Problem:
Score : 300 points

Problem Statement

Given is an integer sequence D_1,...,D_N of N elements. Find the number, modulo 998244353, of trees with N vertices numbered 1 to N that satisfy the following condition:

For every integer i from 1 to N, the distance between Vertex 1 and Vertex i is D_i.

Notes

A tree of N vertices is a connected undirected graph with N vertices and N-1 edges, and the distance between two vertices are the number of edges in the shortest path between them.

Two trees are considered different if and only if there are two vertices x and y such that there is an edge between x and y in one of those trees and not in the other.

Constraints

1 \leq N \leq 10^5

0 \leq D_i \leq N-1

Input

Input is given from Standard Input in the following format:

N
D_1 D_2 ... D_N

Output

Print the answer.

Sample Input 1

4
0 1 1 2

Sample Output 1

2

For example, a tree with edges (1,2), (1,3), and (2,4) satisfies the condition.

Sample Input 2

4
1 1 1 1

Sample Output 2

0

Sample Input 3

7
0 3 2 1 2 2 1

Sample Output 3

24

Solution in Lua (5.3.2) (judge verdict: Accepted)

local mod = 998244353
local function bmul(x, y) return (x * y) % mod end
local function modpow(src, pow)
  local res = 1
  while 0 < pow do
    if pow % 2 == 1 then
      res = bmul(res, src)
      pow = pow - 1
    end
    src = bmul(src, src)
    pow = pow // 2
  end
  return res
end

local function modinv(src)
  return modpow(src, mod - 2)
end

local n = io.read("*n")
local t = {}
for i = 1, n do
  t[i] = 0
end
local tmp = true
local maxlen = 0
for i = 1, n do
  local a = io.read("*n")
  if i == 1 then
    tmp = a == 0
  end
  maxlen = math.max(maxlen, a + 1)
  t[a + 1] = t[a + 1] + 1
end
if not tmp then print(0) os.exit() end
for i = 1, maxlen do
  if t[i] == 0 then
    print(0)
    os.exit()
  end
end
if t[1] ~= 1 then
  print(0)
  os.exit()
end
local ret = 1
for i = 2, maxlen do
  ret = bmul(ret, modpow(t[i - 1], t[i]))
end
print(ret)