O(rand)

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

inputFormat

Input

Input is given from Standard Input in the following format:

N K S T A_1 A_2 ... A_N

outputFormat

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

样例

3 3 0 3
1 2 3

2