Mathtermind is a game where you try to guess ~3~ unique numbers (the winning combination) chosen by the game out of a set of numbers from ~1~ to ~15~. There are at most ~7~ rounds played. For each of these rounds, you guess ~1~ to ~4~ unique numbers. The game will then tell you that ~M~ numbers of your guess match its winning combination.

After guessing ~R~ rounds, you would like to know your chances of winning. Given the current game state, if a winning combination exists and you can determine what it is with ~100\%~ certainty, output that combination in sorted order. If a winning combination exists but you can't determine what it is with ~100\%~ certainty then output the number of possible winning combinations. Lastly, if a winning combination does not exist output -1.

Input Specification

The first line of the input will contain an integer ~R~ ~(1 \le R \le 7)~, the number of rounds.

The next ~2R~ lines will alternate between the following:

A line containing ~N~ ~(1 \le N \le 4)~, the number of guesses, and ~M~ ~(0 \le M \le 3)~, the number of those guesses that match the winning combination.

A line containing ~N~ integers ~G_i~ ~(1 \le G_i \le 15)~, the numbers of your guess.

Output Specification

Output either the winning combination in sorted order, the number of possible winning combinations, or -1 depending on the input.

Sample Input 1

7
4 2
1 2 3 4
3 1
2 7 8
4 0
9 10 11 12
3 0
13 14 15
3 2
3 7 8
2 0
4 8
1 1
1

Sample Output 1

1 3 7

Explanation for Sample Output 1

The number of numbers in your guess that match the winning combination is denoted by >>.

#1: 1, 2, 3, 4
>> 2
#2: 2, 7, 8
>> 1
#3: 9, 10, 11, 12
>> 0
#4: 13, 14, 15
>> 0
#5: 3, 7, 8
>> 2
#6: 4, 8
>> 0
#7: 1
>> 1

For some guesses, the amount of numbers that match the winning combination is ~0~, meaning that those numbers can be eliminated, so the numbers ~4~, ~8~, ~9~, ~10~, ~11~, ~12~, ~13~, ~14~, and ~15~ are not part of the winning combination.

#1: 1, 2, 3
>> 2
#2: 2, 7
>> 1
#5: 3, 7
>> 2
#7: 1
>> 1

By using guesses #~5~ and #~7~, you can figure out that the winning combination is 1, 3, 7.

Sample Input 2

4
4 2
1 3 5 7
3 1
9 10 14
3 1
1 3 9
1 0
7

Sample Output 2

4

Explanation for Sample Output 2

The possible winning combinations are 1, 5, 10, 1, 5, 14, 3, 5, 10, and 3, 5, 14.

Sample Input 3

3
4 3
1 2 3 4
2 2
5 6
1 2
8

Sample Output 3

-1

Explanation for Sample Output 3

These ~3~ guesses imply that ~6~ numbers should appear in the winning combination with ~8~ appearing twice. This is not true because the winning combination can only consist of ~3~ distinct numbers from ~1~ to ~15~.