One day Mr. Takahashi picked up a dictionary containing all of the N! permutations of integers 1 through N. The dictionary has N! pages, and page i (1 ≤ i ≤ N!) contains the i-th permutation in the lexicographical order.

Mr. Takahashi wanted to look up a certain permutation of length N in this dictionary, but he forgot some part of it.

His memory of the permutation is described by a sequence P_1, P_2, ..., P_N. If P_i = 0, it means that he forgot the i-th element of the permutation; otherwise, it means that he remembered the i-th element of the permutation and it is P_i.

He decided to look up all the possible permutations in the dictionary. Compute the sum of the page numbers of the pages he has to check, modulo 10^9 + 7.

Constraints

* 1 ≤ N ≤ 500000
* 0 ≤ P_i ≤ N
* P_i ≠ P_j if i ≠ j (1 ≤ i, j ≤ N), P_i ≠ 0 and P_j ≠ 0.

Input

The input is given from Standard Input in the following format:


N
P_1 P_2 ... P_N


Output

Print the sum of the page numbers of the pages he has to check, as modulo 10^9 + 7.

Examples

Input

4
0 2 3 0


Output

23


Input

3
0 0 0


Output

21


Input

5
1 2 3 5 4


Output

2


Input

1
0


Output

1


Input

10
0 3 0 0 1 0 4 0 0 0


Output

953330050
#include <iostream>
#include <vector>
#include <queue>
#include <math.h>
#include <set>
#define FOR(i, n, m) for(ll i = n; i < (int)m; i++)
#define REP(i, n) FOR(i, 0, n)
#define ALL(v) v.begin(), v.end()
#define pb push_back
using namespace std;
using ll = std::int_fast64_t;
using P = pair<ll, ll>;
constexpr ll inf = 1000000000;
constexpr ll mod = 1000000007;
constexpr long double eps = 1e-15;
template<typename T1, typename T2>
ostream& operator<<(ostream& os, pair<T1, T2> p) {
    os << to_string(p.first) << " " << to_string(p.second);
    return os;
}
template<typename T>
ostream& operator<<(ostream& os, vector<T>& v) {
    REP(i, v.size()) {
        if(i) os << " ";
        os << to_string(v[i]);
    }
    return os;
}

struct modint {
    ll n;
public:
    modint(const ll n = 0) : n((n % mod + mod) % mod) {}
    static modint pow(modint a, int m) {
        modint r = 1;
        while(m > 0) {
            if(m & 1) { r *= a; }
            a = (a * a); m /= 2;
        }
        return r;
    }
    modint &operator++() { *this += 1; return *this; }
    modint &operator--() { *this -= 1; return *this; }
    modint operator++(int) { modint ret = *this; *this += 1; return ret; }
    modint operator--(int) { modint ret = *this; *this -= 1; return ret; }
    modint operator~() const { return (this -> pow(n, mod - 2)); } // inverse
    friend bool operator==(const modint& lhs, const modint& rhs) {
        return lhs.n == rhs.n;
    }
    friend bool operator<(const modint& lhs, const modint& rhs) {
        return lhs.n < rhs.n;
    }
    friend bool operator>(const modint& lhs, const modint& rhs) {
        return lhs.n > rhs.n;
    }
    friend modint &operator+=(modint& lhs, const modint& rhs) {
        lhs.n += rhs.n;
        if (lhs.n >= mod) lhs.n -= mod;
        return lhs;
    }
    friend modint &operator-=(modint& lhs, const modint& rhs) {
        lhs.n -= rhs.n;
        if (lhs.n < 0) lhs.n += mod;
        return lhs;
    }
    friend modint &operator*=(modint& lhs, const modint& rhs) {
        lhs.n = (lhs.n * rhs.n) % mod;
        return lhs;
    }
    friend modint &operator/=(modint& lhs, const modint& rhs) {
        lhs.n = (lhs.n * (~rhs).n) % mod;
        return lhs;
    }
    friend modint operator+(const modint& lhs, const modint& rhs) {
        return modint(lhs.n + rhs.n);
    }
    friend modint operator-(const modint& lhs, const modint& rhs) {
        return modint(lhs.n - rhs.n);
    }
    friend modint operator*(const modint& lhs, const modint& rhs) {
        return modint(lhs.n * rhs.n);
    }
    friend modint operator/(const modint& lhs, const modint& rhs) {
        return modint(lhs.n * (~rhs).n);
    }
};
istream& operator>>(istream& is, modint m) { is >> m.n; return is; }
ostream& operator<<(ostream& os, modint m) { os << m.n; return os; }

