#include <bits/stdc++.h>
using namespace std;

using ll=int64_t;
#define int ll

#define FOR(i,a,b) for(int i=int(a);i<int(b);i++)
#define REP(i,b) FOR(i,0,b)
#define MP make_pair
#define PB push_back
#define ALL(x) x.begin(),x.end()
#ifdef MAROON_LOCAL
#define cerr (cerr<<"-- line "<<__LINE__<<" -- ")
#else
class CerrDummy{}cerrDummy;
template<class T>
CerrDummy& operator<<(CerrDummy&cd,const T&){
	return cd;
}
using charTDummy=char;
using traitsDummy=char_traits<charTDummy>;
CerrDummy& operator<<(CerrDummy&cd,basic_ostream<charTDummy,traitsDummy>&(basic_ostream<charTDummy,traitsDummy>&)){
	return cd;
}
#define cerr cerrDummy
#endif
#define REACH cerr<<"reached"<<endl
#define DMP(x) cerr<<#x<<":"<<x<<endl
#define ZERO(x) memset(x,0,sizeof(x))
#define ONE(x) memset(x,-1,sizeof(x))

using pi=pair<int,int>;
using vi=vector<int>;
using ld=long double;

template<class T,class U>
ostream& operator<<(ostream& os,const pair<T,U>& p){
	os<<"("<<p.first<<","<<p.second<<")";
	return os;
}

template<class T>
ostream& operator <<(ostream& os,const vector<T>& v){
	os<<"[";
	REP(i,(int)v.size()){
		if(i)os<<",";
		os<<v[i];
	}
	os<<"]";
	return os;
}

ll read(){
	ll i;
	scanf("%" SCNd64,&i);
	return i;
}

void printSpace(){
	printf(" ");
}

void printEoln(){
	printf("\n");
}

void print(ll x,int suc=1){
	printf("%" PRId64,x);
	if(suc==1)
		printEoln();
	if(suc==2)
		printSpace();
}

string readString(){
	static char buf[3341000];
	scanf("%s",buf);
	return string(buf);
}

char* readCharArray(){
	static char buf[3341000];
	static int bufUsed=0;
	char* ret=buf+bufUsed;
	scanf("%s",ret);
	bufUsed+=strlen(ret)+1;
	return ret;
}

template<class T,class U>
void chmax(T& a,U b){
	if(a<b)
		a=b;
}

template<class T,class U>
void chmin(T& a,U b){
	if(b<a)
		a=b;
}

template<class T>
T Sq(const T& t){
	return t*t;
}

#define CAPITAL
void Yes(bool ex=true){
	#ifdef CAPITAL
	cout<<"YES"<<endl;
	#else
	cout<<"Yes"<<endl;
	#endif
	if(ex)exit(0);
}
void No(bool ex=true){
	#ifdef CAPITAL
	cout<<"NO"<<endl;
	#else
	cout<<"No"<<endl;
	#endif
	if(ex)exit(0);
}

const ll infLL=LLONG_MAX/3;

#ifdef int
const int inf=infLL;
#else
const int inf=INT_MAX/2-100;
#endif

const int mod=1000000007;
template<class T,class U>
void add(T& a,U b){
	a=((ll)a+b)%mod;
}

template<class T,class U>
void sub(T& a,U b){
	a=((ll)a-b%mod+mod)%mod;
}

template<class T,class U>
void mult(T& a,U b){
	a=((ll)a*b)%mod;
}

ll modPow(ll a,ll p){
	ll s=1;
	while(p){
		if(p&1)mult(s,a);
		mult(a,a);
		p>>=1;
	}
	return s;
}

ll modInv(ll a){
	return modPow(a,mod-2);
}

const int Vmax=1000010;
int fact[Vmax],factInv[Vmax];
void InitFact(){
	fact[0]=1;
	FOR(i,1,Vmax){
		fact[i]=fact[i-1];
		mult(fact[i],i);
	}
	factInv[Vmax-1]=modInv(fact[Vmax-1]);
	for(int i=Vmax-2;i>=0;i--){
		factInv[i]=factInv[i+1];
		mult(factInv[i],i+1);
	}
}
int Choose(int n,int k){
	return ll(fact[n])*factInv[n-k]%mod*factInv[k]%mod;
}
int Binom(int a,int b){
	return ll(fact[a+b])*factInv[a]%mod*factInv[b]%mod;
}

struct BIT{
	vi buf;
	int s;
	BIT(int n){
		buf.assign(s=n,0);
	}
	void Add(int i,int v){
		for(;i<s;i+=((i+1)&(-i-1)))
			add(buf[i],v);
	}
	int Get(int i){
		int res=0;
		for(;i>=0;i-=((i+1)&(-i-1)))
			add(res,buf[i]);
		return res;
	}
	int Range(int b,int e){
		int res=Get(e-1);
		sub(res,Get(b-1));
		return res;
	}
};

signed main(){
	int n=read();
	vi a(n);
	REP(i,n)a[i]=read();
	
	vi cnt(n+1,0);
	for(auto x:a)cnt[x]++;
	for(int i=n-1;i>=1;i--)
		cnt[i]+=cnt[i+1]-1;
	
