# Problem Description
This problem involves a grid where a red triangle and several orange dots are placed. The goal is to determine the number of iterations required to convert all orange dots into purple dots. Each iteration involves drawing a line through the red triangle, and any orange dot that lies on this line is absorbed and turned into a purple dot in the next iteration. The task is to calculate the minimum number of iterations needed to absorb all the orange dots.

# Visual Facts
1. **Grid Structure:**
   - The grid is a 4x4 coordinate system (0-3 on both axes).
   - The origin (0,0) is at the top-left corner.

2. **Components:**
   - One red triangle (fixed position).
   - Multiple orange dots (initial state).
   - Light orange dots (transitioning state).
   - Purple dots (transformed state).
   - Purple lines (transformation medium).

3. **Transformation Sequence:**
   - Initial state shows orange dots and one red triangle.
   - Iterations involve drawing lines through the red triangle.
   - Orange dots on these lines turn into light orange dots in this iteration, and in the next iteration, they transform into purple dots.
   - Each orange dot needs two iterations to turn into a purple dot.
   - The process continues until all orange dots are converted to purple dots.

# Visual Patterns
1. **Line Properties:**
   - Each purple line must pass through the red triangle.
   - Lines can be:
     * Vertical (constant x-coordinate).
     * Horizontal (constant y-coordinate).
     * Diagonal (linear with slope).

2. **Absorption Rules:**
   - Any orange dot lying exactly on a purple line gets absorbed.
   - Each orange dot needs two iterations to turn into a purple dot.
   - Multiple dots can be absorbed in two iterations if they lie on the same line.
   - Each dot can only be absorbed once.
   - The line must pass through both the triangle and at least one orange dot.

3. **Optimization Patterns:**
   - Lines are chosen to maximize the number of dots absorbed per iteration.
   - Dots that share geometric alignment (same x, y, or slope relative to the triangle) should be absorbed together.
   - The order of absorption can affect the total number of iterations.
   - Priority seems to be given to lines that can absorb multiple dots simultaneously.

4. **Mathematical Constraints:**
   - For dots to be absorbed in the same two iterations, they must satisfy one of:
     * Same x-coordinate as the triangle (vertical line).
     * Same y-coordinate as the triangle (horizontal line).
     * Share the same slope relative to the triangle position (diagonal line).
     * Points must be collinear with the triangle.

5. **Iteration Strategy:**
   - Given the red triangle's position (x0, y0):
     - Find the dots with xi=x0; these points can be absorbed in the first two iteration.
     - Then compute all the slopes to the orange dots (yi-y0)/(xi-x0).
     - Count the groups of dots that share the same slope or x/y coordinate.
     - The number of iterations multiplied by two is the number of unique groups (+2 if there are dots with the same x-coordinate as the triangle).

# Solution
def solution(red_triangle: tuple, orange_dots: list) -> int:
    """
    Determine the number of iterations required to absorb all the orange dots.

    Parameters:
    red_triangle (tuple): The coordinates of the red triangle (x0, y0).
    orange_dots (list): A list of tuples, where each tuple represents the coordinates of an orange dot (xi, yi).

    Returns:
    int: The number of iterations.
    """
    x0, y0 = red_triangle
    slopes = set()

    for x, y in orange_dots:
        if x == x0:
            slopes.add('vertical')
        elif y == y0:
            slopes.add('horizontal')
        else:
            slopes.add((y - y0) / (x - x0))

    return len(slopes) * 2

# Test Script

def test():
    test_cases = [
        ((2, 2), [(2, 4)], 2),
        ((3, 0), [(3, 1), (3, 2), (3, 3)], 2),
        ((0, 3), [(1, 3), (2, 3), (4, 3)], 2),
        ((2, 2), [(1, 1), (0, 0), (3, 3)], 2),
        ((2, 2), [(0, 2), (2, 4), (4, 4), (2, 0)], 6),
        ((2, 2), [], 0),
        ((0, 0), [(1, 1), (2, 2), (3, 3), (4, 4)], 2),
        ((2, 0), [(0, 0), (0, 1), (1, 3), (2, 1), (3, 2), (4, 0)], 10),
        ((2, 3), [(0, 3), (2, 1), (4, 3), (3, 4), (1, 5), (0, 5)], 10),
        ((5, 5), [(0, 0), (0, 5), (5, 0), (10, 5), (5, 10), (10, 10)], 6)
    ]

    for i, (red_triangle, orange_dots, expected_output) in enumerate(test_cases):
        try:
            output = solution(red_triangle, orange_dots)
        except Exception as e:
            error_message = (
                f"An exception occurred while running the test case:\n"
                f"`red_triangle`: {red_triangle}\n"
                f"`orange_dots`: {orange_dots}\n\n"
                f"Exception: '{str(e)}'"
            )
            assert False, error_message

        if output != expected_output:
            error_message = (
                f"The following Test Case failed:\n"
                f"`red_triangle`: {red_triangle}\n" 
                f"`orange_dots`: {orange_dots}\n\n"
                f"Actual Output: {output}\n"
                f"Expected Output: {expected_output}"
            )
            assert False, error_message

test()