#define MAX_N 3030303
long long extgcd(long long a, long long b, long long& x, long long& y) {
    long long d = a;
    if (b != 0) {
        d = extgcd(b, a % b, y, x);
        y -= (a / b) * x;
    } else {
        x = 1; y = 0;
    }
    return d;
}
long long mod_inverse(long long a, long long m) {
    long long x, y;
    if(extgcd(a, m, x, y) == 1) return (m + x % m) % m;
    else return -1;
}
vector<long long> fact(MAX_N+1, inf);
long long mod_fact(long long n, long long& e) {
    if(fact[0] == inf) {
        fact[0]=1;
        if(MAX_N != 0) fact[1]=1;
        for(ll i = 2; i <= MAX_N; ++i) {
            fact[i] = (fact[i-1] * i) % mod;
        }
    }
    e = 0;
    if(n == 0) return 1;
    long long res = mod_fact(n / mod, e);
    e += n / mod;
    if((n / mod) % 2 != 0) return (res * (mod - fact[n % mod])) % mod;
    return (res * fact[n % mod]) % mod;
}
// return nCk
long long mod_comb(long long n, long long k) {
    if(n < 0 || k < 0 || n < k) return 0;
    long long e1, e2, e3;
    long long a1 = mod_fact(n, e1), a2 = mod_fact(k, e2), a3 = mod_fact(n - k, e3);
    if(e1 > e2 + e3) return 0;
    return (a1 * mod_inverse((a2 * a3) % mod, mod)) % mod;
}

using mi = modint;

ll N = 505050;
int bit[505050];
void add(int a, ll w) {
    a++;
    for (int x = a; x <= N; x += x & -x) bit[x] += w;
}
ll sum(int a) {
    a++;
    ll ret = 0;
    for (int x = a; x > 0; x -= x & -x) ret += bit[x];
    return ret;
}
// return the sum of [a, b)
ll sum(int a, int b) {
    ll ret = 0;
    if(b == 0) return 0;
    ret += sum(b - 1);
    if(a) ret -= sum(a - 1);
    return ret;
}

ll n, cnt = 0;
vector<mi> fct(505050, 0);
vector<int> p(505050);
vector<int> rev(505050, -1);

mi func1() {
    mi ret = 0;
    mi sum = 0;
    FOR(i, 1, n + 1) if(rev[i] == -1) sum += i - 1;
    REP(i, n) {
        if(p[i]) {
            ret += (p[i] - 1) * fct[cnt] * fct[n - i - 1];
        } else {
            ret += sum * fct[cnt - 1] * fct[n - i - 1];
        }
    }
    return ret;
}

mi func2() {
    mi ret = 0;
    vector<int> c(505050, 0);
    ll tmp = 0;
    FOR(i, 1, n + 1) {
        if(rev[i] == -1) tmp++;
        else c[i] = tmp;
    }
    ll tmp2 = 0;
    REP(i, n) {
        if(p[i]) {
            ret += c[p[i]] * tmp2 * fct[cnt - 1] * fct[n - i - 1];
        } else {
            if(cnt >= 2) ret += cnt * (cnt - 1) / 2 * tmp2 * fct[cnt - 2] * fct[n - i - 1];
            tmp2++;
        }
    }
    return ret;
}

mi func3() {
    mi ret = 0;
    for(int i = 1; i <= n; i++) {
        if(rev[i] != -1) {
            ret += sum(rev[i]) * fct[cnt] * fct[n - rev[i] - 1];
            add(rev[i], 1);
        }
    }
    vector<int> c(505050, 0);
    int tmp = 0;
    for(int i = n; i > 0; i--) {
        if(rev[i] == -1) tmp++;
        else {
            c[i] = tmp;
        }
    }
    ll tmp2 = 0;
    REP(i, n) {
        if(p[i]) {
            tmp2 += c[p[i]];
        } else {
            ret += tmp2 * fct[cnt - 1] * fct[n - i - 1];
        }
    }
    return ret;
}

int main() {
    ios::sync_with_stdio(false);
    cin.tie(0);
    fct[0] = fct[1] = 1;
    FOR(i, 2, 505050) fct[i] = i * fct[i - 1];
    cin >> n;
    REP(i, n) cin >> p[i];
    REP(i, n) rev[p[i]] = i;
    REP(i, n) if(!p[i]) cnt++;
    mi ans = func1();
    ans -= func2();
    ans -= func3();
    ans += fct[cnt];
    cout << ans << endl;
    return 0;
}
