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G. Geolocation

time limit per test

2 secondsmemory limit per test

256 megabytesinput

standard inputoutput

standard outputYou 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.

Codeforces (c) Copyright 2010-2024 Mike Mirzayanov

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