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.*;
import java.util.*;

public class Main {

	int compare(State was, State now) {
		long diff = was.a[0] - now.a[0];
		for (int i = 0; i < was.a.length; i++) {
			if (was.a[i] - now.a[i] != diff) {
				return 0;
			}
		}

		if (was.a[0] >= now.a[0]
				&& was.a[0] + was.spent <= now.a[0] + now.spent) {
			return 1;
		}
		return 0;
	}

	boolean test(int mask, int i) {
		return ((mask >> i) & 1) == 1;
	}

	void submit() {
		int n = nextInt();
		long k = nextLong();
		long[] a = new long[n];
		for (int i = 0; i < n; i++) {
			a[i] = nextLong();
		}
		Arrays.sort(a);
		out.println(go(a, k));
	}

	void preCalc() {

	}

	void stress() {

	}

	boolean kindaSame(State was, State now) {
		long diff = was.a[0] - now.a[0];
		for (int i = 0; i < was.a.length; i++) {
			if (was.a[i] - now.a[i] != diff) {
				return false;
			}
		}
		return true;
	}

	static class State {
		long[] a;
		int spent;

		public State(long[] a, int spent) {
			this.a = a;
			this.spent = spent;
		}

		@Override
		public String toString() {
			return "(" + Arrays.toString(a) + ", " + spent + ")";
		}

		State div() {
			long[] newA = a.clone();
			for (int i = 0; i < newA.length; i++) {
				newA[i] /= 2;
			}
			return new State(newA, spent + 1);
		}

		State divSub() {
			if (a[0] == 0) {
				return null;
			}
			long[] newA = a.clone();
			for (int i = 0; i < newA.length; i++) {
				newA[i] = (a[i] - 1) / 2;
			}
			return new State(newA, spent + 2);
		}
	}

	static final int P = 1_000_000_007;

	static class Arr {
		long[] diff;

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

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

		@Override
		public boolean equals(Object obj) {
			if (this == obj)
				return true;
			if (obj == null)
				return false;
			if (getClass() != obj.getClass())
				return false;
			Arr other = (Arr) obj;
			if (!Arrays.equals(diff, other.diff))
				return false;
			return true;
		}

	}

	int go(long[] a, long k) {
		// a is sorted
		ArrayDeque<State> q = new ArrayDeque<>();
		State init = new State(a, 0);
		q.add(init);
		ArrayList<State> seen = new ArrayList<>();

		HashMap<Arr, ArrayList<Long>> map = new HashMap<>();

		while (!q.isEmpty()) {
			State now = q.poll();
			boolean bad = false;
			for (State was : seen) {
				if (compare(was, now) == 1) {
					bad = true;
					break;
				}
			}
			if (bad) {
				continue;
			}
			seen.add(now);

			Arr arr = new Arr(now.a);
			ArrayList<Long> lst = map.get(arr);
			if (lst == null) {
				lst = new ArrayList<>();
				map.put(arr, lst);
			}
			lst.add(now.a[0]);
			lst.add((long) now.spent);

			State tmp = now.div();
			if (tmp != null) {
				q.add(tmp);
			}
			tmp = now.divSub();
			if (tmp != null) {
				q.add(tmp);
			}
		}

		int ans = 0;
		for (ArrayList<Long> lst : map.values()) {
			ans += solveGroup(lst, k);
			if (ans >= P) {
				ans -= P;
			}
		}
		return ans;
	}

	static class Event implements Comparable<Event> {
		long x;
		int delta;

		public Event(long x, int delta) {
			this.x = x;
			this.delta = delta;
		}

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

	int solveGroup(ArrayList<Long> lst, long k) {
		Event[] a = new Event[lst.size()];
		int ptr = 0;
		for (int i = 0; i < lst.size(); i += 2) {
			long init = lst.get(i);
			long used = lst.get(i + 1);
			if (used > k) {
				continue;
			}
			long low = Math.max(init - (k - used), 0);
			long high = init;
			a[ptr++] = new Event(low, 1);
			a[ptr++] = new Event(high + 1, -1);
		}
		Arrays.sort(a, 0, ptr);
		int sum = 0;
		int ret = 0;
		for (int i = 0; i < ptr - 1; i++) {
			sum += a[i].delta;
			if (sum > 0) {
				ret += (int) ((a[i + 1].x - a[i].x) % P);
				if (ret >= P) {
					ret -= P;
				}
			}
		}
		return ret;
	}

	void test() {
		// go(new long[] { 10, 13, 22 });
		// go(new long[]{228344079825412349L, 478465001534875275L,
		// 398048921164061989L, 329102208281783917L});
		// go(new long[]{3, 4});
		int C = 100;
		for (int tst = 0;; tst++) {
			int n = rand(1, C);
			long[] a = new long[n];
			for (int i = 0; i < n; i++) {
				a[i] = rand(1, C);
			}
			go(a, 0);
			System.err.println(tst);
		}
	}

	Main() throws IOException {
		br = new BufferedReader(new InputStreamReader(System.in));
		out = new PrintWriter(System.out);
		preCalc();
		submit();
		// stress();
//		test();
		out.close();
	}

	static final Random rng = new Random();

	static int rand(int l, int r) {
		return l + rng.nextInt(r - l + 1);
	}

	public static void main(String[] args) throws IOException {
		new Main();
	}

	BufferedReader br;
	PrintWriter out;
	StringTokenizer st;

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

	String nextString() {
		try {
			return br.readLine();
		} catch (IOException e) {
			throw new RuntimeException(e);
		}
	}

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

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

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