Problem: Nezzar buys his favorite snack — n chocolate bars with lengths l_1,l_2,…,l_n. However, chocolate bars might be too long to store them properly! 

In order to solve this problem, Nezzar designs an interesting process to divide them into small pieces. Firstly, Nezzar puts all his chocolate bars into a black box. Then, he will perform the following operation repeatedly until the maximum length over all chocolate bars does not exceed k.

  * Nezzar picks a chocolate bar from the box with probability proportional to its length x. 
  * After step 1, Nezzar uniformly picks a real number r ∈ (0,x) and divides the chosen chocolate bar into two chocolate bars with lengths r and x-r. 
  * Lastly, he puts those two new chocolate bars into the black box. 



Nezzar now wonders, what is the expected number of operations he will perform to divide his chocolate bars into small pieces.

It can be shown that the answer can be represented as P/Q, where P and Q are coprime integers and Q not ≡ 0 (mod 998 244 353). Print the value of P⋅ Q^{-1} mod 998 244 353.

Input

The first line contains two integers n and k (1 ≤ n ≤ 50, 1 ≤ k ≤ 2000).

The second line contains n integers l_1, l_2, …, l_n (1 ≤ l_i, ∑_{i=1}^{n} l_i ≤ 2000).

Output

Print a single integer — the expected number of operations Nezzar will perform to divide his chocolate bars into small pieces modulo 998 244 353.

Examples

Input


1 1
2


Output


4


Input


1 1
1


Output


0


Input


1 5
1234


Output


15630811


Input


2 1
2 3


Output


476014684


Input


10 33
10 20 30 40 50 60 70 80 90 100


Output


675105648

Solution (in CPP):

