```python
def gen_plan(state: DrawState, x: float, y: float, height: float, width: float, cross_height: float, offsets: list):
    # Calculate the points for the letter F based on the input parameters and offsets
    # The letter F is composed of three lines: the vertical spine, the top horizontal line, and the middle cross line.
    # The offsets list contains offsets for each of these points in order: [bottom, top, top-right, middle-right]
    
    # Apply offsets to each point
    bottom = [x + offsets[0][0], y + offsets[0][1]]
    top = [x + offsets[1][0], y + height + offsets[1][1]]
    top_right = [x + width + offsets[2][0], y + height + offsets[2][1]]
    middle_right = [x + cross_height + offsets[3][0], y + (height / 2) + offsets[3][1]]
    
    # Create the actions for drawing the letter F
    actions = [
        Action("draw_line", bottom + top),  # Vertical spine
        Action("draw_line", top + top_right),  # Top horizontal line
        Action("draw_line", [x, y + (height / 2)] + middle_right)  # Middle cross line
    ]
    
    return actions

def gen_initial_guess(initial: DrawState):
    return {
        "x": 0.2,  # Starting x-coordinate
        "y": 0.2,  # Starting y-coordinate
        "height": 0.25,  # Height of the letter F
        "width": 0.175,  # Width of the top horizontal line of the letter F
        "cross_height": 0.125,  # Width of the middle cross line of the letter F
        "offsets": [[0., 0.]] * 4  # Offsets for the bottom, top, top-right, and middle-right points
    }
```