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## 문제

JOI Kingdom is an island surrounded by sea. The land of JOI Kingdom is a grid of square cells consisting of N rows and N columns. The vertical direction is the north-south direction, and the horizontal direction is the west-east direction. The cell in the (r + 1)-th row (0 ≤ r ≤ N − 1) from north and the (c + 1)-th column (0 ≤ c ≤ N − 1) from west is denoted by the cell (r, c).

Anna, the queen of JOI Kingdom, will invite Bruno to a party. She is now choosing the place for the party. She already selected K (= 7) candidate cells of the place for the party. The candidate cells are numbered from 0 through K − 1. The candidate cell i is the cell (Ri, Ci). No candidate cell is adjacent to sea.

The place for the party will be determined on the day of the party.

On the day before the party, in order to help Bruno arrive at the place for the party without getting lost, Anna will place a flag in every cell. On each flag, Anna will write an integer between 1 and 1 000 000 000, inclusive.

On the day of the party, only the index t (0 ≤ t ≤ K − 1) of the candidate cell will be told to Bruno. After that, Bruno will take a helicopter and arrive at a cell which is not adjacent to sea. Then Bruno will start moving to the place for the party.

Bruno does not know where he is, but he knows the directions of north, south, east, and west. Bruno can only see a flag in his current cell and its 8 surrounding cells. In other words, when Bruno is at the cell (a, b) (1 ≤ a ≤ N − 2, 1 ≤ b ≤ N − 2), he can only see the flags in the following 9 cells:

(a − 1, b − 1), (a − 1, b), (a − 1, b + 1), (a, b − 1), (a, b), (a, b + 1), (a + 1, b − 1), (a + 1, b), (a + 1, b + 1)

Bruno can take one of the following 5 actions.

• Action 0: Bruno moves one cell to the east. Namely, he moves from the cell (a, b) to the cell (a, b + 1).
• Action 1: Bruno moves one cell to the west. Namely, he moves from the cell (a, b) to the cell (a, b − 1).
• Action 2: Bruno moves one cell to the south. Namely, he moves from the cell (a, b) to the cell (a + 1, b).
• Action 3: Bruno moves one cell to the north. Namely, he moves from the cell (a, b) to the cell (a − 1, b).
• Action 4: Bruno thinks the party will be held at his current cell, and stays there. He stops moving.

Since it is forbidden to arrive at the party late, Bruno has to move to the place for the party so that the number of actions is the minimum possible. Thus, by the setting of this task, Bruno will never enter into a cell adjacent to sea.

As it is tiresome to write large integers on flags, Anna wants to minimize the maximum integer written on the flags.

Write a program which implements Anna’s strategy and Bruno’s strategy. Anna will write integers on the flags, and Bruno has to arrive at the place for the party with minimum number of actions.

## 구현

You need to submit two files.

The first file is Anna.cpp. It should implement Anna’s strategy. It should implement the following function. The program should include Anna.h using the preprocessing directive #include.

• void Anna(int N, int K, std::vector<int> R, std::vector<int> C) This function implements Anna’s strategy to write integers on the flags. For each scenario (see Scoring), this function is called exactly once in the beginning.
• The parameter N means the land of JOI Kingdom is a grid of square cells consisting of N rows and N columns.
• The parameter K is the number K (= 7) of the candidate cells of the place for the party.
• The parameters R and C are arrays of length K. Here R[i] and C[i] (0 ≤ i ≤ K − 1) denote the cell (Ri, Ci) of the candidate cell i.
• For the range of the values of the parameters N, K, R[i] and C[i] (0 ≤ i ≤ K − 1) see Constraints.

For each function call to Anna, the following function should be called exactly N2 times. It should be called once for every cell.

• void SetFlag(int r, int c, int value)
• The parameters r and c mean Anna writes an integer on the flag at the cell (r, c). Here 0 ≤ r ≤ N−1, 0 ≤ c ≤ N−1 should be satisfied. If this condition is not satisfied, your program is judged as Wrong Answer [1].
• The parameter value is the integer Anna writes on the flag. Here 1 ≤ value ≤ 1 000 000 000 should be satisfied. If this condition is not satisfied, your program is judged as Wrong Answer [2].
• If the function SetFlag is called with the same parameters (r, c) more than once, your program is judged as Wrong Answer [3].
• When the function Anna terminates, if the number of function calls to the function SetFlag is different from N2, your program is judged as Wrong Answer [4].

When a function call to the function SetFlag is considered as a Wrong Answer, your program is terminated immediately

The second file is Bruno.cpp. It should implement Bruno’s strategy. It should implement the following function. The program should include Bruno.h using the preprocessing directive #include.

• std::vector<int> Bruno(int K, std::vector<int> value) This function should describe Bruno’s next action. For each scenario (see Scoring), after the function Anna is called, this function is called exactly once.
• The parameter K is the number of candidate cells K(= 7).
• The parameter value is an array of length 9. It contains the integers written on the flags at Bruno’s current cell and its 8 surrounding cells. More precisely, if Bruno is currently at the cell (a, b) (1 ≤ a ≤ N − 2, 1 ≤ b ≤ N − 2), the values of value[0], value[1], . . . , value[8] are the integers written on the flags in the cells (a−1, b−1), (a−1, b), (a−1, b+1), (a, b−1), (a, b), (a, b+1), (a+1, b−1), (a+1, b), (a+1, b+1), respectively.
• For every t = 0, 1, 2, . . . , K − 1, the function Bruno should determine Bruno’s next action when the party will be held at the candidate cell t. The return value is an array of length K. The (i + 1)-th element (0 ≤ i ≤ K − 1) of the array should be Bruno’s next action for t = i.
• If the return value is not an array of length K, your program is judged as Wrong Answer [5].
• Every element in the array should be one of 0, 1, 2, 3, or 4. If this condition is not satisfied, your program is judged as Wrong Answer [6].
• For every t, the action given by the function Bruno should be Bruno’s next action so that he will move to the place of the party with minimum number of actions. In particular, if his next action is Action 4, his current cell should be the place for the party. If this condition is not satisfied, your program is judged as Wrong Answer [7]. If there are multiple ways to move to the place for the party with the minimum number of actions, the return value can be any one of them.

