Problem:
Score : 500 points

Problem Statement

There are N boxes arranged in a circle. The i-th box contains A_i stones.

Determine whether it is possible to remove all the stones from the boxes by repeatedly performing the following operation:

Select one box. Let the box be the i-th box. Then, for each j from 1 through N, remove exactly j stones from the (i+j)-th box. Here, the (N+k)-th box is identified with the k-th box.

Note that the operation cannot be performed if there is a box that does not contain enough number of stones to be removed.

Constraints

1 ≦ N ≦ 10^5

1 ≦ A_i ≦ 10^9

Input

The input is given from Standard Input in the following format:

N
A_1 A_2 … A_N

Output

If it is possible to remove all the stones from the boxes, print YES. Otherwise, print NO.

Sample Input 1

5
4 5 1 2 3

Sample Output 1

YES

All the stones can be removed in one operation by selecting the second box.

Sample Input 2

5
6 9 12 10 8

Sample Output 2

YES

Sample Input 3

4
1 2 3 1

Sample Output 3

NO

Solution in Ruby (2.3.3) (judge verdict: Time Limit Exceeded)

def solve(n,list)
  total = 0
  list.each do |e|
    total += e
  end

  os = onestep_sum(n)
  return "NO" if total % os != 0

  count = total / os

  loop do
    if count < 1
      return "NO"
    else
      c = update_list2(n,list,count)
      if c == true
        return "YES"
      elsif c == false
        return "NO"
      else
        count -= c
        next
      end
    end
  end
end

def update_list2(n,list,count_limit)

  # puts "updatelist2"

  solved = true
  target_list = []
  list.each_with_index do |e,i|
    ni = i + 1
    ni = 0 if ni == n
    target_list << i if e > list[ni]
  end

  # first diff
  first_target_size = target_list.size
  return false if first_target_size > count_limit

  diff = 0
  target_list.each do |t|
    if t == 0
      diff += n
    else
      diff += n - t
    end
  end

  list[0] -= diff
  solved = false if list[0] != 0
  return false if list[0] < 0

  if target_list.first == 0
    target_list.shift
    diff = diff - n
  end

  # sequence
  (1...n).each do |i|
    diff = diff + first_target_size
    list[i] -= diff

    solved = false if list[i] != 0
    return false if list[i] < 0

    if target_list.first == i
      target_list.shift
      diff = diff - n
    end
  end

  if solved
    return true
  else
    return first_target_size
  end
end

def onestep_sum(n)
  (n * (n + 1) / 2)
end

def main
  n = gets.to_i
  list = gets.split.map {|e| e.to_i}
  puts solve(n,list)
end

main