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
/*
*/

//#pragma comment(linker, "/STACK:16777216")
#define _CRT_SECURE_NO_WARNINGS

#include <fstream>
#include <iostream>
#include <string>
#include <complex>
#include <math.h>
#include <set>
#include <vector>
#include <map>
#include <queue>
#include <stdio.h>
#include <stack>
#include <algorithm>
#include <list>
#include <ctime>
#include <memory.h>
#include <assert.h>

#define y0 sdkfaslhagaklsldk
#define y1 aasdfasdfasdf
#define yn askfhwqriuperikldjk
#define j1 assdgsdgasghsf
#define tm sdfjahlfasfh
#define lr asgasgash
#define norm asdfasdgasdgsd
#define have adsgagshdshfhds

#define eps 1e-9
#define M_PI 3.141592653589793
#define bs 1000000007
#define bsize 350

using namespace std;

const int INF = 1e9;
const int N = 600031;

long long n, ar[N];
int zeros;
long long fact[N];
int used[N];
long long ohead[N];

int brute()
{
	vector<int> order;
	for (int i = 0; i < n; i++)
	{
		order.push_back(i + 1);
	}

	int ans = 0;

	int cnt = 0;

	do
	{
		++cnt;
		int ok = 1;
		for (int i = 0; i < n; i++)
		{
			if (ar[i] != 0 && order[i] != ar[i])
				ok = 0;
		}
		if (ok)
			ans += cnt;

	} while (next_permutation(order.begin(), order.end()));

	return ans;
}

int t[N];

int sum(int r)
{
	int res = 0;
	for (; r >= 0; r = (r&(r + 1)) - 1){
		res += t[r];
		if (res >= bs)
			res -= bs;
	}
	return res;
}

void inc(int i, int delta)
{
	for (; i < n; i = (i | (i + 1)))
		t[i] += delta;
}

int zeros_below[N];

int main(){
	//freopen("fabro.in","r",stdin);
	//freopen("fabro.out","w",stdout);
	//freopen("F:/in.txt", "r", stdin);
	//freopen("F:/output.txt", "w", stdout);
	ios_base::sync_with_stdio(0);
	//cin.tie(0);

	cin >> n;
	for (int i = 0; i < n; i++)
	{
		cin >> ar[i];
		if (ar[i] == 0)
			++zeros;
		else
			used[ar[i]] = 1;
	}

	int unus_sum = 0;

	for (int i = 1; i <= n; i++)
	{
		if (used[i] == 0)
			unus_sum = unus_sum + i - 1;
		unus_sum %= bs;
	}

	fact[0] = 1;
	for (int i = 1; i < N; i++)
	{
		fact[i] = fact[i - 1] * i%bs;
	}

	int cur_ohead_sum = 0;
	int total_ohead_sum = 0;

	for (int i = n; i; --i)
	{
		if (used[i] == 0)
		{
			total_ohead_sum += cur_ohead_sum;
			if (total_ohead_sum >= bs)
				total_ohead_sum %= bs;
			cur_ohead_sum += 1;
			cur_ohead_sum %= bs;
		}
		else
		{
			ohead[i] = cur_ohead_sum;
		}
	}

	int cnt = 0;

	for (int i = 1; i <= n; i++)
	{
		if (used[i] == 0)
			cnt++;
		else
		{
			zeros_below[i] = cnt;
		}
	}

	long long ans = 0;

	//cout << unus_sum << endl;

	int count_zeros = 0;

	int sum_oheads = 0;

	for (int i = 0; i < n; i++)
	{
		if (ar[i] != 0)// easy part
		{
			ans -= fact[zeros] * fact[n - i - 1] % bs*sum(ar[i] - 1) % bs;
			sum_oheads += ohead[ar[i]];
			sum_oheads %= bs;
			if (ans < 0)
				ans += bs;
			inc(ar[i], 1);
			int unus = 0;
			unus = zeros_below[ar[i]];

			ans += (ar[i] - 1) * fact[zeros] % bs*fact[n - i - 1] % bs;
			ans %= bs;
			ans -= count_zeros*fact[zeros - 1] % bs*unus%bs*fact[n - i - 1] % bs;
			if (ans < 0)
				ans += bs;
		}
		else // shit happens
		{
			ans += unus_sum * fact[zeros - 1] % bs*fact[n - i - 1] % bs;
			//	cout << ans << endl;
			ans %= bs;
			ans -= fact[zeros - 1] * fact[n - i - 1] % bs*sum_oheads%bs;
			if (ans < 0)
				ans += bs;

			ans -= total_ohead_sum*fact[n - i - 1] % bs*fact[zeros - 2] % bs*count_zeros%bs;
			if (ans < 0)
				ans += bs;
			count_zeros++;
		}
	}

	//	cout << total_ohead_sum << endl;

	ans += fact[zeros];
	ans %= bs;

	cout << ans << endl;
	
	//	cout << brute() << endl;

	cin.get(); cin.get();
	return 0;
}