Problem: There are N non-negative integers written on the blackboard: A_1, ..., A_N.

Snuke can perform the following two operations at most K times in total in any order:

* Operation A: Replace each integer X on the blackboard with X divided by 2, rounded down to the nearest integer.
* Operation B: Replace each integer X on the blackboard with X minus 1. This operation cannot be performed if one or more 0s are written on the blackboard.



Find the number of the different possible combinations of integers written on the blackboard after Snuke performs the operations, modulo 1,000,000,007.

Constraints

* 1 \leq N \leq 200
* 1 \leq A_i \leq 10^{18}
* 1 \leq K \leq 10^{18}
* A_i and K are integers.

Input

Input is given from Standard Input in the following format:


N K
A_1 A_2 ... A_N


Output

Print the number of the different possible combinations of integers written on the blackboard after Snuke performs the operations, modulo 1,000,000,007.

Examples

Input

2 2
5 7


Output

6


Input

3 4
10 13 22


Output

20


Input

1 100
10


Output

11


Input

10 123456789012345678
228344079825412349 478465001534875275 398048921164061989 329102208281783917 779698519704384319 617456682030809556 561259383338846380 254083246422083141 458181156833851984 502254767369499613


Output

164286011

Solution (in CPP):

Solution:
// #pragma GCC target("avx")  // CPU 処理並列化
// #pragma GCC optimize("O3")  // CPU 処理並列化
// #pragma GCC optimize("unroll-loops")  // 条件処理の呼び出しを減らす
#include<stdio.h>
#include<math.h>
#include<algorithm>
#include<queue>
#include<deque>
#include<stack>
#include<string>
#include<string.h>
#include<vector>
#include<set>
#include<map>
#include<bitset>
#include<stdlib.h>
#include<cassert>
#include<time.h>
#include<bitset>
#include<numeric>
#include<unordered_set>
#include<complex>
using namespace std;
const long long mod=1000000007;
const long long inf=mod*mod;
const long long d2=(mod+1)/2;
const double EPS=1e-10;
const double INF=1e+10;
const double PI=acos(-1.0);
const int C_SIZE = 3100000;
const int UF_SIZE = 210000;
namespace{
	long long fact[C_SIZE];
	long long finv[C_SIZE];
	long long inv[C_SIZE];
	long long Comb(int a,int b){
	 	if(a<b||b<0)return 0;
	 	return fact[a]*finv[b]%mod*finv[a-b]%mod;
	}
	void init_C(int n){
		fact[0]=finv[0]=inv[1]=1;
		for(int i=2;i<n;i++){
			inv[i]=(mod-(mod/i)*inv[mod%i]%mod)%mod;
		}
		for(int i=1;i<n;i++){
			fact[i]=fact[i-1]*i%mod;
			finv[i]=finv[i-1]*inv[i]%mod;
		}
	}
	long long pw(long long a,long long b){
		if(a<0LL)return 0;
		if(b<0LL)return 0;
		long long ret=1;
		while(b){
			if(b%2)ret=ret*a%mod;
			a=a*a%mod;
			b/=2;
		}
		return ret;
	}
	long long pw_mod(long long a,long long b,long long M){
		if(a<0LL)return 0;
		if(b<0LL)return 0;
		long long ret=1;
		while(b){
			if(b%2)ret=ret*a%M;
			a=a*a%M;
			b/=2;
		}
		return ret;
	}
	int pw_mod_int(int a,int b,int M){
		if(a<0)return 0;
		if(b<0)return 0;
		int ret=1;
		while(b){
			if(b%2)ret=(long long)ret*a%M;
			a=(long long)a*a%M;
			b/=2;
		}
		return ret;
	}
	int ABS(int a){return max(a,-a);}
	long long ABS(long long a){return max(a,-a);}
	double ABS(double a){return max(a,-a);}
	int sig(double r) { return (r < -EPS) ? -1 : (r > +EPS) ? +1 : 0; }
	int UF[UF_SIZE];
	void init_UF(int n){
		for(int i=0;i<n;i++)UF[i]=-1;
	}
	int FIND(int a){
		if(UF[a]<0)return a;
		return UF[a]=FIND(UF[a]);
	}
	void UNION(int a,int b){
		a=FIND(a);b=FIND(b);if(a==b)return;
		if(UF[a]>UF[b])swap(a,b);
		UF[a]+=UF[b];UF[b]=a;
	}
}
// ここから編集しろ
map<vector<long long>,int>M;
vector<pair<long long,int> >v[211000];
long long p[210];
long long min_bc(long long L,long long R){
	long long ret=0;
	for(int i=60;i>=0;i--){
		if(L<=ret&&ret<=R){
			break;
		}
		if(ret+(1LL<<i)<=R){
			ret+=(1LL<<i);
		}
	}
	return ret;
}
long long tmp[210];
int main(){
	int a;scanf("%d",&a);
	long long b;scanf("%lld",&b);
	for(int i=0;i<a;i++){
		scanf("%lld",p+i);
	}
	int sz=0;
	std::sort(p,p+a);
	for(int i=0;i<=60;i++){
		long long S=1LL<<i;
		for(int j=0;j<a;j++){
			tmp[j]=(p[j])%(1LL<<i)+1;
		}
		tmp[a]=(1LL<<i);
		tmp[a+1]=0;
		
		std::sort(tmp,tmp+a+2);
		for(int j=0;j<=a;j++){
			long long L=tmp[j];
			long long R=tmp[j+1]-1;
			if(L>R)continue;
			long long fi=min_bc(L,R);
			vector<long long>t(a);
			bool dame=false;
			for(int k=0;k<a;k++){
				if(L>p[k]){
					dame=true;break;
				}
				t[k]=(p[k]-L)/(1LL<<i);
			}
			if(dame)continue;
			long long cur=__builtin_popcountll(fi)+i;
			// printf("%d %d: %lld\n",i,j,cur);
			long long s=t[0];
			for(int k=0;k<a;k++){
				t[k]-=s;
			}
			long long rem=b-cur;
			if(rem<0)continue;
			int ind=0;
			if(!M.count(t)){
				// for(int k=0;k<a;k++){
				// 	printf("%lld ",t[k]);
				// }printf("\n");
				M[t]=ind=sz++;
			}else ind=M[t];
			v[ind].push_back(make_pair(max(0LL,s-rem),1));
			v[ind].push_back(make_pair(s+1,-1));
		}
	}
	long long ret=0;
	for(int i=0;i<sz;i++){
		std::sort(v[i].begin(),v[i].end());
		int now=0;
		for(int j=0;j<v[i].size();j++){
			if(v[i][j].second==1){
				now++;
			}else now--;
			if(now){
				// printf("%d: %lld %lld\n",i,v[i][j].first,v[i][j+1].first);
				ret=(ret+v[i][j+1].first%mod+mod-v[i][j].first%mod)%mod;
			}
		}
		ret%=mod;
	}
	printf("%lld\n",ret);
}