Zero-One (Hard Version)
Description
This is the hard version of this problem. In this version, $n \le 5000$ holds, and this version has no restriction between $x$ and $y$. You can make hacks only if both versions of the problem are solved.
You are given two binary strings $a$ and $b$, both of length $n$. You can do the following operation any number of times (possibly zero).
- Select two indices $l$ and $r$ ($l < r$).
- Change $a_l$ to $(1 - a_l)$, and $a_r$ to $(1 - a_r)$.
- If $l + 1 = r$, the cost of the operation is $x$. Otherwise, the cost is $y$.
You have to find the minimum cost needed to make $a$ equal to $b$ or say there is no way to do so.
The first line contains one integer $t$ ($1 \le t \le 1000$) — the number of test cases.
Each test case consists of three lines. The first line of each test case contains three integers $n$, $x$, and $y$ ($5 \le n \le 5000$, $1 \le x, y \le 10^9$) — the length of the strings, and the costs per operation.
The second line of each test case contains the string $a$ of length $n$. The string only consists of digits $0$ and $1$.
The third line of each test case contains the string $b$ of length $n$. The string only consists of digits $0$ and $1$.
It is guaranteed that the sum of $n$ over all test cases doesn't exceed $5000$.
For each test case, if there is no way to make $a$ equal to $b$, print $-1$. Otherwise, print the minimum cost needed to make $a$ equal to $b$.
Input
The first line contains one integer $t$ ($1 \le t \le 1000$) — the number of test cases.
Each test case consists of three lines. The first line of each test case contains three integers $n$, $x$, and $y$ ($5 \le n \le 5000$, $1 \le x, y \le 10^9$) — the length of the strings, and the costs per operation.
The second line of each test case contains the string $a$ of length $n$. The string only consists of digits $0$ and $1$.
The third line of each test case contains the string $b$ of length $n$. The string only consists of digits $0$ and $1$.
It is guaranteed that the sum of $n$ over all test cases doesn't exceed $5000$.
Output
For each test case, if there is no way to make $a$ equal to $b$, print $-1$. Otherwise, print the minimum cost needed to make $a$ equal to $b$.
6
5 8 9
01001
00101
6 2 11
000001
100000
5 7 2
01000
11011
7 8 3
0111001
0100001
6 3 4
010001
101000
5 10 1
01100
01100
8
10
-1
6
7
0
Note
In the first test case, selecting indices $2$ and $3$ costs $8$, which is the minimum.
In the second test case, we can perform the following operations.
- Select indices $1$ and $2$. It costs $2$, and $a$ is 110001 now.
- Select indices $2$ and $3$. It costs $2$, and $a$ is 101001 now.
- Select indices $3$ and $4$. It costs $2$, and $a$ is 100101 now.
- Select indices $4$ and $5$. It costs $2$, and $a$ is 100011 now.
- Select indices $5$ and $6$. It costs $2$, and $a$ is 100000 now.
The total cost is $10$.
In the third test case, we cannot make $a$ equal to $b$ using any number of operations.
In the fourth test case, we can perform the following operations.
- Select indices $3$ and $6$. It costs $3$, and $a$ is 0101011 now.
- Select indices $4$ and $6$. It costs $3$, and $a$ is 0100001 now.
The total cost is $6$.
In the fifth test case, we can perform the following operations.
- Select indices $1$ and $6$. It costs $4$, and $a$ is 110000 now.
- Select indices $2$ and $3$. It costs $3$, and $a$ is 101000 now.
The total cost is $7$.
In the sixth test case, we don't have to perform any operation.