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 <bits/stdc++.h>
#define syosu(x) fixed<<setprecision(x)
using namespace std;
typedef long long ll;
typedef unsigned long long ull;
typedef pair<int,int> P;
typedef pair<double,double> pdd;
typedef pair<ll,ll> pll;
typedef vector<int> vi;
typedef vector<vi> vvi;
typedef vector<double> vd;
typedef vector<vd> vvd;
typedef vector<ll> vl;
typedef vector<vl> vvl;
typedef vector<string> vs;
typedef vector<P> vp;
typedef vector<vp> vvp;
typedef vector<pll> vpll;
typedef pair<int,P> pip;
typedef vector<pip> vip;
const int inf=1<<30;
const ll INF=1ll<<60;
const double pi=acos(-1);
const double eps=1e-8;
const ll mod=1e9+7;
const int dx[4]={-1,0,1,0},dy[4]={0,-1,0,1};

class Segment_Tree{
	private:
	int n;
	vi date;
	public:
	Segment_Tree(int n_){
		n=1;
		while(n<n_) n*=2;
		date=vi(2*n-1);
	}
	void Update(int k,int x){
		k+=n-1;
		date[k]+=x;
		while(k>0){
			k=(k-1)/2;
			date[k]=date[k*2+1]+date[k*2+2];
		}
	}
	int Query(int a,int b){
		a+=n-1;b+=n-1;
		int m=0;
		while(a<b){
			if(a%2==0) m+=date[a++];
			if(b%2==0) m+=date[--b];
			a/=2;b/=2;
		}
		return m;
	}
	int Open(int k){return date[k+n-1];}
};

const int M=500005;
ll n,F[M],m=0;
vl a,b;

int main(){
	cin>>n;
	F[0]=1;
	a=b=vl(n);
	fill(b.begin(),b.end(),1);
	for(int i=1;i<M;i++) F[i]=F[i-1]*i%mod;
	ll res=0,t=0,sum=0;
	for(int i=0;i<n;i++){
		cin>>a[i];
		a[i]--;
		if(a[i]<0) m++;
		else b[a[i]]--;
	}
	for(int i=1;i<n;i++) b[i]+=b[i-1];
	Segment_Tree st(n);
	for(int i=n-1;i>=0;i--){
		if(i<n-1){
			ll tmp;
			if(a[i]>=0) tmp=(st.Query(0,a[i])*F[m]+(t?t*b[a[i]]%mod*F[m-1]:0))%mod;
			else tmp=(F[m]*(mod/2+1)%mod*t+sum*F[m-1])%mod;
			(res+=tmp*F[n-i-1])%=mod;
		}
		if(a[i]>=0) st.Update(a[i],1),(sum+=m-b[a[i]])%=mod;
		else t++;
	}
	cout<<(res+F[m])%mod<<endl;
}