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 LuaJIT (2.0.4) (judge verdict: Accepted)

local mma = math.max
local mfl, mce, mmi = math.floor, math.ceil, math.min

local n = io.read("*n")
local a = {}
for i = 1, n do
  a[i] = io.read("*n")
end
if n == 1 then print("YES") os.exit() end

local sum = 0
local mod = mfl(n * (n + 1) / 2)
local cnt = 0
for i = 1, n do
  cnt = cnt + mfl((sum + a[i]) / mod)
  sum = (sum + a[i]) % mod
end
if sum ~= 0 then print("NO") os.exit() end

local b = {}
for i = 1, n - 1 do
  b[i] = a[i + 1] - a[i]
end
b[n] = a[1] - a[n]

for i = 1, n do
  b[i] = b[i] - cnt
end

for i = 1, n do
  if 0 < b[i] then cnt = -1 break end
  local v = -b[i]
  if v % n ~= 0 then cnt = -1 break end
  cnt = cnt - mfl(v / n)
end
print(cnt == 0 and "YES" or "NO")