```python
def gen_plan(state: DrawState, center_x: float, center_y: float, side_length: float, offsets: list):
    import math

    # Calculate the coordinates of the hexagon's vertices
    vertices = []
    for i in range(6):
        angle_deg = 60 * i - 30  # Hexagon starts with flat top; -30 degrees to rotate it
        angle_rad = math.radians(angle_deg)
        x = center_x + side_length * math.cos(angle_rad)
        y = center_y + side_length * math.sin(angle_rad)
        vertices.append([x, y])

    # Add offsets to each vertex
    vertices_with_offsets = [
        [vx + dx, vy + dy] for (vx, vy), (dx, dy) in zip(vertices, offsets)
    ]

    # Create draw_line actions for each edge of the hexagon
    actions = []
    for i in range(6):
        x0, y0 = vertices_with_offsets[i]
        x1, y1 = vertices_with_offsets[(i + 1) % 6]  # Wrap around to close the hexagon
        actions.append(Action("draw_line", [x0, y0, x1, y1]))

    return actions

def gen_initial_guess(initial: DrawState):
    return {
        "center_x": 0.2,  # Center of the drawing board (X coordinate)
        "center_y": 0.35,  # Center of the drawing board (Y coordinate)
        "side_length": 0.15,  # Side length of the hexagon
        "offsets": [[0., 0.]] * 6  # 2D point offsets for the hexagon (6 points)
    }
```