201463: [AtCoder]ARC146 D - >=<

Problem Statement

A fantastic IS is an integer sequence of length $N$ whose every element is between $1$ and $M$ (inclusive) that satisfies the following condition.

• For every integer $i$ such that $1 \le i \le K$, one of the following holds.
  • $A_{P_i} < X_i$ and $A_{Q_i} < Y_i$;
  • $A_{P_i} = X_i$ and $A_{Q_i} = Y_i$;
  • $A_{P_i} > X_i$ and $A_{Q_i} > Y_i$.

Determine whether a fantastic IS exists. If it does, find the minimum possible sum of the elements in a fantastic IS.

Constraints

• $1 \le N,M,K \le 2 \times 10^5$
• $1 \le P_i,Q_i \le N$
• $1 \le X_i,Y_i \le M$
• $P_i \neq Q_i$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$ $K$
$P_1$ $X_1$ $Q_1$ $Y_1$
$P_2$ $X_2$ $Q_2$ $Y_2$
$\vdots$
$P_K$ $X_K$ $Q_K$ $Y_K$

Output

If a fantastic IS exists, print the minimum possible sum of the elements in a fantastic IS; otherwise, print $-1$.

Sample Input 1

3 4 3
3 1 1 2
1 1 2 2
3 4 1 4

Sample Output 1

6

$A=(2,3,1)$ fully satisfies the condition and thus is a fantastic IS, whose sum of the elements is $6$.

There is no fantastic IS whose sum of the elements is less than $6$, so the answer is $6$.

Sample Input 2

2 2 2
1 1 2 2
2 1 1 2

Sample Output 2

-1

There is no fantastic IS, so $-1$ should be printed.

Sample Input 3

5 10 10
4 1 2 7
5 1 3 2
2 9 4 4
5 4 2 9
2 9 1 9
4 8 3 10
5 7 1 5
3 5 1 2
3 8 2 10
2 9 4 8

Sample Output 3

12