	FOR(i,1,n+1)
		if(cnt[i]==0){
			print(0);
			return 0;
		}
	
	vi w{1},zs;
	FOR(i,1,n+1){
		if(cnt[i]==1){
			w.PB(1);
			zs.PB(i);
		}else{
			int x=w.back();
			mult(x,cnt[i]-1);
			mult(x,modInv(cnt[i]));
			w.PB(x);
		}
	}
	zs.PB(n+1);
	
	int ans=0;
	
	{
		BIT bit(n+1);
		REP(i,n){
			auto itr=upper_bound(ALL(zs),a[i]);
			itr--;
			int b=*itr;
			int x=bit.Range(b,a[i]+1);
			int y=w[a[i]];
			mult(y,x);
			mult(y,modInv(2));
			add(ans,y);
			bit.Add(a[i],modInv(w[a[i]]));
		}
	}
	{
		BIT bit(n+1),bit2(n+1);
		int invs=0,revs=0;
		REP(i,n){
			auto itr=upper_bound(ALL(zs),a[i]);
			int e=*itr;
			int x=bit.Range(a[i]+1,e);
			int y=modInv(w[a[i]]);
			mult(y,x);
			mult(y,modInv(2));
			add(revs,y);
			bit.Add(a[i],w[a[i]]);
			
			add(invs,bit2.Range(a[i]+1,n+1));
			bit2.Add(a[i],1);
		}
		add(ans,invs);
		sub(ans,revs);
	}
	
	FOR(i,1,n+1)
		mult(ans,cnt[i]);
	print(ans);
}

### Prompt

Please provide a Cpp coded solution to the problem described below:

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 <bits/stdc++.h>
using namespace std;

using ll=int64_t;
#define int ll

#define FOR(i,a,b) for(int i=int(a);i<int(b);i++)
#define REP(i,b) FOR(i,0,b)
#define MP make_pair
#define PB push_back
#define ALL(x) x.begin(),x.end()
#ifdef MAROON_LOCAL
#define cerr (cerr<<"-- line "<<__LINE__<<" -- ")
#else
class CerrDummy{}cerrDummy;
template<class T>
CerrDummy& operator<<(CerrDummy&cd,const T&){
	return cd;
}
using charTDummy=char;
using traitsDummy=char_traits<charTDummy>;
CerrDummy& operator<<(CerrDummy&cd,basic_ostream<charTDummy,traitsDummy>&(basic_ostream<charTDummy,traitsDummy>&)){
	return cd;
}
#define cerr cerrDummy
#endif
#define REACH cerr<<"reached"<<endl
#define DMP(x) cerr<<#x<<":"<<x<<endl
#define ZERO(x) memset(x,0,sizeof(x))
#define ONE(x) memset(x,-1,sizeof(x))

using pi=pair<int,int>;
using vi=vector<int>;
using ld=long double;

template<class T,class U>
ostream& operator<<(ostream& os,const pair<T,U>& p){
	os<<"("<<p.first<<","<<p.second<<")";
	return os;
}

template<class T>
ostream& operator <<(ostream& os,const vector<T>& v){
	os<<"[";
	REP(i,(int)v.size()){
		if(i)os<<",";
		os<<v[i];
	}
	os<<"]";
	return os;
}

ll read(){
	ll i;
	scanf("%" SCNd64,&i);
	return i;
}

void printSpace(){
	printf(" ");
}

void printEoln(){
	printf("\n");
}

void print(ll x,int suc=1){
	printf("%" PRId64,x);
	if(suc==1)
		printEoln();
	if(suc==2)
		printSpace();
}

string readString(){
	static char buf[3341000];
	scanf("%s",buf);
	return string(buf);
}

char* readCharArray(){
	static char buf[3341000];
	static int bufUsed=0;
	char* ret=buf+bufUsed;
	scanf("%s",ret);
	bufUsed+=strlen(ret)+1;
	return ret;
}

