# Problem Description
This is a geometric transformation problem on a grid where we need to simulate the absorption of orange dots by purple lines emanating from a red triangle. The goal is to calculate the minimum number of iterations needed to convert all orange dots into purple dots, following specific geometric rules of absorption.

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

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

3. Transformation Sequence:
- Initial state shows orange dots and one red triangle
- Iter1: Vertical line absorption
- Iter2: Diagonal line absorption
- Iter3: Horizontal line absorption
- Iter4: Final diagonal absorption
- Process completes in exactly 4 iterations in the example

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

2. Absorption Rules:
- Any orange dot lying exactly on a purple line gets absorbed
- Multiple dots can be absorbed in a single iteration 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 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 iteration, they must satisfy one of:
  * Same x-coordinate as triangle (vertical line)
  * Same y-coordinate as triangle (horizontal line)
  * Share the same slope relative to 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 iteration
  - Then compute all the slopes to the orange dots (yi-y0)/(xi-x0)
  - Count the groups that dots share the same slope or x/y coordinate
  - the total iteration is the number of groups (+1 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)

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

    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()