PROGRAMMING:Mobius Ring
Does everyone know the Mobius Ring? The Mobius Ring is a structure that loops indefinitely. Here shows the picture of it:
Little Gyro is fond of the Mobius Ring. One day, Little Gyro found that Derrick was studying with his girlfriend, so he called another teammate Onlystar for his brand new game which was recently invented. And they were happily playing games on the Mobius Ring.
In this game, there are $$N$$ positions numbered from $$0$$ to $$N-1$$ arranged in a circle(in the clockwise order). Position numbered $$0$$ is a black hole and others are all empty. And there's a huge monster in one of the empty positions. Both Little Gyro and Onlystar don't know the initial position of the monster yet, they only know it's not at the black hole. Before the game starts, the system will inform them of the monster's position randomly. But now, they want to be well prepared for every possible scenario.
At the beginning of the game, each person has a set of numbers between $$1$$ and $$N-1$$ (inclusive). Little Gyro's set is named $$s_ 1$$ and contains $$k_ 1$$ elements while Onlystar's is named $$s_ 2$$ and contains $$k_ 2$$ elements. One of them goes first and the players change alternatively. In each player's turn, the player should choose an arbitrary number such as $$x$$ from his set and then the monster will move to his next $$x$$-th position from its current position(also in the clockwise order). If and only if after the move that the monster falls into the black hole, the game ends while the person who makes the last move wins the game.
Given two number sets $$s_ 1$$, $$s_ 2$$, for each player goes first, you should determine if the starter wins, loses, or the game will stuck in an infinite loop for every initial position of the monster from $$1$$ to $$N-1$$. Please note that, in case a player may lose the game or make the game go infinity, it's more profitable to control the game never end.
### Input Specification:
There are multiple test cases. The first line of the input contains an integer $$T$$, indicating the number of test cases. For each test case:
The first line contains a single integer $$N$$ (2 ≤ $$N$$ ≤ 7000), indicating the total number of positions in game.
The second line contains an integer $$k_ 1$$ followed by $$k_ 1$$ distinct integers $$s_{ 1,1}$$, $$s_{ 1,2}$$,..., $$s_{ 1,k_ 1}$$, indicating Little Gyro's set.
The third line contains an integer $$k_ 2$$ followed by $$k_ 2$$ distinct integers $$s_{ 2,1}$$, $$s_{ 2,2}$$,..., $$s_{ 2,k_ 2}$$, indicating Onlystar's set.
It's guaranteed that 1 ≤ $$k_ 1$$,$$k_ 2$$ ≤ $$N-1$$ and 1 ≤ $$s_{ i,1}$$, $$s_{ i,2}$$,..., $$s_{ i,k}$$ ≤ $$N-1$$, and in the huge data file, $$T$$ is exactly 1.
### Output Specification:
For each case, in the first line output $$N-1$$ words separated by a space where $$i$$-th word is "Win"(without quotes) if it's in the scenario that Little Gyro plays first and monster is initially at the position numbered $$i+1$$ as he wins, output "Lose"(without quotes) if he loses or "Loop"(without quotes) if the game will never end.
Similarly, in the second line print $$N-1$$ words separated by a space where $$i$$-th word is "Win"(without quotes) if it's in the scenario that Onlystar plays first and monster is initially at the position numbered $$i+1$$ he wins, "Lose"(without quotes) if he loses and "Loop"(without quotes) if the game will never end.
**Note: Please, DO NOT output extra spaces at the end of each line, or your answer may be considered incorrect!**
### Sample Input:
```in
two
five
2 3 2
3 1 2 3
eight
4 6 2 3 4
2 3 6
```
### Sample Output:
```out
Lose Win Win Loop
Loop Win Win Win
Win Win Win Win Win Win Win
Lose Win Lose Lose Win Lose Lose
```
answer：If there is no answer, please comment
First of all, define ans [x] [y] as the winning or losing situation when the monster is in position X and Y is in the first place, then ans [0] [0] and ans [0] [1] are the inevitable losing states
Secondly, consider the strategy of two people: if one step operation can make one's own ans [x] [y] reach the other's inevitable state ans [x '] [y'], then ans [x] [y] is the inevitable state, and only one's own operations will reach the other's inevitable state ans [x '] [y'], one's own ans [x] [y] is the inevitable state, and the rest are cyclic states.
Finally, the state transition is considered
For the inevitable state ans [x] [y] of Y, if there is an operation s of Y 'which can make X' + S = = x, then ans [x '] [y'] is the inevitable state;
For the winning state ans [x] [y] of Y, if there is an operation s of Y 'which can make X' + S = = x, then there is less possibility of one operation in ans [x '] [y']. We record it as an outdegree deg. when the outdegree of a point is equal to its own operands, the point is a must state;
Except for 0,0 and 0,1, all the points are initially set to circular state. By using queue operation, the points that are still in circular state after a BFS execution are the real circular state.
