
 from math import atan2, sqrt

 """
 The first thing you had to realize in order to solve this challenge is that you
 could simply mirror the room and check if the beam gets where you want. Let me
 show you:
 In our case let us make a room of dimension [3,2], with the position of the
 target at [1,2]. For the sake of simplicity let us imagine the beam is shot at
 [1,2] and so we are trying to shot ourselves. We get something like this:
 
 *xx
 xxx
 
 If we shoot with a vector of [1,0], aka straight ahead, we will get this result:
 
 *~~
 xxx
 
 We can see that the beam gets back to you directly, thanks to the reflection,
 and that the goal is achieved! But there is another way to do that:
  
 *~~ ~~*
 xxx xxx
  
 We can here realize that, by putting a mirror version of the room where the beam
 gets, we can make a straight line instead of calculating the reflection and see
 that it gets to our original target, [1,2], being ourselves!
 The good thing with that is that it also allows us to determine the distance
 needed by the beam to get through, and thus we can mirror the rooms up to the
 maximum distance easily.

 The next thing to realize was that room mirrors in a pattern, see the below
 diagram:
 
 *xx
 xxx
 
 [...]  [...]
 xxx
 *xx
 
     
 *xx xx* *xx xx* *xx
 xxx xxx xxx xxx xxx
     
 
 xxx
 *xx
 
 [...]  [...]
 *xx
 xxx
 
 x always repeats in the same way, and so does y
 Thus we need to figure out how to calculate how to mirror one room twice and we
 can make an entire atlas of mirrored rooms!
 In our case though the function below only calculates the mirrors of the point
 of coordinates node across an atlas of length (distance*2)^2
 """
 def mirror_atlas(node, dimensions, distance):
 node_mirrored=[]
 for i in range(len(node)):
 points=[]
 # since it is floored we add 1, and python stops before the upper bound
 # so +2 in our case
 for j in range((distance//dimensions[i])1, (distance//dimensions[i]+2)):
 points.append(get_mirror(j, node[i], dimensions[i]))
 node_mirrored.append(points)
 return node_mirrored

 # this simply calculates the mirror nth of a point in a 1D space
 def get_mirror(mirror, coordinates, dimensions):
 res=coordinates
 mirror_rotation=[2*coordinates, 2*(dimensionscoordinates)]
 if(mirror<0):
 for i in range(mirror, 0):
 res=mirror_rotation[(i+1)%2]
 else:
 for i in range(mirror, 0, 1):
 res+=mirror_rotation[i%2]
 return res

 # pretty simple too: we get the angle at which the beam is needed to be to get
 # to the guard and store its distance. First we do that to yourself, then to the
 # guard and if the distance for the guard is lower than yourself then he'll be
 # hit first by the beam.
 def answer(dimensions, your_position, guard_position, distance):
 mirrored = [mirror_atlas(your_position, dimensions,
 distance),mirror_atlas(guard_position, dimensions, distance)]
 res=set()
 angles_dist={}
 for i in range(0, len(mirrored)):
 for j in mirrored[i][0]:
 for k in mirrored[i][1]:
 # position of the beam to touch the node, either yourself or the
 # guard
 beam=atan2((your_position[1]k), (your_position[0]j))
 # pythagorean theorem gets you the hypothenuse given two points
 # of a triangle, which is basically the distance between those
 # two points
 l=sqrt((your_position[0]j)**2 + (your_position[1]k)**2)
 if [j,k] != your_position and distance >= l:
 if((beam in angles_dist and angles_dist[beam] > l) or beam not in angles_dist):
 if i == 0:
 angles_dist[beam] = l
 else:
 angles_dist[beam] = l
 # we could find a better result later so let's just
 # put that in a set to sort out duplicates
 res.add(beam)
 return len(res)
