Problem:
Score : 1200 points

Problem Statement

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.

Sample Input 1

2 2
5 7

Sample Output 1

6

There are six possible combinations of integers on the blackboard: (1, 1), (1, 2), (2, 3), (3, 5), (4, 6) and (5, 7).
For example, (1, 2) can be obtained by performing Operation A and Operation B in this order.

Sample Input 2

3 4
10 13 22

Sample Output 2

20

Sample Input 3

1 100
10

Sample Output 3

11

Sample Input 4

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

Sample Output 4

164286011

Solution in Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)

import java.io.*;
import java.util.*;

public class Main {
    private FastScanner in;
    private PrintWriter out;

    long smallestNumberOfBits(long from, long to) {
        long res = 0;
        for (int i = 61; i >= 0; i--) {
            long x = 1L << i;
            if ((to & x) == 0) {
                continue;
            }
            if ((from & x) != (to & x)) {
                if (from != res)
                    res |= x;
                break;
            }
            res |= x;
        }
        return res;
    }

    final int mod = (int) 1e9 + 7;

    private void solve() {
        int n = in.nextInt();
        long k = in.nextLong();
        long[] a = new long[n];
        for (int i = 0; i < n; i++) {
            a[i] = in.nextLong();
        }
        HashMap<Diffs, ArrayList<Segment>> answers = new HashMap<>();
        Arrays.sort(a);
        for (int shift = 0; shift <= 60; shift++) {
            long mod = 1L << shift;
            ArrayList<Long> pos = new ArrayList<>();
            pos.add(0L);
            pos.add(mod);
            for (long value : a) {
                long cur = value & (mod - 1);
                pos.add(cur + 1);
            }
            Collections.sort(pos);
            for (int i = 0; i + 1 < pos.size(); i++) {
                long from = pos.get(i), to = pos.get(i + 1);
                if (from == to) {
                    continue;
                }
                if (from > a[0]) {
                    continue;
                }
                long sub = smallestNumberOfBits(from, to - 1);
                int cost = Long.bitCount(sub) + shift;
                if (cost > k) {
                    continue;
                }
                long[] nvals = new long[n];
                for (int j = 0; j < n; j++) {
                    nvals[j] = (a[j] - sub) >> shift;
                }
                Diffs d = new Diffs(nvals);
                ArrayList<Segment> ar = answers.get(d);
                if (ar == null) {
                    ar = new ArrayList<>();
                }
                long fr = Math.max(0, nvals[0] - (k - cost));
                ar.add(new Segment(fr, nvals[0]));
                answers.put(d, ar);
            }
        }
        long ans = 0;
        for (Map.Entry<Diffs, ArrayList<Segment>> entry : answers.entrySet()) {
            ArrayList<Segment> segs = entry.getValue();
            Collections.sort(segs);
            long to = -1;
            for (Segment s : segs) {
                if (s.to <= to) {
                    continue;
                }
                ans += s.to - Math.max(to + 1, s.from) + 1;
                to = s.to;
            }
        }
        out.println(ans % mod);
    }

    class Segment implements Comparable<Segment> {
        long from, to;

        public Segment(long from, long to) {
            this.from = from;
            this.to = to;
        }

        @Override
        public int compareTo(Segment o) {
            return Long.compare(from, o.from);
        }
    }

    class Diffs {
        long[] d;

        Diffs(long[] vals) {
            d = new long[vals.length - 1];
            for (int i = 0; i + 1 < vals.length; i++) {
                d[i] = vals[i + 1] - vals[i];
            }
        }

        @Override
        public boolean equals(Object o) {
            if (this == o) return true;
            if (o == null || getClass() != o.getClass()) return false;

            Diffs diffs = (Diffs) o;

            return Arrays.equals(d, diffs.d);
        }

        @Override
        public int hashCode() {
            return Arrays.hashCode(d);
        }
    }

    private void run() {
        try {
            in = new FastScanner(new File("ARC_086_D.in"));
            out = new PrintWriter(new File("ARC_086_D.out"));

            solve();

            out.close();
        } catch (FileNotFoundException e) {
            e.printStackTrace();
        }
    }

    private void runIO() {
        in = new FastScanner(System.in);
        out = new PrintWriter(System.out);

        solve();

        out.close();
    }

    private class FastScanner {
        BufferedReader br;
        StringTokenizer st;

        FastScanner(File f) {
            try {
                br = new BufferedReader(new FileReader(f));
            } catch (FileNotFoundException e) {
                e.printStackTrace();
            }
        }

        FastScanner(InputStream f) {
            br = new BufferedReader(new InputStreamReader(f));
        }

        String next() {
            while (st == null || !st.hasMoreTokens()) {
                String s = null;
                try {
                    s = br.readLine();
                } catch (IOException e) {
                    e.printStackTrace();
                }
                if (s == null)
                    return null;
                st = new StringTokenizer(s);
            }
            return st.nextToken();
        }

        boolean hasMoreTokens() {
            while (st == null || !st.hasMoreTokens()) {
                String s = null;
                try {
                    s = br.readLine();
                } catch (IOException e) {
                    e.printStackTrace();
                }
                if (s == null)
                    return false;
                st = new StringTokenizer(s);
            }
            return true;
        }

        int nextInt() {
            return Integer.parseInt(next());
        }

        long nextLong() {
            return Long.parseLong(next());
        }

        double nextDouble() {
            return Double.parseDouble(next());
        }
    }

    public static void main(String[] args) {
        new Main().runIO();
    }
}