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 JAVA):

Solution:
import java.io.ByteArrayInputStream;
import java.io.IOException;
import java.io.InputStream;
import java.io.PrintWriter;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.Collections;
import java.util.Comparator;
import java.util.InputMismatchException;
import java.util.List;

public class Main {
	static InputStream is;
	static PrintWriter out;
	static String INPUT = "";
	
	static void solve()
	{
		int n = ni();
		long K = nl();
		long[] a = new long[n];
		for(int i = 0;i  < n;i++)a[i] = nl();
		Arrays.sort(a);
		
		
		List<Datum> all = new ArrayList<>();
		for(int d = 0;K-d >= 0 && d <= 60;d++){
			long[][] lows = new long[n][];
			for(int i = 0;i < n;i++){
				lows[i] = new long[]{a[i]&(1L<<d)-1, i};
			}
			Arrays.sort(lows, new Comparator<long[]>() {
				public int compare(long[] a, long[] b) {
					return Long.compare(a[0], b[0]);
				}
			});
			
			long[] b = new long[n];
			for(int i = 0;i < n;i++){
				b[i] = a[i]>>>d;
			}
			if(K-d >= 0){
				long sup = Math.min(b[0], K-d);
				all.add(new Datum(sub(b), b[0]-sup, b[0]));
			}
			
			for(int i = 0;i < n;){
				int j = i;
				while(j < n && lows[j][0] == lows[i][0])j++;
				
				for(int k = i;k < j;k++){
					b[(int)lows[k][1]]--;
				}
				if(b[0] >= 0 && j < n){
					long llow = lows[i][0]+1;
					long lhigh = lows[j][0];
					int num = Long.bitCount(lhigh);
					for(;lhigh >= llow;lhigh &= lhigh-1){
						num--;
					}
					num++;
//					int num = 
//							llow == lhigh ? Long.bitCount(llow) : 
//							Math.min(Long.bitCount(llow), Long.bitCount(llow / Long.highestOneBit(llow^lhigh)) + 1);
					
					if(K-d-num >= 0){
//						tr(d, sub(b), llow, lhigh);
						long sup = Math.min(b[0], K-d-num);
						all.add(new Datum(sub(b), b[0]-sup, b[0]));
					}
				}
				
				i = j;
			}
		}
		
		Collections.sort(all, (x, y) -> {
			for(int i = 0;i < x.sub.length;i++){
				if(x.sub[i] != y.sub[i])return Long.compare(x.sub[i], y.sub[i]);
			}
			return Long.compare(x.inf, y.inf);
		});
		long len = 0;
		for(int i = 0;i < all.size();){
			int j = i;
			while(j < all.size() && Arrays.equals(all.get(j).sub, all.get(i).sub))j++;
			
			long sup = -3;
			for(int k = i;k < j;k++){
//				tr(all.get(k));
				if(sup < all.get(k).inf){
					sup = all.get(k).sup;
					len += sup-all.get(k).inf+1;
				}else{
					if(all.get(k).sup > sup){
						len += all.get(k).sup - sup;
						sup = all.get(k).sup;
					}
				}
			}
			len %= 1000000007;
			
			i = j;
		}
		out.println(len%1000000007);
	}
	
	static long[] sub(long[] u)
	{
		int n = u.length;
		long[] v = new long[n-1];
		for(int i = 0;i < n-1;i++){
			v[i] = u[i+1] - u[i];
		}
		return v;
	}
	
	static class Datum
	{
		long[] sub;
		long inf, sup;
		public Datum(long[] sub, long inf, long sup) {
			this.sub = sub;
			this.inf = inf;
			this.sup = sup;
		}
		@Override
		public String toString() {
			return "Datum [" + (sub != null ? "sub=" + Arrays.toString(sub) + ", " : "") + "inf=" + inf + ", sup=" + sup
					+ "]";
		}
		
	}
	
	public static void main(String[] args) throws Exception
	{
		long S = System.currentTimeMillis();
		is = INPUT.isEmpty() ? System.in : new ByteArrayInputStream(INPUT.getBytes());
		out = new PrintWriter(System.out);
		
		solve();
		out.flush();
		long G = System.currentTimeMillis();
		tr(G-S+"ms");
	}
	
	private static boolean eof()
	{
		if(lenbuf == -1)return true;
		int lptr = ptrbuf;
		while(lptr < lenbuf)if(!isSpaceChar(inbuf[lptr++]))return false;
		
		try {
			is.mark(1000);
			while(true){
				int b = is.read();
				if(b == -1){
					is.reset();
					return true;
				}else if(!isSpaceChar(b)){
					is.reset();
					return false;
				}
			}
		} catch (IOException e) {
			return true;
		}
	}
	
	private static byte[] inbuf = new byte[1024];
	static int lenbuf = 0, ptrbuf = 0;
	
	private static int readByte()
	{
		if(lenbuf == -1)throw new InputMismatchException();
		if(ptrbuf >= lenbuf){
			ptrbuf = 0;
			try { lenbuf = is.read(inbuf); } catch (IOException e) { throw new InputMismatchException(); }
			if(lenbuf <= 0)return -1;
		}
		return inbuf[ptrbuf++];
	}
	
	private static boolean isSpaceChar(int c) { return !(c >= 33 && c <= 126); }
//	private static boolean isSpaceChar(int c) { return !(c >= 32 && c <= 126); }
	private static int skip() { int b; while((b = readByte()) != -1 && isSpaceChar(b)); return b; }
	
	private static double nd() { return Double.parseDouble(ns()); }
	private static char nc() { return (char)skip(); }
	
	private static String ns()
	{
		int b = skip();
		StringBuilder sb = new StringBuilder();
		while(!(isSpaceChar(b))){
			sb.appendCodePoint(b);
			b = readByte();
		}
		return sb.toString();
	}
	
	private static char[] ns(int n)
	{
		char[] buf = new char[n];
		int b = skip(), p = 0;
		while(p < n && !(isSpaceChar(b))){
			buf[p++] = (char)b;
			b = readByte();
		}
		return n == p ? buf : Arrays.copyOf(buf, p);
	}
	
	private static char[][] nm(int n, int m)
	{
		char[][] map = new char[n][];
		for(int i = 0;i < n;i++)map[i] = ns(m);
		return map;
	}
	
	private static int[] na(int n)
	{
		int[] a = new int[n];
		for(int i = 0;i < n;i++)a[i] = ni();
		return a;
	}
	
	private static int ni()
	{
		int num = 0, b;
		boolean minus = false;
		while((b = readByte()) != -1 && !((b >= '0' && b <= '9') || b == '-'));
		if(b == '-'){
			minus = true;
			b = readByte();
		}
		
		while(true){
			if(b >= '0' && b <= '9'){
				num = num * 10 + (b - '0');
			}else{
				return minus ? -num : num;
			}
			b = readByte();
		}
	}
	
	private static long nl()
	{
		long num = 0;
		int b;
		boolean minus = false;
		while((b = readByte()) != -1 && !((b >= '0' && b <= '9') || b == '-'));
		if(b == '-'){
			minus = true;
			b = readByte();
		}
		
		while(true){
			if(b >= '0' && b <= '9'){
				num = num * 10 + (b - '0');
			}else{
				return minus ? -num : num;
			}
			b = readByte();
		}
	}
	
	private static void tr(Object... o) { if(INPUT.length() != 0)System.out.println(Arrays.deepToString(o)); }
}