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: Wrong Answer)

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

def update_list2(n,list)
  # puts "updatelist2"
  # p list:list
  solved = true
  target_list = []
  list.each_with_index do |e,i|
    ni = (i + 1) % n
    target_list << i if e > list[ni]
  end
  p target_list:target_list
  (0...n).each do |i|
    diff = 0
    target_list.each do |t|
      if t == i
        diff += n
      else
        diff += (((i + n) - t) % n)
      end
    end
    list[i] -= diff
    solved = false if list[i] != 0
  end
  # p list:list
  if solved
    return true
  else
    return target_list.size
  end
end

def update_list(n,list)
  target = nil
  same_count = 0
  list.each_with_index do |e,i|
    ni = (i + 1) % n
    if e > list[ni]
      diff = e - list[ni]
      same_count = diff / (n-1)
      target = i
      break
    end
  end
  raise "system error" if target == nil
  (1..n).each do |j|
    idx = (j + target) % n
    list[idx] -= j
  end
end

def onestep_check(n,list)
  target = 0
  list.each_with_index do |e,i|
    if e == n
      target = i
      break
    end
  end
  (1..n).each do |j|
    idx = (target + j) % n
    unless list[idx] == j
      return false
    end
  end
  true
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