Problem: 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.

Examples

Input

3 3 0 3
1 2 3


Output

2


Input

5 3 1 7
3 4 9 1 5


Output

2


Input

5 4 0 15
3 4 9 1 5


Output

3

Solution (in CPP):

Solution:
#include <bits/stdc++.h>
using namespace std;

int A[55];
bool boleh[55];
int B[55];
int C[55];
int indx[55];
long long ncr[55][55];

long long ncrUpTo[55][55];

int main(){
    int N, K, S, T;
    scanf("%d%d%d%d", &N, &K, &S, &T);

    if( ((S&T) == S) && ((S|T) == T) ){
    }else{
        printf("0");
        return 0;
    }

    memset(ncr, 0, sizeof(ncr));
    ncr[0][0] = 1;
    for(int i = 1; i <= N; i ++){
        ncr[i][0] = 1;
        for(int j = 1; j <= i; j ++){
            ncr[i][j] = ncr[i-1][j] + ncr[i-1][j-1];
        }
    }

    ncrUpTo[0][0] = 0;
    for(int i = 1; i <= N; i ++){
        ncrUpTo[i][0] = 0;
        for(int j = 1; j <= N; j ++){
            ncrUpTo[i][j] = ncrUpTo[i][j-1] + ncr[i][j];
        }
    }

    for(int i = 0; i < N; i ++){
        scanf("%d", &A[i]);
        boleh[i] = true;
        //printf("A[%d]=%d\n", i, A[i]);
    }

    int L = 0;
    for(int i = 0; i < 20; i ++){
        if( (S&(1<<i)) == (T&(1<<i)) ){
            for(int j = 0; j < N; j ++){
                if( (S&(1<<i)) != (A[j]&(1<<i)) ){
                    boleh[j] = false;
                    //printf("boleh[%d]=false\n", j);
                }
            }
        }else{
            indx[L ++] = i;
        }
    }

    for(int i = 0; i < L; i ++){
        for(int j = 0; j < N; j ++){
            if(A[j]&(1<<indx[i])){
                B[j] |= (1<<i);
            }
        }
    }

    int M = 0;
    for(int i = 0; i < N; i ++){
        if(boleh[i]){
            C[M ++] = B[i];
        }
    }

    for(int i = 0; i < M; i ++){
        //printf("C[%d]=%d\n", i, C[i]);
    }

    long long ans = 0;
    for(int U = 0; U < (1<<L); U ++){
        long long f = 0;

        set<int> s;
        multiset<int> ms;
        for(int i = 0; i < M; i ++){
            s.insert(U&C[i]);
            ms.insert(U&C[i]);
        }

        for(int j: s){
            int cnt = ms.count(j);
            f += ncrUpTo[cnt][K];
        }

        int temp = __builtin_popcount(U);
        if(temp&1){
            ans -= f;
        }else{
            ans += f;
        }
    }

    printf("%lld", ans);
    return 0;
}