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 C++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include "bits/stdc++.h"

// Begin {{{
using namespace std;

#ifndef DEBUG
#define dump(...)
#endif

template <class A, class B> inline bool chmax(A &a, const B &b) { return b > a && (a = b, true); }
template <class A, class B> inline bool chmin(A &a, const B &b) { return b < a && (a = b, true); }

template <class T>
inline vector<T> make_v(const T &initvalue, size_t sz) {
    return vector<T>(sz, initvalue);
}

template <class T, class... Args>
inline auto make_v(const T &initvalue, size_t sz, Args... args) {
    return vector<decltype(make_v<T>(initvalue, args...))>(sz, make_v<T>(initvalue, args...));
}

constexpr int INF = 0x3f3f3f3f;
constexpr int64_t LINF = 0x3f3f3f3f3f3f3f3fLL;

constexpr size_t    operator""_sz(unsigned long long value) { return value; };
constexpr intmax_t  operator""_im(unsigned long long value) { return value; };
constexpr uintmax_t operator""_um(unsigned long long value) { return value; };
// }}} End

constexpr int MOD = 998244353;

intmax_t power(intmax_t x, intmax_t n, const intmax_t M) {
    if (n == 0) return 1;
    if (n % 2 == 0) {
        intmax_t t = power(x, n / 2, M);
        return t * t % M;
    }
    return x * power(x, n - 1, M) % M;
}

signed main() {
    ios_base::sync_with_stdio(false);
    cin.tie(nullptr);

    size_t N;
    cin >> N;

    vector<intmax_t> cnt(N);
    for (size_t i = 0; i < N; ++i) {
        int d; cin >> d;
        if (i == 0 && d != 0) {
            cout << 0 << "\n";
            return 0;
        }
        cnt[d]++;
    }

    if (cnt[0] != 1) {
        cout << 0 << "\n";
        return 0;
    }

    intmax_t res = 1;
    for (size_t d = 1; d < N; ++d) {
        if (cnt[d]) {
            res *= power(cnt[d], cnt[d + 1], MOD);
            res %= MOD;
        }
    }

    cout << res << "\n";

    return 0;
}