CF1220G Geolocation

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.

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 $ .

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 $ .

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.