Tree of Life (hard)

To add insult to injury, the zombies have taken all but two drawings from Heidi! Please help her recover the Tree of Life from only these two drawings.

Input

The input format is the same as in the medium version, except that now the bound on n is 2 ≤ n ≤ 1000 and that k = 2.

Output

The same as in the medium version.

Example

Input

1 9 2 6 4 3 5 4 6 1 8 6 8 2 7 1 5 8 6 8 7 8 1 7 3 5 1

Output

YES 2 1 3 1 5 4 6 5 7 6 8 5 9 8 1 4

inputFormat

Input

The input format is the same as in the medium version, except that now the bound on n is 2 ≤ n ≤ 1000 and that k = 2.

outputFormat

Output

The same as in the medium version.

Example

Input

1 9 2 6 4 3 5 4 6 1 8 6 8 2 7 1 5 8 6 8 7 8 1 7 3 5 1

Output

YES 2 1 3 1 5 4 6 5 7 6 8 5 9 8 1 4

样例

1
9 2
6
4 3
5 4
6 1
8 6
8 2
7 1
5
8 6
8 7
8 1
7 3
5 1

YES
2 1
3 1
4 1
5 4
6 5
7 6
8 5
9 8

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