Boardle

The Boardle game is played on an $N \times M$ grid. The two opposite corners of the grid are $(1, 1)$ and $(N, M)$. The treasure is hidden in one of the $NM$ cells, and you win the game by finding it.

To find the treasure, you asked $Q$ questions of the following form.

• Relative to the point $(x, y)$, in which direction is the treasure?

The answer is one of east, west, south, north, northeast, northwest, southeast, southwest, or same point. East is $+x$, west is $-x$, north is $+y$, and south is $-y$. Use the answers to the $Q$ questions to count the cells where the treasure could still be.

The first line contains two integers $N$ and $M$, the size of the grid.

The second line contains the number of questions $Q$.

From the third line to line $Q+2$, line $i+2$ gives the information for the $i$-th question in the form $x_i$ $y_i$ $d_i$. $x_i$ and $y_i$ are the position of the point asked about in the $i$-th question. $d_i$ is an integer from 1 to 9 and corresponds to each direction as shown in the table below.

If $1 \le d_i \le 4$, the treasure is in exactly that direction. For example, if $d_i = 4$, the treasure is directly north of the chosen point, not northwest or northeast of it.

Print the number of cells where the treasure can be. It is guaranteed that at least one candidate cell exists.

• $1 \le N, M \le 10^9$
• $1 \le Q \le 10^5$
• $1 \le x_i \le N$ ($1 \le i \le Q$)
• $1 \le y_i \le M$ ($1 \le i \le Q$)
• $1 \le d_i \le 9$ ($1 \le i \le Q$)