Description

Translated from Chinese version of ROIR 2021 Day1 T4 Антенна on this website.

There are $n$ ropes, among which the $i$-th rope is $s_i$ centimeter long. Besides, $m_i$ nodes are on that rope, and the $j$-th node is $p_{i,~j}$ centimeter away from the leftmost side of the rope.

Your task is to find a solution to connect ropes from left to right satisfying that all distances between adjacent nodes equals after connecting $q_i$-th and $q_{i+1}$-th ropes, whereas $q$ represents the solution and would obviously be a permutation of $1\sim n$.

Output No if such solution does not exist.

Input Format

The first line will be an integer $n$.

In the following $2n$ lines:

• The $(2i)$-th line would be two integers, $m_i$ and $s_i$
• The $(2i+1)$-th line would be $m_i$ integers, respectively $p_{i,j}$

Note that the value of $i$ above satisfies $1\le i\le n$.

Output Format

If the solution could be found, output Yes in the first line and $n$ integers $q_i$ in the following line.

Otherwise, output No.

3
1 7
3
1 8
6
2 8
1 6

Yes
2 1 3

1
1 7
5

Yes
1

1
3 10
2 5 9

No

3
1 5
3
1 3
3
1 6
3

No

4
1 5
0
1 0
0
1 3
3
1 0
0

Yes
3 2 4 1

Constraints

It is guaranteed that for all subtasks, there are $1\le n\le 10^5$, $1\le m_i\le 10^5$, $0\le s_i\le 10^9$, $0\le p_{i,1}<p_{i,2}<\cdots<p_{i,m_i}\le s_i$, $\sum m_i\le 10^5$.