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 $$$10^6$$$.

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.

Idx	Name	Rating
A	Cards	800
B	Multiplication Table	1300
C	Substring Game in the Lesson	1300
D	Alex and Julian	1900
E	Tourism	2200
F	Gardener Alex	2700
G	Geolocation	3400

Codeforces Round 586 (Div. 1 + Div. 2)