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문제

마법사 이재환은 V개의 방과 V-1개의 문으로 이루어진 미로에 갇혀있다. 문은 방을 양방향으로 연결하고 있고, 한 방에서 다른 방으로 가는 경로는 항상 존재한다. 또, C개의 서로 다른 색상으로 이루어진 C개의 자물쇠와 C개의 열쇠가 있다. 문은 최대 한 개의 자물쇠로 잠겨있을 수 있고, 방에는 최대 한 개의 열쇠가 놓여져 있다.

재환이는 마법사이기 때문에, 자물쇠를 마법으로 풀면 된다. 하지만, 주문책을 놓고 왔기 때문에, 마법을 전혀 할 수 없다. 

재환이는 현재 방 X에 있고, 방 Y에 있는 주문북을 찾으러 가려고 한다. 재환이는 현재 있는 방과 인접한 방으로 문을 통해 한 번에 이동할 수 있다. 문이 잠겨있는 경우에는 자물쇠의 색상과 같은 색상의 열쇠가 있어야 문을 열 수 있다.

재환이는 한 번에 열쇠 하나만 들고 있을 수 있다. 또, 열쇠를 집은 이후에 나중에 다시 사용하기 위해서 열쇠를 다른 방에 내려 놓을 수 없다. 자물쇠로 잠겨있는 문을 열게 되면, 열쇠는 사라지게 된다.

미로와 열쇠 C개와 자물쇠 C개의 위치가 주어진다. 이때, X에서 Y로 이동하는 것이 가능한지 알아내는 프로그램을 작성하시오. 경로의 길이는 4·(C+1)·V를 넘을 수 없다.

입력

입력은 여러 개의 테스트 케이스로 이루어져 있다.

첫째 줄에는 네 정수가 주어진다. 방의 개수 V (1 ≤ V ≤ 1500), 자물쇠의 수 (0 ≤ C < V), 방 번호 X와 Y가 주어진다. 방 번호는 0번부터 V-1가지 번호가 매겨져 있다. 다음 C개 줄에는 색의 번호가 증가하는 순서대로 키가 들어있는 방의 번호가 주어진다. 그 다음 V-1개 줄에는 미로의 정보 A B L이 주어진다. 0 ≤ L < C인 경우에는 방 A와 B를 연결하는 문을 잠구는 자물쇠의 색은 L이라는 뜻이고, -1인 경우에는 잠겨있지 않다는 뜻이다.

입력의 마지막 줄은 V,C,X,Y = 0,0,0,0 이다.

출력

각 테스트 케이스마다 한 줄씩 출력한다. 답이 없는 경우에는 Impossible을 출력한다. 그렇지 않으면 L: V0 ... VL 형식으로 경로를 출력한다. L ≤ 4(C+1)V 은 X에서 Y로 가는 경로이고, X=V0,V1,...,VL-1,VL = Y는 X에서 Y로 가기 위해 지나가야 하는 방의 번호 L+1개이다. 각 정점을 출력하기 전에 공백을 하나 출력해야 한다.

예제 입력 1

1 0 0 0

3 1 0 2
1
0 1 -1
0 2 0

3 2 0 2
1 2
0 1 1
0 2 0

5 3 0 4
2 0 3
0 1 0
0 2 -1
1 3 1
2 4 2

0 0 0 0

예제 출력 1

0: 0
3: 0 1 0 2
Impossible
10: 0 2 0 1 0 1 3 1 0 2 4
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V<sub>L<\/sub> \ud615\uc2dd\uc73c\ub85c \uacbd\ub85c\ub97c \ucd9c\ub825\ud55c\ub2e4. L &le; 4(C+1)V \uc740 X\uc5d0\uc11c Y\ub85c \uac00\ub294 \uacbd\ub85c\uc774\uace0, X=V<sub>0<\/sub>,V<sub>1<\/sub>,...,V<sub>L-1<\/sub>,V<sub>L<\/sub> = Y\ub294 X\uc5d0\uc11c Y\ub85c \uac00\uae30 \uc704\ud574 \uc9c0\ub098\uac00\uc57c \ud558\ub294 \ubc29\uc758 \ubc88\ud638 L+1\uac1c\uc774\ub2e4. \uac01 \uc815\uc810\uc744 \ucd9c\ub825\ud558\uae30 \uc804\uc5d0 \uacf5\ubc31\uc744 \ud558\ub098 \ucd9c\ub825\ud574\uc57c \ud55c\ub2e4.<\/p>\r\n","hint":"","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"3853","problem_lang":"1","title":"Locks and keys","description":"<p>A wizard is in a labyrinth where there are V rooms and V - 1 doors connecting some pairs of rooms in both directions, in such a way that there is always a sequence of doors one can traverse to go from a room to any other room. Additionally, there are C locks and C keys of C di erent colours (one of each) in some of the doors and rooms of the maze, respectively; each door has at most one lock, and there is at most one key placed in each room. It should be an easy matter for the wizard to bypass the lock system, were it not for the fact that he forgot his spell book, without which his wizardry is utterly useless. The wizard is currently in room X, and he wants to get his spell book, located in room Y , without taking too long. At every step he may go to an adjacent room through one of the doors. Of course, if the door is locked, he needs to be carrying the key of the same colour as the lock (unless, of course, that door has already been unlocked). The wizard can carry only one key at a time and after picking up a key it is not possible for him to drop it somewhere in the maze in order to take it again afterwards. Once a door has been unlocked, the key is thrown away since it is no longer any use.<\/p>\r\n\r\n<p>Given the maze and the positions of the C keys and C locks, determine how to reach Y from X, if possible. Any path whose length does not exceed 4 &middot; (C + 1) &middot; V is acceptable.<\/p>\r\n","input":"<p>The first line of each case contains four integers: the number V of rooms in the maze (1 &le; V &le; 1 500), the number C of locks (0 &le; C &lt; V ), and rooms X and Y numbered 0,1, ..., V - 1. Then comes a (possibly empty) line with C integers indicating the location of each of the keys, in order of increasing colour. The next V - 1 lines describe the maze: each contains three integers A B L, meaning that there is a door between rooms A and B which can be unlocked with the key of colour L, if 0 &le; L &lt; C, a value of 1 for L indicates that no lock is needed.<\/p>\r\n\r\n<p>The last line has V, C, X, Y = 0,0,0,0.<\/p>\r\n","output":"<p>There is one line of output per test case. If there is no solution, output Impossible. Otherwise print the full path corresponding to your solution in the format L: V<sub>0<\/sub> ... V<sub>L<\/sub>, where L &le; 4(C + 1)V is the length of a path from X to Y, and X = V<sub>0<\/sub>, V<sub>1<\/sub>, ..., V<sub>L-1<\/sub>, VL = Y is the sequence of L + 1 vertices visited in the right order. A single space must precede each vertex in the path; see sample output for clari cation.<\/p>\r\n","hint":"","original":"1","problem_lang_code":"\uc601\uc5b4"}]