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Ex - Constrained Sums

Score : $600$ points

Problem Statement

Determine whether there is a sequence of $N$ integers $X = (X_1, X_2, \ldots ,X_N)$ that satisfies all of the following conditions, and construct one such sequence if it exists.

• $0 \leq X_i \leq M$ for every $1 \leq i \leq N$.
  
• $L_i \leq X_{A_i} + X_{B_i} \leq R_i$ for every $1 \leq i \leq Q$.

Constraints

• $1 \leq N \leq 10000$
• $1 \leq M \leq 100$
• $1 \leq Q \leq 10000$
• $1 \leq A_i, B_i \leq N$
• $0 \leq L_i \leq R_i \leq 2 \times M$
• All values in the input are integers.

Input

The input is given from Standard Input in the following format:

$N$ $M$ $Q$

$A_1$ $B_1$ $L_1$ $R_1$

$A_2$ $B_2$ $L_2$ $R_2$

$\vdots$

$A_Q$ $B_Q$ $L_Q$ $R_Q$

Output

If there is an integer sequence that satisfies all of the conditions in the Problem Statement, print the elements $X_1, X_2, \ldots, X_N$ of one such sequence, separated by spaces. Otherwise, print -1.

4 5 3
1 3 5 7
1 4 1 2
2 2 3 8

2 4 3 0

For $X = (2,4,3,0)$, we have $X_1 + X_3 = 5$, $X_1 + X_4 = 2$, and $X_2 + X_2 = 8$, so all conditions are satisfied. There are other sequences, such as $X = (0,2,5,2)$ and $X = (1,3,4,1)$, that satisfy all conditions, and those will also be accepted.

3 7 3
1 2 3 4
3 1 9 12
2 3 2 4

-1

No sequence $X$ satisfies all conditions.