How to find the vertex of a square by hough line transform?

Question

The image on the left is the original image, and the right is the image detected through houghlineP. What I want is to find the coordinates of each vertex of Sudoku through the information on the right, as shown in the figure below.

I don't know what else to do to detect vertices after houghlineP.

def hough_line_segments(src) :
    src = cv2.resize(src, (800, 800))
    edge = cv2.Canny(src, 75, 200)
    lines = cv2.HoughLinesP(edge, rho=5, theta=math.pi / 180, threshold=250, minLineLength=30, maxLineGap=30)
    dst = cv2.cvtColor(edge, cv2.COLOR_GRAY2BGR)
    
    red_lines = np.zeros([src.shape[0], src.shape[1], 3], dtype=np.uint8)
    print(lines)
    gradients = []
    if lines is not None:
        for i in range(lines.shape[0]):
            pt1 = (lines[i][0][0], lines[i][0][1])
            pt2 = (lines[i][0][2], lines[i][0][3])
            gradients.append(calculate_slope(pt1[0], pt1[1], pt2[0], pt2[1]))
            # cv2.line(dst, pt1, pt2, (0, 0, 255), 2, cv2.LINE_AA)
            cv2.line(red_lines, pt1, pt2, (0, 0, 255), 2, cv2.LINE_AA)
        group1, group2 = divide_numbers_by_average(gradients)
        for i in range(lines.shape[0]):
            if gradients in group1:
                cv2.line(dst, pt1, pt2, (0, 255, 0), 2, cv2.LINE_AA)  # green line
            elif gradients in group2:
                cv2.line(dst, pt1, pt2, (0, 0, 255), 2, cv2.LINE_AA)  # red line
    cv2.imshow('src', src)
    cv2.imshow('dst', dst)
    cv2.imshow('red_line', red_lines)
    cv2.waitKey(0)
    cv2.destroyAllWindows()

src = cv2.imread('../img/board6.png', cv2.IMREAD_GRAYSCALE)

hough_line_segments(src)