200863: [AtCoder]ARC086 F - Shift and Decrement

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 $0$s 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