12. In a game, two players take turns to "eat squares" from a $5 \times 7$ grid chessboard. To "eat a square," a player selects an uneaten square and moves the piece to that square, then all the squares in the quadrant formed (along the left edge of the square upwards, and along the bottom edge of the square to the right) are eaten. For example, in the right figure, moving the piece to the shaded square results in the shaded square and the four squares marked with $\times$ being eaten (the squares with lines in them were previously eaten). The goal of the game is to make the opponent eat the last square. The figure above shows a situation that can occur during the game. How many different situations can appear at most during the game process?
12. In this game, the height of the table formed by the un-eaten squares from left to right is not constant (as shown in the right figure). It is easy to prove that this situation is both sufficient and necessary. Moreover, each shape can be completely described by a 12-step zigzag line. This zigzag line starts from the top-left corner of the original shape to the bottom-right corner, following the boundary formed by the eaten and uneaten squares. This zigzag line can be described by a sequence of 12 letters, consisting of 7 $H$s (where $H$ represents the top edge of an uneaten square or the bottom edge of an eaten square) and 5 $V$s (where $V$ represents moving vertically down one unit from the top-right vertex of an uneaten square to the vertex of the adjacent lower square). For example, the situation given in the original figure can be described as $H H H^{2} H^{V} V H H H V \square$, and the situation shown above can be described as $V V_{H} H H H V H V Y H H$. Therefore, there are $\frac{12!}{7!5!}=792$ possible sequences of $H$ and $V$. These include the sequences $H H H H H H H V V V V V$ and $V V V V V H H H H H H H$ (i.e., the full and empty figures).