Problem:
Score : 400 points

Problem Statement

There are two persons, numbered 0 and 1, and a variable x whose initial value is 0.
The two persons now play a game.
The game is played in N rounds. The following should be done in the i-th round (1 \leq i \leq N):

Person S_i does one of the following:

Replace x with x \oplus A_i, where \oplus represents bitwise XOR.

Do nothing.

Person 0 aims to have x=0 at the end of the game, while Person 1 aims to have x \neq 0 at the end of the game.

Determine whether x becomes 0 at the end of the game when the two persons play optimally.

Solve T test cases for each input file.

Constraints

1 \leq T \leq 100

1 \leq N \leq 200

1 \leq A_i \leq 10^{18}

S is a string of length N consisting of 0 and 1.

All numbers in input are integers.

Input

Input is given from Standard Input in the following format.
The first line is as follows:

T

Then, T test cases follow.
Each test case is given in the following format:

N
A_1 A_2 \cdots A_N
S

Output

For each test case, print a line containing 0 if x becomes 0 at the end of the game, and 1 otherwise.

Sample Input 1

3
2
1 2
10
2
1 1
10
6
2 3 4 5 6 7
111000

Sample Output 1

1
0
0

In the first test case, if Person 1 replaces x with 0 \oplus 1=1, we surely have x \neq 0 at the end of the game, regardless of the choice of Person 0.

In the second test case, regardless of the choice of Person 1, Person 0 can make x=0 with a suitable choice.

Solution in PyPy3 (7.3.0) (judge verdict: Wrong Answer)

def main():
    # rankを出力、modを使って実装
    def matrix_rank(a, mod=10**9+7):
        n, m = len(a), len(a[0])
        M = [[j for j in i] for i in a]
        rank = min(n, m)
        for i in range(min(n, m)):
            if M[i][i] == 0:
                for j in range(i+1, n):
                    if M[j][i] != 0:
                        for k in range(i, m):
                            M[i][k] = (M[i][k]+M[j][k]) % mod
                        break
                else:
                    for j in range(i+1, m):
                        if M[i][j] != 0:
                            for k in range(i, n):
                                M[k][i] = (M[k][i]+M[k][j]) % mod
                            break
                    else:
                        rank -= 1
            pinv = pow(M[i][i], mod-2, mod)
            for j in range(m):
                M[i][j] = M[i][j]*pinv % mod
            M2 = M[i]
            for j in range(i+1, n):
                M3 = M[j][i]
                M[j][i] = 0
                for k in range(i+1, m):
                    M[j][k] = (M[j][k]-M3*M2[k]) % mod
        return rank

    t = int(input())
    for _ in range(t):
        n = int(input())
        a = list(map(int, input().split()))
        s = input()
        r = 0
        m = []
        for i in range(n-1, -1, -1):
            ai = a[i]
            b = [[(ai >> j) & 1 for j in range(64)]]
            newr = matrix_rank(m+b)
            if s[i] == "0":
                if newr > r:
                    if newr != r+1:
                        print(1/0)
                    r += 1
                    m += b
            else:
                if newr > r:
                    if newr != r+1:
                        print(1/0)
                    print(1)
                    break
        else:
            print(0)


main()