Background

Special for beginners, ^_^

Description

This problem is based on the game of Black Vienna. In this version there are three players and 18 cards labeled A-R. Three of the cards are set aside (hidden) and form the "Black Vienna" gang. The remaining cards are shuffled and dealt to the players so that each player has 5 cards. Players never reveal their cards to each other. There is a separate deck of "interrogation cards" which contain three distinct letters in ascending order, like ACG or BHR. Turns rotate through players 1, 2, and 3. On each player's turn, that player selects an interrogation card, puts it face up in front of another player, and that other player must indicate the total number of these cards being held, without saying which ones. All players see the result of the "interrogation". The play continues until a player deduces the three cards in the "gang". For example, suppose the cards are distributed as follows, and the game then proceeds:

Player 1: DGJLP; Player 2: EFOQR; Player 3: ACHMN; Gang: BIK

Turn 1: Player 1 interrogates player 2 with BJK; answer 0 Turn 2: Player 2 interrogates player 3 with ABK; answer 1 Turn 3: Player 3 interrogates player 2 with DEF; answer 2 Turn 4: Player 1 interrogates player 2 with EIL; answer 1 Turn 5: Player 2 interrogates player 3 with FIP; answer 0 Turn 6: Player 3 interrogates player 1 with GMO; answer 1 Turn 7: Player 1 interrogates player 2 with OQR; answer 3 Turn 8: Player 2 interrogates player 3 with ADQ; answer 1 Turn 9: Player 3 interrogates player 1 with EGJ; answer 2

In fact, the game does not need to get to turn 9. With enough thought, player 1 can deduce after turn 8 that the gang is BIK. It is your job to analyse records of games and deduce the earliest time that the gang could be determined for sure.

Black Vienna游戏是当前很流行的桌面游戏，规则是这样的：有18张牌，分别标着A-R，游戏有三个人。游戏开始的时候每个人拿5张牌，剩下三张牌隐藏起来，大家都只能看见自己手中的牌。之后，从第一个人到第三个人轮流拿一张“疑问牌”去问另外的一个人，“疑问牌”上有三个字母，那个人必须诚实地回答自己手中有多少疑问牌中的字母。哪个人能够最先推断出隐藏起来的牌是什么，那个人就取得胜利。现在你的任务是，给出三个人手上的牌和询问情况，你需要计算出最早在哪一次询问之后，有人能够推断出隐藏的牌。

Format

Input

The input will consist of one to twelve data sets, followed by a line containing only 0.The first line of a dataset contains the number, t, of turns reported, 2 ≤ t ≤ 15.The next line contains four blank separated strings for the hands of players 1, 2, and 3, followed by the cards for the gang.The remaining t lines of the data set contain the data for each turn in order. Each line contains three blank separated tokens: the number of the player interrogated, the string of interrogation letters, and the answer provided.All letter strings will contain only capital letters from A to R, in strictly increasing alphabetical order. The same interrogation string may appear in more than one turn of a game.

Output

There is one line of output for each data set. The line contains the single character "?" if no player can be sure of the gang after all the turns listed. If a player can determine the gang, the line contains the earliest turn after which one or more players can be sure of the answer.

Samples

9
DGJLP EFOQR ACHMN BIK
2 BJK 0
3 ABK 1
2 DEF 2
2 EIL 1
3 FIP 0
1 GMO 1
2 OQR 3
3 ADQ 1
1 EGJ 2
3
ABCDE FGHIJ KLMNO PQR
3 BKQ 1
1 ADE 3
2 CHJ 2
0

8
?

Limitation

对于样例一的解释

               从第一个人角度看：

1组: 2 BJK 0

第二人：无DGJLPBJK

第三人：无DGJLP

2组: 3 ABK 1

第二人：无DGJLPBJK

第三人：无DGJLP ABK中有一个

3组: 2 DEF 2

第二人：无DGJLPBJK 有EF

第三人：无DGJLPEF ABK中有一个

4组: 2 EIL 1

第二人：无DGJLPBJKIL 有EF

第三人：无DGJLP EF ABK中有一个

5组: 3 FIP 0

第二人：无DGJLPBJKIL 有EF

第三人：无DGJLP EFIP ABK中有一个

6组: 1 GMO 1 （对自己没有利用价值）

7组：2 OQR 3

第二人：无DGJLPBJKIL 有EFOQR

第三人：无DGJLP EFIPOQR ABK中有一个

8组：3 ADQ 1

第二人：无DGJLPBJKIL 有EFOQR

第三人：无DGJLP EFIPOQR [ ABK中有一个 ADK中有一个]

由括号部分可知：

第二人：无DGJLPBJKIL 有EFOQR

第三人：无DGJLP EFIPOQRBK 有A

此时，第一人知道自己和2,3都无BIK，所以隐藏牌为BIK。

（2,3人视角同上，前8步无法得出结论）