Problem:
Score : 600 points

Problem Statement

Given are two sequences a=\{a_0,\ldots,a_{N-1}\} and b=\{b_0,\ldots,b_{N-1}\} of N non-negative integers each.

Snuke will choose an integer k such that 0 \leq k < N and an integer x not less than 0, to make a new sequence of length N, a'=\{a_0',\ldots,a_{N-1}'\}, as follows:

a_i'= a_{i+k \mod N}\ XOR \ x

Find all pairs (k,x) such that a' will be equal to b.

What is \mbox{ XOR }?

The XOR of integers A and B, A \mbox{ XOR } B, is defined as follows:

When A \mbox{ XOR } B is written in base two, the digit in the 2^k's place (k \geq 0) is 1 if either A or B, but not both, has 1 in the 2^k's place, and 0 otherwise.

For example, 3 \mbox{ XOR } 5 = 6. (In base two: 011 \mbox{ XOR } 101 = 110.)

Constraints

1 \leq N \leq 2 \times 10^5

0 \leq a_i,b_i < 2^{30}

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
a_0 a_1 ... a_{N-1}
b_0 b_1 ... b_{N-1}

Output

Print all pairs (k, x) such that a' and b will be equal, using one line for each pair, in ascending order of k (ascending order of x for pairs with the same k).

If there are no such pairs, the output should be empty.

Sample Input 1

3
0 2 1
1 2 3

Sample Output 1

1 3

If (k,x)=(1,3),

a_0'=(a_1\ XOR \ 3)=1

a_1'=(a_2\ XOR \ 3)=2

a_2'=(a_0\ XOR \ 3)=3

and we have a' = b.

Sample Input 2

5
0 0 0 0 0
2 2 2 2 2

Sample Output 2

0 2
1 2
2 2
3 2
4 2

Sample Input 3

6
0 1 3 7 6 4
1 5 4 6 2 3

Sample Output 3

2 2
5 5

Sample Input 4

2
1 2
0 0

Sample Output 4

No pairs may satisfy the condition.

Solution in PyPy3 (2.4.0) (judge verdict: Accepted)

def main():
    mod = 2**31-1
    pow3 = [1]*200001
    p = 1
    i3 = pow(3, mod-2, mod)
    for i in range(1, 200001):
        p = p*3 % mod
        pow3[i] = p

    def rolling_hash(seq):
        h = 0
        for i, j in enumerate(seq):
            h += (j+1)*pow3[i]
        h %= mod
        H0s = [0]*n
        for i, j in enumerate(seq):
            H0s[i] = h
            h = (h-j-1+(j+1)*pn)*i3 % mod
        return H0s

    def rolling_hash2(seq):
        p = 0
        p2 = 0
        for i, j in enumerate(seq):
            p += (j+1)*pow3[i]
            p2 += (2-j)*pow3[i]
        return p % mod, p2 % mod

    n = int(input())
    a = list(map(int, input().split()))
    b = list(map(int, input().split()))
    m = max(a+b)
    a1 = [0]*n
    b1 = [0]*n
    memo = [0]*n
    cnt = n
    pn = pow3[n]

    for k in range(len(bin(m))-2):
        k2 = 2**k
        for i, j in enumerate(a):
            a1[i] = (j & k2) >> k
        for i, j in enumerate(b):
            b1[i] = (j & k2) >> k
        b_hash, b_hash2 = rolling_hash2(b1)
        A1 = rolling_hash(a1)
        for i, (j, k) in enumerate(zip(memo, A1)):
            if j is None:
                continue
            elif k == b_hash:
                pass
            elif k == b_hash2:
                memo[i] += k2
            else:
                memo[i] = None
                cnt -= 1
        if cnt == 0:
            return

    for i, j in enumerate(memo):
        if j is not None:
            print(i, j)


main()