Problem: You are given integers N, K, and an integer sequence A of length M.

An integer sequence where each element is between 1 and K (inclusive) is said to be colorful when there exists a contiguous subsequence of length K of the sequence that contains one occurrence of each integer between 1 and K (inclusive).

For every colorful integer sequence of length N, count the number of the contiguous subsequences of that sequence which coincide with A, then find the sum of all the counts. Here, the answer can be extremely large, so find the sum modulo 10^9+7.

Constraints

* 1 \leq N \leq 25000
* 1 \leq K \leq 400
* 1 \leq M \leq N
* 1 \leq A_i \leq K
* All values in input are integers.

Input

Input is given from Standard Input in the following format:


N K M
A_1 A_2 ... A_M


Output

For every colorful integer sequence of length N, count the number of the contiguous subsequences of that sequence which coincide with A, then print the sum of all the counts modulo 10^9+7.

Examples

Input

3 2 1
1


Output

9


Input

4 2 2
1 2


Output

12


Input

7 4 5
1 2 3 1 2


Output

17


Input

5 4 3
1 1 1


Output

0


Input

10 3 5
1 1 2 3 3


Output

1458


Input

25000 400 4
3 7 31 127


Output

923966268


Input

9954 310 12
267 193 278 294 6 63 86 166 157 193 168 43


Output

979180369

Solution (in CPP):

Solution:
#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;
	a+=b;
	if(a>=mod)a-=mod;
}

template<class T,class U>
void sub(T& a,U b){
	//a=((ll)a-b%mod+mod)%mod;
	a-=b;
	if(a<0)a+=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;
}

const int Nmax=25010;
const int Kmax=410;
int dp[Nmax][2][Kmax];

int len(vi x){
	set<int> pre;
	int cur=0;
	for(auto v:x){
		if(pre.find(v)!=pre.end())
			return cur;
		pre.insert(v);
		cur++;
	}
	return cur;
}

bool HasPerm(int k,vi a){
	if(int(a.size())<k)return false;
	vi cnt(k,0);
	int nonzero=0;
	auto Add=[&](int x){
		nonzero-=bool(cnt[x]);
		cnt[x]++;
		nonzero+=bool(cnt[x]);
	};
	auto Del=[&](int x){
		nonzero-=bool(cnt[x]);
		cnt[x]--;
		nonzero+=bool(cnt[x]);
	};
	REP(i,k)
		Add(a[i]);
	FOR(i,k,a.size()){
		if(nonzero==k)return true;
		Del(a[i-k]);
		Add(a[i]);
	}
	return nonzero==k;
}

signed main(){
	InitFact();
	
	int n=read(),k=read(),m=read();
	vi a(m);
	REP(i,m)a[i]=read()-1;
	
	int ans=modPow(k,n-m);
	mult(ans,n-m+1);
	
	int res=0;
	
	if(!HasPerm(k,a)){
		dp[1][0][1]=k;
		if(m==1)dp[1][1][1]=k;
		FOR(i,1,n+1){
			REP(u,2)
				for(int c=k-1;c>=1;c--){
					add(dp[i][u][c],dp[i][u][c+1]);
					dp[i+1][u][c]=dp[i][u][c];
					int w=dp[i][u][c-1];
					mult(w,k-c+1);
					add(dp[i+1][u][c],w);
				}
			FOR(c,m,k)
				add(dp[i+1][1][c],dp[i+1][0][c]);
		}
		int l1=len(a);
		reverse(ALL(a));
		int l2=len(a);
		if(l1==m){
			res=dp[n][1][1];
			mult(res,factInv[k]);
			mult(res,fact[k-m]);
		}else{
			REP(i,n-m+1){
				int j=n-m-i;
				int w=dp[i+l1][0][l1];
				mult(w,factInv[k]);
				mult(w,fact[k-l1]);
				mult(w,dp[j+l2][0][l2]);
				mult(w,factInv[k]);
				mult(w,fact[k-l2]);
				add(res,w);
			}
		}
	}
	
	sub(ans,res);
	print(ans);
}