers[next_position] == "B":
                return True

            next_position += x

        next_position = position - x
        while next_position >= 0:

            if pos2x[next_position] > x and answers[next_position] == "B":
                return True

            next_position -= x

        return False

    if can_go_to_looser():
        answers[position] = "A"
    else:
        answers[position] = "B"

print(''.join(answers))