You are working for the Gryzzl company, headquartered in Pawnee, Indiana.

The new national park has been opened near Pawnee recently and you are to implement a geolocation system, so people won't get lost. The concept you developed is innovative and minimalistic. There will be $n$ antennas located somewhere in the park. When someone would like to know their current location, their Gryzzl hologram phone will communicate with antennas and obtain distances from a user's current location to all antennas.

Knowing those distances and antennas locations it should be easy to recover a user's location... Right? Well, almost. The only issue is that there is no way to distinguish antennas, so you don't know, which distance corresponds to each antenna. Your task is to find a user's location given as little as all antennas location and an unordered multiset of distances.


-----Input-----

The first line of input contains a single integer $n$ ($2 \leq n \leq 10^5$) which is the number of antennas.

The following $n$ lines contain coordinates of antennas, $i$-th line contain two integers $x_i$ and $y_i$ ($0 \leq x_i,y_i \leq 10^8$). It is guaranteed that no two antennas coincide.

The next line of input contains integer $m$ ($1 \leq n \cdot m \leq 10^5$), which is the number of queries to determine the location of the user.

Following $m$ lines contain $n$ integers $0 \leq d_1 \leq d_2 \leq \dots \leq d_n \leq 2 \cdot 10^{16}$ each. These integers form a multiset of squared distances from unknown user's location $(x;y)$ to antennas.

For all test cases except the examples it is guaranteed that all user's locations $(x;y)$ were chosen uniformly at random, independently from each other among all possible integer locations having $0 \leq x, y \leq 10^8$.


-----Output-----

For each query output $k$, the number of possible a user's locations matching the given input and then output the list of these locations in lexicographic order.

It is guaranteed that the sum of all $k$ over all points does not exceed $10^6$.


-----Examples-----
Input
3
0 0
0 1
1 0
1
1 1 2

Output
1 1 1 

Input
4
0 0
0 1
1 0
1 1
2
0 1 1 2
2 5 5 8

Output
4 0 0 0 1 1 0 1 1 
4 -1 -1 -1 2 2 -1 2 2 



-----Note-----

As you see in the second example, although initially a user's location is picked to have non-negative coordinates, you have to output all possible integer locations.

Difficulty: competition

Starter code:

Solutions:
import sys
import math

n = int(input())

x = [0]*n
y = [0]*n

for i in range(n):
    x[i], y[i] = list(map(int, input().split()))

sx = sum(x)
sy = sum(y)
    
for i in range(n):
    x[i] = n * x[i] - sx
    y[i] = n * y[i] - sy

m = int(input())

d = [0]*n
e = [0]*n

HD = 0

def check(a, b):
    nonlocal HD
    HE = 0
    for i in range(n):
        HE ^= hash((a-x[i])*(a-x[i])+(b-y[i])*(b-y[i]))
    return HD == HE

def sqrt(x):
    nn = int(x)
    if nn == 0:
        return 0
    fa, fb = divmod(nn.bit_length(), 2)
    x = 2**(fa+fb)
    while True:
        y = (x + nn//x)//2
        if y >= x:
            return x
        x = y

def hash(x):
    return x * 9991 + 43

pans = []

def solve():
    nonlocal d
    d = list(map(int, input().split()))
    c = 0
    d = [p * n * n for p in d]
    for i in range(n):
        c += d[i] - x[i] * x[i] - y[i] * y[i]

    assert(c % n == 0)
    c //= n
    ans = []
    ax = x[0]
    ay = y[0]
    if ax is 0 and ay is 0:
        ax = x[1]
        ay = y[1]
    rev = 0
    if ay == 0:
        ay = ax
        ax = 0
        rev = 1
    d.sort()
    nonlocal HD
    HD = 0
    for p in d:
        HD ^= hash(p)
    old = -1
    for p in d:
        if (p == old):
            continue
        old = p
        a = c + ax * ax + ay * ay - p
        if (a % 2 != 0):
            continue
        a //= 2
        A = ax * ax + ay * ay
        B = a * ax
        C = a * a - ay * ay * c
        D = B * B - A * C
        if (D < 0):
            continue
        sD = sqrt(D)
        if D != sD * sD:
            continue
        if (B + sD) % A == 0:
            qx = (B + sD) // A
            qy = (a - ax * qx) // ay
            if rev:
                t = qx
                qx = qy
                qy = t
            if ((qx + sx) % n == 0 and (qy + sy) % n == 0 and check(qx, qy)):
                qx = (qx + sx) // n
                qy = (qy + sy) // n
                ans.append([qx, qy])
        if sD == 0:
            continue
        if (B - sD) % A == 0:
            qx = (B - sD) // A
            qy = (a - ax * qx) // ay
            if rev:
                t = qx
                qx = qy
                qy = t
            if ((qx + sx) % n == 0 and (qy + sy) % n == 0 and check(qx, qy)):
                qx = (qx + sx) // n
                qy = (qy + sy) // n
                ans.append([qx, qy])
                
    ans.sort()
    buf=[]
    buf.append(len(ans))
    for p in ans:
            buf.append(p[0])
            buf.append(p[1])
    nonlocal pans
    pans.append(" ".join(map(str,buf)))

while m > 0:
    m -= 1
    solve()
    
sys.stdout.write("\n".join(pans))