ABABABAB

Consider the following formula:

$A + B * A + B * A + B * A + B$

We can obtain many different values by replacing each $A$ with an arbitrary number from set $\mathcal A$ , and each $B$ with an arbitrary number from set $\mathcal B$ .

Even more, in this problem we are allowed to place parentheses in any way we want. For example, $(A+B)*(A+B)*(A+B)*(A+B)$ can be a very big number. We don’t like very big numbers.

Output the number of ways we can obtain a result which is at most $M$ .

Input

The first line of input contains four numbers: $N, M, Q_A, Q_B$ . We have $1 \leq N \leq 16$ , $1 \leq M \leq 1000$ , $1 \leq Q_A, Q_B \leq 1000$ . $N$ is the number of operands (8 in the formula above, it always starts with $A$ ).

The second line contains $Q_A$ non-negative integers — these are the elements of $\mathcal A$ . Each of them is different, and in range from $0$ to $10000$ .

The third line contains $Q_B$ non-negative integers — these are the elements of $\mathcal B$ . Each of them is different, and in range from $0$ to $10000$ .

Output

Output the number of ways of obtaining at most $M$ , modulo 1000003.

Problem information

Author: Eryk Kopczynski

Generation: 2026-01-25T22:30:02.330Z