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:
//Create Time: 2018-05-05 16:45:20
#include <iostream>
#include <cstdio>
#include <cstring>
#include <string>
#include <cmath>
#include <algorithm>
#include <vector>
#include <cstdlib>
#include <sstream>
#include <fstream>
#include <functional>
#include <list>
#include <deque>
#include <queue>
#include <stack>
#include <map>
#include <set>
#include <bitset>
#include <cctype>
#include <ctime>
#include <utility>
#include <complex>
#include <cassert>
#include <climits>
#include <iterator>
#include <tuple>
using namespace std;
#define snuke(i,x) for (__typeof((x).begin()) i = (x).begin(); i != (x).end(); ++i)
#define rep(i,a,n) for (int i=(a);i<(n);i++)
#define per(i,n,a) for (int i=(n)-1;i>=(a);i--)
// #define OUT(x) std::cout<<(#x)<<":"<<(x)<<std::endl
// #define TST(x) std::cout<<(#x)<<":"<<(x)<<" "
#define OUT(args...) {string _s=#args;replace(_s.begin(),_s.end(),',',' ');stringstream _ss(_s);istream_iterator<string> _it(_ss);OUT_TST("\n",_it,args);}
#define TST(args...) {string _s=#args;replace(_s.begin(),_s.end(),',',' ');stringstream _ss(_s);istream_iterator<string> _it(_ss);OUT_TST("",_it,args);}
#define SZ(v) ((int)(v).size())
#define all(v) (v).begin(), (v).end()
#define unq(x) sort(all(x)),(x).erase(unique(all(x)),(x).end())
#define mem(f,x) memset((f), (x), sizeof(f))
#define clr(f) mem(f,0)
#define x first
#define y second
#define mp make_pair
#define pb push_back
#define mt make_tuple
#define eb emplace_back
#define DEBUG
void OUT_TST(string s,istream_iterator<string> it) {cerr<<s;}
template<typename T, typename... Args> void OUT_TST(string s,istream_iterator<string> it,T a,Args... args){cerr<<*it<<"="<<a<<" ";OUT_TST(s,++it,args...);}
template<class T> void rd(T&ret){ret=0;bool ok=0,u=0;for(;;){int c=getchar();if(c>='0'&&c<='9')ret=(ret<<3)+(ret<<1)+c-'0',ok=1;else if(c=='-')u=1;else if(ok){if(u)ret*=-1;return;}}}
long long powmod(long long p,long long n,long long mod){long long ret=1;for(;n;n>>=1){if(n&1)ret=ret*p%mod;p=p*p%mod;}return ret;}
template <class T> bool chmin(T& a, const T &b) {return b < a? a = b, 1: 0;}
template <class T> bool chmax(T& a, const T &b) {return b > a? a = b, 1: 0;}
int gcd(int a,int b){return b?gcd(b,a%b):a;}
typedef long long ll;
typedef pair<int, int> pii;
typedef vector<int> vi;
typedef vector<pii > vii;
typedef tuple<int,int,int> tp;
/****head****/
typedef pair<ll,ll> pll;
const ll mod=(ll)1e9+7;
ll an[222],cn[222],stk[222];
int n,top;
ll K;
map<vector<ll>,vector<pll> > has;
bool ok(ll x,int t,ll op){
    rep(i,0,t){
        if(op&1){
            if(x<1)return 0;
            x-=1;
        }
        x>>=1;
        op>>=1;
    }
    return 1;
}
int count(ll a,ll b){
    if(a==-1)return 0;
    while(a!=b){
        a>>=1;
        b>>=1;
    }
    return __builtin_popcount(a)+1;
}
int main(){
    cin>>n>>K;
    rep(i,0,n)cin>>an[i];
    sort(an,an+n);
    vector<ll> bn(n,0);
    rep(t,0,K+1){
        ll p2=1ll<<t;
        if(p2/2>an[n-1]) break;
        top=0;
        rep(i,0,n){
            // bn[i]=an[i]/p2;
            // cn[i]=an[i]%p2;
            stk[top++]=an[i]%p2;
        }
        stk[top++]=-1;stk[top++]=p2-1;
        sort(stk,stk+top);
        top=unique(stk,stk+top)-stk;
        rep(i,0,top-1){
            ll yu=stk[i]+1;
            if(!ok(an[0],t,yu))continue;
            ll left=K-t-count(stk[i],stk[i+1]);
            if(left<0)continue;
            rep(j,0,n)bn[j]=(an[j]-yu)/p2;
            rep(j,1,n)bn[j]-=bn[0];
            ll r=bn[0],l=r-left;chmax(l,0ll);
            bn[0]=0;
            auto it=has.find(bn);
            if(it==has.end()) {
                has.emplace(bn,vector<pll>());
                it=has.find(bn);
            }
            it->y.emplace_back(-r,l);
        }
    }
    ll ans=0;
    for(auto i:has){
        // puts("---------------------------------------------");
        // for(auto j:i.x)printf("%lld ",j);puts("");
        ll last=(ll)2e18;
        auto &p=i.y;
        sort(all(p));
        rep(j,0,SZ(p)){
            if(j && p[j-1].x==p[j].x)continue;
            ll r=-p[j].x,l=p[j].y;
            chmin(r,last-1);
            if(l<=r)ans+=(r-l+1)%mod;
            chmin(last,l);
            // OUT(-p[j].x,p[j].y,ans);
        }
    }
    ans%=mod;
    cout<<ans<<endl;
    return 0;
}