def solution(dimentions, your_position, trainer_position, distance):
    direction_target_map = {}
    for reflected_position, direction, length, is_trainer in generate_reflected_positions(dimentions, your_position, trainer_position, distance):
        if (direction in direction_target_map):
            if length < direction_target_map[direction][0]:
                direction_target_map[direction] = (length, is_trainer)
        else:
            direction_target_map[direction] = (length, is_trainer)
    return len([k for k, (d, is_trainer) in direction_target_map.items() if is_trainer])

def generate_room_id(max):
    for room_id in range(max):
        yield room_id
    for room_id in range(max):
        yield - room_id - 1

def generate_reflected_positions(dimentions, your_position, trainer_position, distance):
    memo = {}
    room_width, room_height = dimentions
    x_range = distance // room_width + 1
    y_range = distance // room_height + 1
    for room_x in generate_room_id(x_range):
        for room_y in generate_room_id(y_range):
            x_reflected = room_x % 2 == 1
            y_reflected = room_y % 2 == 1
            x_base, y_base = room_x*room_width, room_y*room_height
            your_x_offset, your_y_offset = your_position
            trainer_x_offset, trainer_y_offset = trainer_position
            if (x_reflected):
                your_x_offset = room_width - your_x_offset
                trainer_x_offset = room_width - trainer_x_offset
            if (y_reflected):
                your_y_offset = room_height - your_y_offset
                trainer_y_offset = room_height - trainer_y_offset
            reflected_your_position = x_base + your_x_offset, y_base + your_y_offset
            reflected_trainer_position = x_base + trainer_x_offset, y_base + trainer_y_offset
            reflected_your_direction = calc_direction(your_position, reflected_your_position)
            reflected_trainer_direction = calc_direction(your_position, reflected_trainer_position)
            if reflected_your_direction not in memo:
                reflected_your_distance = line_length(your_position, reflected_your_position)
                if reflected_your_distance <= distance:
                    yield reflected_your_position, reflected_your_direction, reflected_your_distance, False
            if reflected_trainer_direction not in memo:
                reflected_trainer_distance = line_length(your_position, reflected_trainer_position)
                if reflected_trainer_distance <= distance:
                    yield reflected_trainer_position, reflected_trainer_direction, reflected_trainer_distance, True
            memo[reflected_your_direction] = 0
            memo[reflected_trainer_direction] = 0

def abs_gcd(x, y):
    m, n = abs(x), abs(y)
    while n != 0:
        t = m % n
        m, n = n, t
    return abs(m)

def calc_direction(origin, target):
    return reduce_direction((target[0] - origin[0], target[1] - origin[1]))

def reduce_direction(direction):
    if direction[0] == 0 and direction[1] == 0:
        return 0, 0
    elif direction[0] == 0:
        return 0, direction[1] / abs(direction[1])
    elif direction[1] == 0:
        return direction[0] / abs(direction[0]), 0
    gcd = abs_gcd(direction[0], direction[1])
    return direction[0]/gcd, direction[1]/gcd

def line_length(origin, target):
    return ((target[0] - origin[0])**2 + (target[1] - origin[1])**2) ** 0.5

tests = [
    (([3, 2], [1, 1], [2, 1], 4), 7),
    (([300,275], [150,150], [185,100], 500), 9)
]

for i, o in tests:
    result = solution(*i)
    print(i, result == o, result, o)