#include <iostream>
#include <vector>
#include <algorithm>
#include <cmath>
#include <string>
#include <queue>
#include <stack>
#include <set>
#include <map>
#include <iomanip>
#include <utility>
#include <tuple>
#include <functional>
#include <bitset>
#include <cassert>
#include <complex>
#include <stdio.h>
#include <time.h>
#include <numeric>
#include <random>
#include <unordered_map>
#include <unordered_set>
#define all(a) a.begin(),a.end()
#define rep(i, n) for (ll i = 0; i < (n); i++)
#define pb push_back
#define debug(x) cerr << #x << ':' << x << '\n';
#pragma GCC optimize("O3")
#pragma GCC optimize("unroll-loops")
using namespace std;
typedef long long ll;
typedef unsigned long long ull;
typedef long double ld;
typedef pair<ll, ll> P;
typedef complex<ld> com;
constexpr int inf = 1000000010;
constexpr ll INF = 1000000000000000010;
constexpr ld eps = 1e-12;
constexpr ld pi = 3.141592653589793238;
template<class T, class U> inline bool chmax(T &a, const U &b) { if (a < b) { a = b; return true; } return false; }
template<class T, class U> inline bool chmin(T &a, const U &b) { if (a > b) { a = b; return true; } return false; }

constexpr ll mod = 1000000007;
constexpr ll modsize = 0;
vector<ll> fac(modsize);
vector<ll> inv(modsize);
vector<ll> facinv(modsize);

void modcalc() {
	if (modsize == 0) abort();
	fac[0] = 1; fac[1] = 1; inv[1] = 1;
	facinv[0] = 1; facinv[1] = 1;
	for (ll i = 2; i < modsize; i++) {
		fac[i] = fac[i - 1] * i % mod;
		inv[i] = mod - inv[mod % i] * (mod / i) % mod;
		facinv[i] = facinv[i - 1] * inv[i] % mod;
	}
}

ll modinv(ll a) {
	if (a == 0) abort();
	ll b = mod, u = 1, v = 0;
	while (b) {
		ll t = a / b;
		a -= t * b; swap(a, b);
		u -= t * v; swap(u, v);
	}
	u %= mod;
	if (u < 0) u += mod;
	return u;
}

ll modpow(ll a, ll b) {
	ll ans = 1;
	a %= mod;
	while (b) {
		if (b & 1) ans = ans * a % mod;
		a = a * a % mod;
		b >>= 1;
	}
	return ans;
}

ll modcomb(ll n, ll k) {
	if (n < 0 || k < 0 || n < k) return 0;
	return fac[n] * facinv[k] % mod * facinv[n - k] % mod;
}

ll modperm(ll n, ll k) {
	if (n < 0 || k < 0 || n < k) return 0;
	return fac[n] * facinv[n - k] % mod;
}

ll modhom(ll n, ll k) {
	if (n < 0 || k < 0 || n == 0 && k > 0) return 0;
	if (n == 0 && k == 0) return 1;
	return fac[n + k - 1] * facinv[k] % mod * facinv[n - 1] % mod;
}

template<class T>
class lazysegtree {
	int n;
	vector<T> data, lazy;
	vector<bool> lazyflag;
	T id = 0;
	T operation(T a, T b) { return (a + b) % mod; };
public:
	lazysegtree(int _n) {
		n = 1;
		while (n < _n + 2) { n <<= 1; }
		data = lazy = vector<T>(2 * n - 1, id);
		lazyflag = vector<bool>(2 * n - 1, false);
	}
	void eval(int k, int l, int r) {
		if (lazyflag[k]) {
			data[k] = lazy[k];
			if (r - l > 1) {
				lazy[2 * k + 1] = lazy[k];
				lazy[2 * k + 2] = lazy[k];
				lazyflag[2 * k + 1] = lazyflag[2 * k + 2] = true;
			}
			lazyflag[k] = false;
		}
	}
	void lazychange(int a, int b, int x, int k = 0, int l = 0, int r = -1) {
		if (r < 0) r = n;
		eval(k, l, r);
		if (b <= l || r <= a) return;
		if (a <= l && r <= b) {
			lazy[k] = x;
			lazyflag[k] = true;
			eval(k, l, r);
		}
		else {
			lazychange(a, b, x, 2 * k + 1, l, (l + r) / 2);
			lazychange(a, b, x, 2 * k + 2, (l + r) / 2, r);
			data[k] = operation(data[2 * k + 1], data[2 * k + 2]);
		}
	}
	void change(int i, T x) { lazychange(i, i + 1, x); }
	void add(int i, T x) { change(i, (data[i + n - 1] + x) % mod); }
	T get(int a, int b, int k = 0, int l = 0, int r = -1) {
		if (r < 0) r = n;
		eval(k, l, r);
		if (b <= l || r <= a) return id;
		if (a <= l && r <= b) return data[k];
		int vl = get(a, b, 2 * k + 1, l, (l + r) / 2);
		int vr = get(a, b, 2 * k + 2, (l + r) / 2, r);
		return operation(vl, vr);
	}
	T operator[](int i) { return get(i, i + 1); }
};

