Score : $1400$ points

Problem Statement

There is a non-negative integer $X$ initially equal to $S$. One can perform the following operations, in any order, any number of times:

• Add $1$ to $X$. This has a cost of $A$.
• Choose $i$ between $1$ and $N$, and replace $X$ with $X \oplus Y_i$. This has a cost of $C_i$. Here, $\oplus$ denotes bitwise $\mathrm{XOR}$.

Find the smallest total cost to make $X$ equal to a given non-negative integer $T$, or report that this is impossible.

Constraints

• $1 \leq N \leq 8$
• $0 \leq S, T < 2^{40}$
• $0 \leq A \leq 10^5$
• $0 \leq Y_i < 2^{40}$ ($1 \leq i \leq N$)
• $0 \leq C_i \leq 10^{16}$ ($1 \leq i \leq N$)
• All values in the input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $S$ $T$ $A$

$Y_1$ $C_1$

$\vdots$

$Y_N$ $C_N$

Output

If the task is impossible, print -1. Otherwise, print the smallest total cost.

1 15 0 1
8 2

5

You can make $X$ equal to $T$ in the following manner:

• Choose $i=1$ and replace $X$ with $X \oplus 8$ to get $X=7$. This has a cost of $2$.
• Add $1$ to $X$ to get $X=8$. This has a cost of $1$.
• Choose $i=1$ and replace $X$ with $X \oplus 8$ to get $X=0$. This has a cost of $2$.

3 21 10 100
30 1
12 1
13 1

3

1 2 0 1
1 1

-1

8 352217 670575 84912
239445 2866
537211 16
21812 6904
50574 8842
380870 5047
475646 8924
188204 2273
429397 4854

563645