Score : 800 points

Problem Statement

Given are N pairwise distinct non-negative integers A_1,A_2,\ldots,A_N. Find the number of ways to choose a set of between 1 and K numbers (inclusive) from the given numbers so that the following two conditions are satisfied:

  • The bitwise AND of the chosen numbers is S.
  • The bitwise OR of the chosen numbers is T.

Constraints

  • 1 \leq N \leq 50
  • 1 \leq K \leq N
  • 0 \leq A_i < 2^{18}
  • 0 \leq S < 2^{18}
  • 0 \leq T < 2^{18}
  • A_i \neq A_j (1 \leq i < j \leq N)

Input

Input is given from Standard Input in the following format:

N K S T
A_1 A_2 ... A_N

Output

Print the answer.

Sample Input 1

3 3 0 3
1 2 3

Sample Output 1

2

The conditions are satisfied when we choose \{1,2\} or \{1,2,3\}.

Sample Input 2

5 3 1 7
3 4 9 1 5

Sample Output 2

2

Sample Input 3

5 4 0 15
3 4 9 1 5

Sample Output 3

3