Jumbled Trees

题意翻译

你在一个无向图中选择若干生成树，并给每棵生成树的所有边的边权增加一个值 $v$，使得每条边的边权都能够达到它的目标值 $x_i$（对质数 $p$ 取模）。

题目描述

You are given an undirected connected graph with $ n $ vertices and $ m $ edges. Each edge has an associated counter, initially equal to $ 0 $ . In one operation, you can choose an arbitrary spanning tree and add any value $ v $ to all edges of this spanning tree. Determine if it's possible to make every counter equal to its target value $ x_i $ modulo prime $ p $ , and provide a sequence of operations that achieves it.

输入输出格式

输入格式

The first line contains three integers $ n $ , $ m $ , and $ p $ — the number of vertices, the number of edges, and the prime modulus ( $ 1 \le n \le 500 $ ; $ 1 \le m \le 1000 $ ; $ 2 \le p \le 10^9 $ , $ p $ is prime). Next $ m $ lines contain three integers $ u_i $ , $ v_i $ , $ x_i $ each — the two endpoints of the $ i $ -th edge and the target value of that edge's counter ( $ 1 \le u_i, v_i \le n $ ; $ 0 \le x_i < p $ ; $ u_i \neq v_i $ ). The graph is connected. There are no loops, but there may be multiple edges between the same two vertices.

输出格式

If the target values on counters cannot be achieved, print -1. Otherwise, print $ t $ — the number of operations, followed by $ t $ lines, describing the sequence of operations. Each line starts with integer $ v $ ( $ 0 \le v < p $ ) — the counter increment for this operation. Then, in the same line, followed by $ n - 1 $ integers $ e_1 $ , $ e_2 $ , ... $ e_{n - 1} $ ( $ 1 \le e_i \le m $ ) — the edges of the spanning tree. The number of operations $ t $ should not exceed $ 2m $ . You don't need to minimize $ t $ . Any correct answer within the $ 2m $ bound is accepted. You are allowed to repeat spanning trees.

输入输出样例

输入样例 #1

3 3 101
1 2 30
2 3 40
3 1 50

输出样例 #1

3
10 1 2
20 1 3
30 2 3

输入样例 #2

2 2 37
1 2 8
1 2 15

输出样例 #2

2
8 1
15 2

输入样例 #3

5 4 5
1 3 1
2 3 2
2 5 3
4 1 4

输出样例 #3

-1