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 LuaJIT (2.0.4) (judge verdict: Accepted)

local mod = 998244353
local mfl = math.floor
local function bmul(x, y)
  local x0, y0 = x % 31596, y % 31596
  local x1, y1 = mfl(x / 31596), mfl(y / 31596)
  return (x1 * y1 * 62863 + (x1 * y0 + x0 * y1) * 31596 + x0 * y0) % mod
end

local function badd(x, y)
  return (x + y) % mod
end

local function bsub(x, y)
  return x < y and x - y + mod or x - y
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 = mfl(pow / 2)
  end
  return res
end

local n = io.read("*n")
local d = {}
local mlen = 0
local mma = math.max
for i = 1, n do
  d[i] = io.read("*n")
  mlen = mma(mlen, d[i])
end
local valid = true
if d[1] ~= 0 then
  valid = false
end
for i = 2, n do
  if d[i] == 0 then
    valid = false
  end
end
local t = {}
for i = 1, mlen do
  t[i] = 0
end
if valid then
  for i = 2, n do
    t[d[i]] = t[d[i]] + 1
  end
end
for i = 1, mlen do
  if t[i] == 0 then
    valid = false
  end
end
if not valid then
  print(0)
else
  local ret = 1
  for i = 2, mlen do
    local v = modpow(t[i - 1], t[i])
    ret = bmul(ret, v)
  end
  print(ret)
end