int main() {
	cin.tie(0);
	ios::sync_with_stdio(false);
	cout << fixed << setprecision(20);

	ll n;
	cin >> n;
	vector<P> p(n);
	rep(i, n) {
		ll a;
		cin >> a;
		p[i] = { a,i };
	}
	sort(all(p)); reverse(all(p));
	lazysegtree<ll> seg(n), seg2(n);
	ll ans = 0, cnt = 1, coeff = 1;
	rep(i, n) {
		ll num = p[i].first; ll idx = p[i].second;
		ll pat = num - (n - i) + 1;
		if (pat <= 0) {
			cout << "0\n";
			return 0;
		}
		ll start = ans + cnt * seg2.get(0, idx); start %= mod;
		seg2.add(idx, 1);
		ll d = cnt - 2 * seg.get(0, idx) * coeff; d %= mod; if (d < 0) d += mod;
		coeff *= pat - 1; coeff %= mod;
		cnt *= pat; cnt %= mod;
		if (coeff) seg.add(idx, cnt * modinv(coeff) % mod * modinv(pat) % mod);
		else {
			seg.lazychange(0, n, 0);
			seg.add(idx, cnt);
			coeff = 1;
		}
		ans = start * pat + pat * (pat - 1) / 2 % mod * d % mod;
		ans %= mod;
	}
	cout << ans << '\n';
}
### Prompt

Generate a cpp solution to the following problem:

Snuke has an integer sequence A whose length is N. He likes permutations of (1, 2, ..., N), P, that satisfy the following condition:

* P_i \leq A_i for all i ( 1 \leq i \leq N ).



Snuke is interested in the inversion numbers of such permutations. Find the sum of the inversion numbers over all permutations that satisfy the condition. Since this can be extremely large, compute the sum modulo 10^9+7.

Constraints

* 1 \leq N \leq 2 \times 10^5
* 1 \leq A_i \leq N ( 1 \leq i \leq N )
* All values in input are integers.

Input

Input is given from Standard Input in the following format:


N
A_1 A_2 ... A_N


Output

Print the sum of the inversion numbers over all permutations that satisfy the condition.

Examples

Input

3
2 3 3


Output

4


Input

6
4 2 5 1 6 3


Output

7


Input

5
4 4 4 4 4


Output

0


Input

30
22 30 15 20 10 29 11 29 28 11 26 10 18 28 22 5 29 16 24 24 27 10 21 30 29 19 28 27 18 23


