In the top row of an $8 \times 8$ chessboard, Tom writes the values 1, 2, 4, 8, 16, 32, 64, 128. In the leftmost column, Tom writes the values 1, 3, 9, 27, 81, 243, 729, 2187. In each of the remaining empty squares, Tom writes the product of the leftmost number in that square's row and the topmost number in that square's column. What is the sum of all the numbers on the chessboard