Little Gyro is fond of the Mobius Ring. One day, Little Gyro found that Derrick was studying with his girlfriend, so he called another teammate Onlystar for his brand new game which was recently invented. And they were happily playing games on the Mobius Ring.
In this game, there are $$N$$ positions numbered from $$0$$ to $$N-1$$ arranged in a circle(in the clockwise order). Position numbered $$0$$ is a black hole and others are all empty. And there's a huge monster in one of the empty positions. Both Little Gyro and Onlystar don't know the initial position of the monster yet, they only know it's not at the black hole. Before the game starts, the system will inform them of the monster's position randomly. But now, they want to be well prepared for every possible scenario.
At the beginning of the game, each person has a set of numbers between $$1$$ and $$N-1$$ (inclusive). Little Gyro's set is named $$s_ 1$$ and contains $$k_ 1$$ elements while Onlystar's is named $$s_ 2$$ and contains $$k_ 2$$ elements. One of them goes first and the players change alternatively. In each player's turn, the player should choose an arbitrary number such as $$x$$ from his set and then the monster will move to his next $$x$$-th position from its current position(also in the clockwise order). If and only if after the move that the monster falls into the black hole, the game ends while the person who makes the last move wins the game.
Given two number sets $$s_ 1$$, $$s_ 2$$, for each player goes first, you should determine if the starter wins, loses, or the game will stuck in an infinite loop for every initial position of the monster from $$1$$ to $$N-1$$. Please note that, in case a player may lose the game or make the game go infinity, it's more profitable to control the game never end.
### Input Specification:
There are multiple test cases. The first line of the input contains an integer $$T$$, indicating the number of test cases. For each test case:
The first line contains a single integer $$N$$ (2 ≤ $$N$$ ≤ 7000), indicating the total number of positions in game.
The second line contains an integer $$k_ 1$$ followed by $$k_ 1$$ distinct integers $$s_{ 1,1}$$, $$s_{ 1,2}$$,..., $$s_{ 1,k_ 1}$$, indicating Little Gyro's set.
The third line contains an integer $$k_ 2$$ followed by $$k_ 2$$ distinct integers $$s_{ 2,1}$$, $$s_{ 2,2}$$,..., $$s_{ 2,k_ 2}$$, indicating Onlystar's set.
It's guaranteed that 1 ≤ $$k_ 1$$,$$k_ 2$$ ≤ $$N-1$$ and 1 ≤ $$s_{ i,1}$$, $$s_{ i,2}$$,..., $$s_{ i,k}$$ ≤ $$N-1$$, and in the huge data file, $$T$$ is exactly 1.
### Output Specification:
For each case, in the first line output $$N-1$$ words separated by a space where $$i$$-th word is "Win"(without quotes) if it's in the scenario that Little Gyro plays first and monster is initially at the position numbered $$i+1$$ as he wins, output "Lose"(without quotes) if he loses or "Loop"(without quotes) if the game will never end.
Similarly, in the second line print $$N-1$$ words separated by a space where $$i$$-th word is "Win"(without quotes) if it's in the scenario that Onlystar plays first and monster is initially at the position numbered $$i+1$$ he wins, "Lose"(without quotes) if he loses and "Loop"(without quotes) if the game will never end.
**Note: Please, DO NOT output extra spaces at the end of each line, or your answer may be considered incorrect!**
### Sample Input:
```in
two
five
2 3 2
3 1 2 3
eight
4 6 2 3 4
2 3 6
```
### Sample Output:
```out
Lose Win Win Loop
Loop Win Win Win
Win Win Win Win Win Win Win
Lose Win Lose Lose Win Lose Lose
```
answer：If there is no answer, please comment
First of all, define ans [x] [y] as the winning or losing situation when the monster is in position X and Y is in the first place, then ans [0] [0] and ans [0] [1] are the inevitable losing states
Secondly, consider the strategy of two people: if one step operation can make one's own ans [x] [y] reach the other's inevitable state ans [x '] [y'], then ans [x] [y] is the inevitable state, and only one's own operations will reach the other's inevitable state ans [x '] [y'], one's own ans [x] [y] is the inevitable state, and the rest are cyclic states.
Finally, the state transition is considered
For the inevitable state ans [x] [y] of Y, if there is an operation s of Y 'which can make X' + S = = x, then ans [x '] [y'] is the inevitable state;
For the winning state ans [x] [y] of Y, if there is an operation s of Y 'which can make X' + S = = x, then there is less possibility of one operation in ans [x '] [y']. We record it as an outdegree deg. when the outdegree of a point is equal to its own operands, the point is a must state;
Except for 0,0 and 0,1, all the points are initially set to circular state. By using queue operation, the points that are still in circular state after a BFS execution are the real circular state.