Too Many Constraints
Description
You are asked to build an array $a$, consisting of $n$ integers, each element should be from $1$ to $k$.
The array should be non-decreasing ($a_i \le a_{i+1}$ for all $i$ from $1$ to $n-1$).
You are also given additional constraints on it. Each constraint is of one of three following types:
- $1~i~x$: $a_i$ should not be equal to $x$;
- $2~i~j~x$: $a_i + a_j$ should be less than or equal to $x$;
- $3~i~j~x$: $a_i + a_j$ should be greater than or equal to $x$.
Build any non-decreasing array that satisfies all constraints or report that no such array exists.
The first line contains a single integer $t$ ($1 \le t \le 10^4$) — the number of testcases.
The first line of each testcase contains three integers $n, m$ and $k$ ($2 \le n \le 2 \cdot 10^4$; $0 \le m \le 2 \cdot 10^4$; $2 \le k \le 10$).
The $i$-th of the next $m$ lines contains a description of a constraint. Each constraint is of one of three following types:
- $1~i~x$ ($1 \le i \le n$; $1 \le x \le k$): $a_i$ should not be equal to $x$;
- $2~i~j~x$ ($1 \le i < j \le n$; $2 \le x \le 2 \cdot k$): $a_i + a_j$ should be less than or equal to $x$;
- $3~i~j~x$ ($1 \le i < j \le n$; $2 \le x \le 2 \cdot k$): $a_i + a_j$ should be greater than or equal to $x$.
The sum of $n$ over all testcases doesn't exceed $2 \cdot 10^4$. The sum of $m$ over all testcases doesn't exceed $2 \cdot 10^4$.
For each testcase, determine if there exists a non-decreasing array that satisfies all conditions. If there is no such array, then print -1. Otherwise, print any valid array — $n$ integers from $1$ to $k$.
Input
The first line contains a single integer $t$ ($1 \le t \le 10^4$) — the number of testcases.
The first line of each testcase contains three integers $n, m$ and $k$ ($2 \le n \le 2 \cdot 10^4$; $0 \le m \le 2 \cdot 10^4$; $2 \le k \le 10$).
The $i$-th of the next $m$ lines contains a description of a constraint. Each constraint is of one of three following types:
- $1~i~x$ ($1 \le i \le n$; $1 \le x \le k$): $a_i$ should not be equal to $x$;
- $2~i~j~x$ ($1 \le i < j \le n$; $2 \le x \le 2 \cdot k$): $a_i + a_j$ should be less than or equal to $x$;
- $3~i~j~x$ ($1 \le i < j \le n$; $2 \le x \le 2 \cdot k$): $a_i + a_j$ should be greater than or equal to $x$.
The sum of $n$ over all testcases doesn't exceed $2 \cdot 10^4$. The sum of $m$ over all testcases doesn't exceed $2 \cdot 10^4$.
Output
For each testcase, determine if there exists a non-decreasing array that satisfies all conditions. If there is no such array, then print -1. Otherwise, print any valid array — $n$ integers from $1$ to $k$.
Samples
4
4 0 4
2 2 3
3 1 2 3
1 2 2
3 3 2
1 1 1
2 2 3 2
3 2 3 2
5 5 5
3 2 5 7
2 4 5 10
3 4 5 6
3 3 4 7
2 1 5 7
1 2 3 4
1 3
-1
1 2 2 5 5