Solution:
#include<cstdio>
#include<cstring>
#include<algorithm>
using std::swap;
const int mod=998244353;
inline int add(int a,int b)
{
	return (a+=b)>=mod?a-mod:a;
}
inline int sub(int a,int b)
{
	return (a-=b)<0?a+mod:a;
}
inline int mul(int a,int b)
{
	return (long long)a*b%mod;
}
inline int qpow(int a,int b)
{
	int res=1;
	for(;b;a=mul(a,a),b>>=1)
		if(b&1)
			res=mul(res,a);
	return res;
}
const int N=55,M=2005;
int fact[M],ifact[M],inv[M];
inline void init(int n)
{
	register int i;
	fact[0]=1;
	for(i=1;i<=n;i++)
		fact[i]=mul(fact[i-1],i);
	ifact[n]=qpow(fact[n],mod-2);
	for(i=n-1;i>=0;i--)
		ifact[i]=mul(ifact[i+1],i+1);
	for(i=1;i<=n;i++)
		inv[i]=mul(ifact[i],fact[i-1]);
	return;
}
int rev[M<<2];
int *F[M<<2],*G[M<<2];
inline void ntt(int **f,int n,int m,int p)
{
	int w,wi,u,t;
	register int v,i,j,k;
	for(v=0;v<n;v++)
	{
		for(i=0;i<m;i++)
			if(i<(rev[i]=i&1?rev[i^1]|m>>1:rev[i>>1]>>1))
				swap(f[v][i],f[v][rev[i]]);
		for(i=1;wi=qpow(p^1?inv[3]:3,(mod-1)/(i<<1)),i<<1<=m;i<<=1)
			for(j=0;w=1,j<m;j+=i<<1)
				for(k=0;k<i;w=mul(w,wi),k++)
					u=f[v][j+k],t=mul(w,f[v][j+k+i]),f[v][j+k]=add(u,t),f[v][j+k+i]=sub(u,t);
		if(!~p)
			for(w=qpow(m,mod-2),i=0;i<m;i++)
				f[v][i]=mul(w,f[v][i]);
	}
	for(v=0;v<m;v++)
	{
		for(i=0;i<n;i++)
			if(i<(rev[i]=i&1?rev[i^1]|n>>1:rev[i>>1]>>1))
				swap(f[i][v],f[rev[i]][v]);
		for(i=1;wi=qpow(p^1?inv[3]:3,(mod-1)/(i<<1)),i<<1<=n;i<<=1)
			for(j=0;w=1,j<n;j+=i<<1)
				for(k=0;k<i;w=mul(w,wi),k++)
					u=f[j+k][v],t=mul(w,f[j+k+i][v]),f[j+k][v]=add(u,t),f[j+k+i][v]=sub(u,t);
		if(!~p)
			for(w=qpow(n,mod-2),i=0;i<n;i++)
				f[i][v]=mul(w,f[i][v]);
	}
	return;
}
inline void poly_mul(int **f,int nf,int mf,int **g,int ng,int mg,int **h)
{
	int n,m;
	register int i,j;
	for(n=1;n<nf||n<ng;n<<=1);
	for(m=1;m<mf||m<mg;m<<=1);
	for(i=0;i<n<<1;i++)
		memset(F[i],0,sizeof(int)*(m<<1)),memset(G[i],0,sizeof(int)*(m<<1));
	for(i=0;i<nf;i++)
		memcpy(F[i],f[i],sizeof(int)*mf);
	for(i=0;i<ng;i++)
		memcpy(G[i],g[i],sizeof(int)*mg);
	ntt(F,n<<1,m<<1,1);ntt(G,n<<1,m<<1,1);
	for(i=0;i<n<<1;i++)
		for(j=0;j<m<<1;j++)
			F[i][j]=mul(F[i][j],G[i][j]);
	ntt(F,n<<1,m<<1,-1);
	for(i=0;i<nf+ng-1;i++)
		memcpy(h[i],F[i],sizeof(int)*(mf+mg-1));
	return;
}
int n,K,L,omega,ans;
int l[N];
int **pool[N<<2],nlen[N<<2],mlen[N<<2];
int a[M][N];
void run(int x,int lp,int rp)
{
	register int i,j;
	if(lp==rp)
	{
		int w=mul(l[lp],qpow(L,mod-2)),wi;
		nlen[x]=l[lp]/K+1;mlen[x]=2;
		pool[x]=new int *[l[lp]/K+1];
		for(i=0;i<=l[lp]/K;i++)
			pool[x][i]=new int [2],pool[x][i][0]=pool[x][i][1]=0;
		pool[x][0][0]=1;
		for(i=1;i<=l[lp]/K;i++)
		{
			wi=sub(w,mul(i,omega));
			if(i&1)
				pool[x][i][0]=sub(0,mul(qpow(wi,i),ifact[i])),pool[x][i][1]=sub(0,mul(wi?qpow(wi,i-1):0,ifact[i-1]));
			else
				pool[x][i][0]=mul(qpow(wi,i),ifact[i]),pool[x][i][1]=mul(wi?qpow(wi,i-1):0,ifact[i-1]);
		}
		return;
	}
	int mid=(lp+rp)>>1;
	run(x<<1,lp,mid);run(x<<1|1,mid+1,rp);
	nlen[x]=nlen[x<<1]+nlen[x<<1|1]-1;mlen[x]=mlen[x<<1]+mlen[x<<1|1]-1;
	pool[x]=new int *[nlen[x]];
	for(i=0;i<nlen[x];i++)
		pool[x][i]=new int [mlen[x]];
	poly_mul(pool[x<<1],nlen[x<<1],mlen[x<<1],pool[x<<1|1],nlen[x<<1|1],mlen[x<<1|1],pool[x]);
	return;
}
inline int sum(int k,int c)
{
	return mul(fact[k],qpow(qpow(sub(1,c),mod-2),k+1));
}
signed main()
{
	int w;
	register int i,j;
	for(i=0;i<M<<2;i++)
		F[i]=new int[N<<2],G[i]=new int[N<<2];
	scanf("%d%d",&n,&K);
	for(i=1;i<=n;i++)
		scanf("%d",&l[i]),L+=l[i];
	init(L);
	omega=mul(K,qpow(L,mod-2));
	run(1,1,n);
	for(i=0;i<nlen[1];i++)
		for(j=0;j<mlen[1];j++)
			a[i][j]=sub(0,pool[1][i][j]);
	a[0][0]=add(a[0][0],1);
	for(i=0;i<nlen[1];i++)
		for(j=0;j<mlen[1];j++)
			if(a[i][j])
				ans=add(ans,mul(a[i][j],sum(i-j,sub(1,mul(i,omega)))));
	printf("%d\n",ans);
	return 0;
}