Problem Statement

You are given a positive integer N and sequences of N positive integers each: $A=(A_1,A_2,\dots,A_N)$ and $B=(B_1,B_2,\dots,B_N)$.
We have an $N \times N$ grid. The square at the i-th row from the top and the j-th column from the left is called the square (i,j). For each pair of integers (i,j) such that $1 \le i,j \le N$, the square (i,j) has the integer A_i + B_j written on it. Process Q queries of the following form.

• You are given a quadruple of integers h_1,h_2,w_1,w_2 such that $1 \le h_1 \le h_2 \le N,1 \le w_1 \le w_2 \le N.$ Find the greatest common divisor of the integers contained in the rectangle region whose top-left and bottom-right corners are (h_1,w_1) and (h_2,w_2), respectively.

Input

Input is given from Standard Input in the following format:
N Q
$A_1 A_2 \dots A_N$
$B_1 B_2 \dots B_N$
$\mathrm{query}_1$
$\mathrm{query}_2$
$\vdots$
$\mathrm{query}_Q$

Each query is in the following format:
h_1 h_2 w_1 w_2

Output

Print Q lines. The i-th line should contain the answer to $\mathrm{query}_i$.

Constraints

• $1 \le N,Q \le 2 \times 10^5$
• $1 \le A_i,B_i \le 10^9$
• $1 \le h_1 \le h_2 \le N$
• $1 \le w_1 \le w_2 \le N$
• All values in input are integers.

Sample Input 1

3 5
3 5 2
8 1 3
1 2 2 3
1 3 1 3
1 1 1 1
2 2 2 2
3 3 1 1

Sample Output 1

2
1
11
6
10

Let C_{i,j} denote the integer on the square (i,j).
For the 1-st query, we have C_{1,2}=4,C_{1,3}=6,C_{2,2}=6,C_{2,3}=8, so the answer is their greatest common divisor, which is 2.

Sample Input 2

1 1
9
100
1 1 1 1

Sample Output 2

109

见题面 Input 部分

见题面 Output 部分