Forest for the Trees

You have sent a robot into a forest, and it has gotten lost. The robot has a sensor that detects all trees around it, regardless of occlusions. Unfortunately, all trees in this forest look alike. You have a map of all the trees as $(x,y)$ points. Every tree is at an integer coordinate, but not every integer coordinate is occupied, since this used to be a tree farm.

The robot always places itself at integer coordinates and aligned with the positive or negative $x$ or $y$ axis. It is never at the same location as a tree. Its heading relative to the map is unknown. Each sensor reading is a tuple (distance to the right of the robot, distance forward of the robot). Either value can be negative, because the sensor covers all directions. Find where the robot is.

The first line contains three integers $n_t$, $n_s$, and $r_{max}$: the number of trees in the forest, the number of trees sensed by the robot, and the maximum Manhattan distance (sum of the absolute $x$ and $y$ distances) of any sensor reading. Each of the next $n_t$ lines contains two integers $x$ and $y$, the location of a tree in the global coordinate system. Each of the final $n_s$ lines contains two integers. The first integer $s_{i,x}$ is the distance along the axis perpendicular to the robot's heading, and the second integer $s_{i,y}$ is the distance along the axis parallel to the robot's heading.

It is guaranteed that $|s_{i,x}|+|s_{i,y}| \leq r_{max}$ for all $i$. Also, $0 < n_t \leq 5000$, $0 < n_s \leq 1000$, $0 < r_{max} \leq 1000$, and $-100000 \leq x,y \leq 100000$ for every tree.

Print the $x$ and $y$ location of the robot as two integers separated by a space. If no location on the map could produce the given sensor values, print "Impossible". If two or more distinct locations and/or orientations could produce the given sensor values, print "Ambiguous".