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
// {{{ by shik
#include <bits/stdc++.h>
#include <unistd.h>
#define SZ(x) ((int)(x).size())
#define ALL(x) begin(x),end(x)
#define REP(i,n) for ( int i=0; i<int(n); i++ )
#define REP1(i,a,b) for ( int i=(a); i<=int(b); i++ )
#define FOR(it,c) for ( auto it=(c).begin(); it!=(c).end(); it++ )
#define MP make_pair
#define PB push_back
using namespace std;
typedef long long LL;
typedef pair<int,int> PII;
typedef vector<int> VI;

#ifdef SHIK
template<typename T>
void _dump( const char* s, T&& head ) { cerr<<s<<"="<<head<<endl; }

template<typename T, typename... Args>
void _dump( const char* s, T&& head, Args&&... tail ) {
    int c=0;
    while ( *s!=',' || c!=0 ) {
        if ( *s=='(' || *s=='[' || *s=='{' ) c++;
        if ( *s==')' || *s==']' || *s=='}' ) c--;
        cerr<<*s++;
    }
    cerr<<"="<<head<<", ";
    _dump(s+1,tail...);
}

#define dump(...) do { \
    fprintf(stderr, "%s:%d - ", __PRETTY_FUNCTION__, __LINE__); \
    _dump(#__VA_ARGS__, __VA_ARGS__); \
} while (0)

template<typename Iter>
ostream& _out( ostream &s, Iter b, Iter e ) {
    s<<"[";
    for ( auto it=b; it!=e; it++ ) s<<(it==b?"":" ")<<*it;
    s<<"]";
    return s;
}

template<typename A, typename B>
ostream& operator <<( ostream &s, const pair<A,B> &p ) { return s<<"("<<p.first<<","<<p.second<<")"; }
template<typename T>
ostream& operator <<( ostream &s, const vector<T> &c ) { return _out(s,ALL(c)); }
template<typename T, size_t N>
ostream& operator <<( ostream &s, const array<T,N> &c ) { return _out(s,ALL(c)); }
template<typename T>
ostream& operator <<( ostream &s, const set<T> &c ) { return _out(s,ALL(c)); }
template<typename A, typename B>
ostream& operator <<( ostream &s, const map<A,B> &c ) { return _out(s,ALL(c)); }
#else
#define dump(...)
#endif

template<typename T>
void _R( T &x ) { cin>>x; }
void _R( int &x ) { scanf("%d",&x); }
void _R( long long &x ) { scanf("%" PRId64,&x); }
void _R( double &x ) { scanf("%lf",&x); }
void _R( char &x ) { scanf(" %c",&x); }
void _R( char *x ) { scanf("%s",x); }

void R() {}
template<typename T, typename... U>
void R( T& head, U&... tail ) {
    _R(head);
    R(tail...);
}

template<typename T>
void _W( const T &x ) { cout<<x; }
void _W( const int &x ) { printf("%d",x); }
template<typename T>
void _W( const vector<T> &x ) {
    for ( auto i=x.cbegin(); i!=x.cend(); i++ ) {
        if ( i!=x.cbegin() ) putchar(' ');
        _W(*i);
    }
}

void W() {}
template<typename T, typename... U>
void W( const T& head, const U&... tail ) {
    _W(head);
    putchar(sizeof...(tail)?' ':'\n');
    W(tail...);
}

#ifdef SHIK
#define FILEIO(...)
#else
#define FILEIO(name) do {\
    freopen(name ".in","r",stdin); \
    freopen(name ".out","w",stdout); \
} while (0)
#endif

// }}}

template<typename T>
struct BIT {
    int n;
    vector<T> a;
    void init(int _n) {
        n = _n;
        a.resize(n + 1);
        a.clear();
    }
    void ins(int x, T v) {
        for (int i = x; i <= n; i += i & -i) {
            a[i] += v;
        }
    }
    T ask(int x) {
        T s = 0;
        for (int i = x; i; i -= i & -i) {
            s += a[i];
        }
        return s;
    }
    T ask(int l, int r) {
        auto s = ask(r);
        if (l > 0) {
            s += ask(l - 1);
        }
        return s;
    }
};

BIT<int> bit;

const int N=5e5+10;
const LL MOD=1e9+7;

int inv[N],fac[N],ifac[N];
void predo() {
	inv[1]=1;
	REP1(i,2,N-1) inv[i]=1LL*inv[MOD%i]*(MOD-MOD/i)%MOD;
	fac[0]=fac[1]=ifac[0]=ifac[1]=1;
	REP1(i,2,N-1) fac[i]=1LL*fac[i-1]*i%MOD;
	REP1(i,2,N-1) ifac[i]=1LL*ifac[i-1]*inv[i]%MOD;
}

LL C( int n, int m ) {
    if ( m>n ) return 0;
    return 1LL*fac[n]*ifac[m]%MOD*ifac[n-m]%MOD;
}

int n,a[N],c[N],gt[N];
LL sgt[N];
int main() {
    predo();
    R(n);
    REP1(i,1,n) R(a[i]);
    REP1(i,1,n) if ( a[i] ) c[a[i]]++;
    REP1(i,1,n) c[i]+=c[i-1];
    LL sz=0;
    REP1(i,1,n) if ( a[i]==0 ) sz++;
    REP1(i,1,n) if ( a[i] ) {
        int lt=a[i]-c[a[i]];
        gt[i]=sz-lt;
    }
    for ( int i=n; i>=1; i-- ) sgt[i]=(sgt[i+1]+gt[i])%MOD;
    bit.init(n);
    REP1(i,1,n) if ( a[i] ) bit.ins(a[i],1);
    LL ans=0;
    LL z=sz;
    REP1(i,1,n) {
        LL e;
        if ( a[i]==0 ) {
            z--;
            e=z*inv[2]%MOD;
            e=(e+sgt[i]*inv[sz])%MOD;
        } else {
            bit.ins(a[i],-1);
            LL base=bit.ask(a[i]);
            if ( z==0 ) {
                e=base;
            } else {
                LL lt=a[i]-c[a[i]];
                e=(base+lt*inv[sz]%MOD*z)%MOD;
            }
        }
        LL now=e*fac[sz]%MOD;
        now=now*fac[n-i]%MOD;
        ans+=now;
        dump(i,a[i],e*fac[sz]%MOD,now);
    }
    ans+=fac[sz];
    ans%=MOD;
    W(ans);
    return 0;
}