template<class T,class U>
void chmax(T& a,U b){
	if(a<b)
		a=b;
}

template<class T,class U>
void chmin(T& a,U b){
	if(b<a)
		a=b;
}

template<class T>
T Sq(const T& t){
	return t*t;
}

#define CAPITAL
void Yes(bool ex=true){
	#ifdef CAPITAL
	cout<<"YES"<<endl;
	#else
	cout<<"Yes"<<endl;
	#endif
	if(ex)exit(0);
}
void No(bool ex=true){
	#ifdef CAPITAL
	cout<<"NO"<<endl;
	#else
	cout<<"No"<<endl;
	#endif
	if(ex)exit(0);
}

const ll infLL=LLONG_MAX/3;

#ifdef int
const int inf=infLL;
#else
const int inf=INT_MAX/2-100;
#endif

const int mod=1000000007;
template<class T,class U>
void add(T& a,U b){
	a=((ll)a+b)%mod;
}

template<class T,class U>
void sub(T& a,U b){
	a=((ll)a-b%mod+mod)%mod;
}

template<class T,class U>
void mult(T& a,U b){
	a=((ll)a*b)%mod;
}

ll modPow(ll a,ll p){
	ll s=1;
	while(p){
		if(p&1)mult(s,a);
		mult(a,a);
		p>>=1;
	}
	return s;
}

ll modInv(ll a){
	return modPow(a,mod-2);
}

const int Vmax=1000010;
int fact[Vmax],factInv[Vmax];
void InitFact(){
	fact[0]=1;
	FOR(i,1,Vmax){
		fact[i]=fact[i-1];
		mult(fact[i],i);
	}
	factInv[Vmax-1]=modInv(fact[Vmax-1]);
	for(int i=Vmax-2;i>=0;i--){
		factInv[i]=factInv[i+1];
		mult(factInv[i],i+1);
	}
}
int Choose(int n,int k){
	return ll(fact[n])*factInv[n-k]%mod*factInv[k]%mod;
}
int Binom(int a,int b){
	return ll(fact[a+b])*factInv[a]%mod*factInv[b]%mod;
}

struct BIT{
	vi buf;
	int s;
	BIT(int n){
		buf.assign(s=n,0);
	}
	void Add(int i,int v){
		for(;i<s;i+=((i+1)&(-i-1)))
			add(buf[i],v);
	}
	int Get(int i){
		int res=0;
		for(;i>=0;i-=((i+1)&(-i-1)))
			add(res,buf[i]);
		return res;
	}
	int Range(int b,int e){
		int res=Get(e-1);
		sub(res,Get(b-1));
		return res;
	}
};

signed main(){
	int n=read();
	vi a(n);
	REP(i,n)a[i]=read();
	
	vi cnt(n+1,0);
	for(auto x:a)cnt[x]++;
	for(int i=n-1;i>=1;i--)
		cnt[i]+=cnt[i+1]-1;
	
	FOR(i,1,n+1)
		if(cnt[i]==0){
			print(0);
			return 0;
		}
	
	vi w{1},zs;
	FOR(i,1,n+1){
		if(cnt[i]==1){
			w.PB(1);
			zs.PB(i);
		}else{
			int x=w.back();
			mult(x,cnt[i]-1);
			mult(x,modInv(cnt[i]));
			w.PB(x);
		}
	}
	zs.PB(n+1);
	
	int ans=0;
	
	{
		BIT bit(n+1);
		REP(i,n){
			auto itr=upper_bound(ALL(zs),a[i]);
			itr--;
			int b=*itr;
			int x=bit.Range(b,a[i]+1);
			int y=w[a[i]];
			mult(y,x);
			mult(y,modInv(2));
			add(ans,y);
			bit.Add(a[i],modInv(w[a[i]]));
		}
	}
	{
		BIT bit(n+1),bit2(n+1);
		int invs=0,revs=0;
		REP(i,n){
			auto itr=upper_bound(ALL(zs),a[i]);
			int e=*itr;
			int x=bit.Range(a[i]+1,e);
			int y=modInv(w[a[i]]);
			mult(y,x);
			mult(y,modInv(2));
			add(revs,y);
			bit.Add(a[i],w[a[i]]);
			
			add(invs,bit2.Range(a[i]+1,n+1));
			bit2.Add(a[i],1);
		}
		add(ans,invs);
		sub(ans,revs);
	}
	
	FOR(i,1,n+1)
		mult(ans,cnt[i]);
	print(ans);
}

```