Robogolf

The World Robot Olympiad holds robogolf competitions. The playing field is a rectangle of $n \times m$ unit squares. Rows are numbered 1 to $n$, and columns are numbered 1 to $m$. Each square is identified by its row $r$ and column $c$.

Two robots take turns moving a special chip across the field. Some squares contain traps. When the chip is on square $(r, c)$, the robot whose turn it is can move it to $(r+1, c)$ or to $(r, c+1)$. If that square does not exist because the chip is in the last row or the last column, the robot can move the chip off the field. The game ends when the chip is off the field or on a square with a trap.

Each trap has an integer price $v_i$. The game result is the price of the trap where the game ended, or 0 if the chip left the field. The robot that moves first wants to minimize the result, and the robot that moves second wants to maximize it.

Let the chip start on square $(r, c)$. The guaranteed result $g(r, c)$ is the smallest result the first robot can force, no matter how the opponent plays. Since the starting square is unknown in advance, the developers want the sum of $g(r, c)$ over all squares. Given the positions and prices of the traps, compute this sum.

The first line contains three integers $n$, $m$, and $k$ ($1 \le n, m \le 10^9$; $1 \le k \le \min(n \cdot m, 10^5)$), the number of rows, columns, and traps. Each of the next $k$ lines contains three integers $r_i$, $c_i$, and $v_i$ ($1 \le r_i \le n$, $1 \le c_i \le m$, $-10^9 \le v_i \le 10^9$), the row, column, and price of the $i$-th trap. No two traps share a square.

Print one integer, the sum of $g(r, c)$ over all squares modulo $998\,244\,353$. The remainder $r = a \bmod b$ for a possibly negative $a$ is the value in $[0, b-1]$ such that $a = qb + r$ for some integer $q$. For example, $13 \bmod 4 = 1$, $-13 \bmod 4 = 3$, $12 \bmod 4 = 0$, and $-12 \bmod 4 = 0$.

In the first example, the guaranteed results for all squares are shown in the figure below, with trap squares highlighted.

The sum of the results is $1 + 1 - 2 + 0 - 2 - 2 + 3 + 0 + 0 = -1$. The answer is $(-1) \bmod 998\,244\,353 = 998\,244\,352$.

In the second example, the guaranteed results for all squares are shown in the figure below, with trap squares highlighted.

The sum of the results is $1 + 2 + 0 - 3 + 1 + 0 - 3 - 3 = -5$. The answer is $(-5) \bmod 998\,244\,353 = 998\,244\,348$.