Output

848414012

### Response

```cpp
#include <iostream>
#include <vector>
#include <algorithm>
#include <cmath>
#include <string>
#include <queue>
#include <stack>
#include <set>
#include <map>
#include <iomanip>
#include <utility>
#include <tuple>
#include <functional>
#include <bitset>
#include <cassert>
#include <complex>
#include <stdio.h>
#include <time.h>
#include <numeric>
#include <random>
#include <unordered_map>
#include <unordered_set>
#define all(a) a.begin(),a.end()
#define rep(i, n) for (ll i = 0; i < (n); i++)
#define pb push_back
#define debug(x) cerr << #x << ':' << x << '\n';
#pragma GCC optimize("O3")
#pragma GCC optimize("unroll-loops")
using namespace std;
typedef long long ll;
typedef unsigned long long ull;
typedef long double ld;
typedef pair<ll, ll> P;
typedef complex<ld> com;
constexpr int inf = 1000000010;
constexpr ll INF = 1000000000000000010;
constexpr ld eps = 1e-12;
constexpr ld pi = 3.141592653589793238;
template<class T, class U> inline bool chmax(T &a, const U &b) { if (a < b) { a = b; return true; } return false; }
template<class T, class U> inline bool chmin(T &a, const U &b) { if (a > b) { a = b; return true; } return false; }

constexpr ll mod = 1000000007;
constexpr ll modsize = 0;
vector<ll> fac(modsize);
vector<ll> inv(modsize);
vector<ll> facinv(modsize);

void modcalc() {
	if (modsize == 0) abort();
	fac[0] = 1; fac[1] = 1; inv[1] = 1;
	facinv[0] = 1; facinv[1] = 1;
	for (ll i = 2; i < modsize; i++) {
		fac[i] = fac[i - 1] * i % mod;
		inv[i] = mod - inv[mod % i] * (mod / i) % mod;
		facinv[i] = facinv[i - 1] * inv[i] % mod;
	}
}

ll modinv(ll a) {
	if (a == 0) abort();
	ll b = mod, u = 1, v = 0;
	while (b) {
		ll t = a / b;
		a -= t * b; swap(a, b);
		u -= t * v; swap(u, v);
	}
	u %= mod;
	if (u < 0) u += mod;
	return u;
}

ll modpow(ll a, ll b) {
	ll ans = 1;
	a %= mod;
	while (b) {
		if (b & 1) ans = ans * a % mod;
		a = a * a % mod;
		b >>= 1;
	}
	return ans;
}

ll modcomb(ll n, ll k) {
	if (n < 0 || k < 0 || n < k) return 0;
	return fac[n] * facinv[k] % mod * facinv[n - k] % mod;
}

ll modperm(ll n, ll k) {
	if (n < 0 || k < 0 || n < k) return 0;
	return fac[n] * facinv[n - k] % mod;
}

ll modhom(ll n, ll k) {
	if (n < 0 || k < 0 || n == 0 && k > 0) return 0;
	if (n == 0 && k == 0) return 1;
	return fac[n + k - 1] * facinv[k] % mod * facinv[n - 1] % mod;
}

template<class T>
class lazysegtree {
	int n;
	vector<T> data, lazy;
	vector<bool> lazyflag;
	T id = 0;
	T operation(T a, T b) { return (a + b) % mod; };
public:
	lazysegtree(int _n) {
		n = 1;
		while (n < _n + 2) { n <<= 1; }
		data = lazy = vector<T>(2 * n - 1, id);
		lazyflag = vector<bool>(2 * n - 1, false);
	}
	void eval(int k, int l, int r) {
		if (lazyflag[k]) {
			data[k] = lazy[k];
			if (r - l > 1) {
				lazy[2 * k + 1] = lazy[k];
				lazy[2 * k + 2] = lazy[k];
				lazyflag[2 * k + 1] = lazyflag[2 * k + 2] = true;
			}
			lazyflag[k] = false;
		}
	}
	void lazychange(int a, int b, int x, int k = 0, int l = 0, int r = -1) {
		if (r < 0) r = n;
		eval(k, l, r);
		if (b <= l || r <= a) return;
		if (a <= l && r <= b) {
			lazy[k] = x;
			lazyflag[k] = true;
			eval(k, l, r);
		}
		else {
			lazychange(a, b, x, 2 * k + 1, l, (l + r) / 2);
			lazychange(a, b, x, 2 * k + 2, (l + r) / 2, r);
			data[k] = operation(data[2 * k + 1], data[2 * k + 2]);
		}
	}
	void change(int i, T x) { lazychange(i, i + 1, x); }
	void add(int i, T x) { change(i, (data[i + n - 1] + x) % mod); }
	T get(int a, int b, int k = 0, int l = 0, int r = -1) {
		if (r < 0) r = n;
		eval(k, l, r);
		if (b <= l || r <= a) return id;
		if (a <= l && r <= b) return data[k];
		int vl = get(a, b, 2 * k + 1, l, (l + r) / 2);
		int vr = get(a, b, 2 * k + 2, (l + r) / 2, r);
		return operation(vl, vr);
	}
	T operator[](int i) { return get(i, i + 1); }
};

int main() {
	cin.tie(0);
	ios::sync_with_stdio(false);
	cout << fixed << setprecision(20);

	ll n;
	cin >> n;
	vector<P> p(n);
	rep(i, n) {
		ll a;
		cin >> a;
		p[i] = { a,i };
	}
	sort(all(p)); reverse(all(p));
	lazysegtree<ll> seg(n), seg2(n);
	ll ans = 0, cnt = 1, coeff = 1;
	rep(i, n) {
		ll num = p[i].first; ll idx = p[i].second;
		ll pat = num - (n - i) + 1;
		if (pat <= 0) {
			cout << "0\n";
			return 0;
		}
		ll start = ans + cnt * seg2.get(0, idx); start %= mod;
		seg2.add(idx, 1);
		ll d = cnt - 2 * seg.get(0, idx) * coeff; d %= mod; if (d < 0) d += mod;
		coeff *= pat - 1; coeff %= mod;
		cnt *= pat; cnt %= mod;
		if (coeff) seg.add(idx, cnt * modinv(coeff) % mod * modinv(pat) % mod);
		else {
			seg.lazychange(0, n, 0);
			seg.add(idx, cnt);
			coeff = 1;
		}
		ans = start * pat + pat * (pat - 1) / 2 % mod * d % mod;
		ans %= mod;
	}
	cout << ans << '\n';
}
```