In this task, each test case consists of Q scenarios. For each scenario, each of the function Anna and the function Bruno is called exactly once. Therefore, for each test case, each of the function Anna and the function Bruno is called Q times. These functions are called alternately. See Scoring for details.

## 점수

Each test case consists of Q scenarios. The scenarios are numbered from 0 through Q − 1. For each scenario, the following values are fixed. For the range of these values, see Constraints.

• The vertical and the horizontal size N of JOI Kingdom.
• The number of candidate cells K(= 7).
• The candidate cells (R0, C0), (R1, C1), . . . , (RK−1, CK−1) of the place for the party.
• Bruno’s current cell (a, b).

The function Anna is called for each scenario. For given parameters, Anna should write integers on flags. The function Bruno is also called for each scenario. It should determine Bruno’s next action. The function Anna and the function Bruno are called in the following way.

1. For each k = 0, 1, 2, . . . , Q − 1, in order, the following 2. and 3. are performed, in this order.
2. The function Anna is called. The parameters for the scenario k are given as in Implementation Details.
3. The function Bruno is called. The parameters for the scenario k are given as in Implementation Details.

If your program is judged as Wrong Answer during this process, your program is terminated immediately, and it is considered as a Wrong Answer for the test case.

If your program is judged as Wrong Answer in any one of the test cases, your score for this task is 0 point.

If your program is judged as correct for every test case, your score for this task is calculated as follows. Let L be the maximum integer written on the flags among all test cases.

• If 70 001 ≤ L ≤ 1 000 000 000, your score is 7 points.
• If 10 001 ≤ L ≤ 70 000, your score is 13 points.
• If 2 001 ≤ L ≤ 10 000, your score is 19 points.
• If 21 ≤ L ≤ 2 000, your score is 50 − 12.5 × log10(L/20) points, rounded down to the nearest integer.

If L ≤ 20, your score is given by the following table.

 L Score 20 19 18 17 16 15 14 13 ≤ 12 50 53 56 60 64 69 75 85 100

## 입력

The sample grader reads the following data from the standard input. Given values are all integers.

Q
(Input for scenario 0)
.
.
.
(Input for scenario Q − 1)

The input for each scenario is given as follows.

N K
R0 C0
.
.
.
RK−1 CK−1
a b

As the input for the sample grader, you can set the value of N in the range 3 ≤ N ≤ 100, and the value of K in the range 1 ≤ K ≤ 7. Note that these ranges are different from the actual constraints of this problem.

## 출력

When the program terminates successfully, the sample grader writes the following information to the standard output (quotes for clarity).

• If the answer is correct, it writes the maximum number written on the flags by the function Anna as “Accepted : Maximum value = 12’.
• If your program is judged as Wrong Answer, it writes its type as “Wrong Answer [1]”.

If your program is judged as multiple types of Wrong Answer, the sample grader reports only one of them.

## 제한

• 1 ≤ Q ≤ 300.
• 5 ≤ N ≤ 100.
• K = 7.
• 1 ≤ Ri ≤ N − 2 (0 ≤ i ≤ K − 1).
• 1 ≤ Ci ≤ N − 2 (0 ≤ i ≤ K − 1).
• (Ri, Ci) ≠ (Rj, Cj) (0 ≤ i < j ≤ K − 1).
• 1 ≤ a ≤ N − 2.
• 1 ≤ b ≤ N − 2.

## 예제

Here is a sample input for the sample grader and corresponding function calls. In the following example, the integers written by Anna on the 25 flags are as in the following table.

Sample Input 1 Integers written by Anna on the flags
1
5 7
1 1
1 2
2 1
2 2
2 3
3 2
3 3
1 1
47 15 63 56 71
10 46 52 18 67
63 56 71 19 48
52 18 67 99 26
71 19 48 60 89
Call to Anna Call to Bruno Return value
Anna(5,7,[1,1,2,...,3],[1,2,1,...,3])
SetFlag(0,0,47)
SetFlag(0,1,15)
SetFlag(0,2,63)
...
SetFlag(4,4,89)
Bruno(7,[47,15,63,...,71]) [4,0,2,2,2,0,0]

In this sample input, (a, b) = (1, 1). When the party is held at the candidate cell 0, 1, 2, or 3, Bruno’s next action should be as follows, and the function Bruno should return the actions accordingly.

• If the candidate 0 is chosen, the place of the party will be the cell (1, 1), and Bruno has to take Action 4.
• If the candidate 1 is chosen, the place of the party will be the cell (1, 2), and Bruno has to take Action 0.
• If the candidate 2 is chosen, the place of the party will be the cell (2, 1), and Bruno has to take Action 2.
• If the candidate 3 is chosen, the place of the party will be the cell (2, 2), and Bruno has to take either Action 0 or Action 2.

In this sample communication, the return values are [4,0,2,2,2,0,0]. There can be multiple ways to move to the place for the party with the minimum number of actions. For example, your program is judged as correct as well if the return values are [4,0,2,0,2,0,2].

Sample Input 2
1
100 7
3 21
16 9
44 36
44 78
45 78
67 59
90 22
84 59

In this sample input, your program is judged as correct if the return values of the function Bruno are [3,1,1,0,0,3,2], for example.

## 제출할 수 있는 언어

C++17, C++14, C++20, C++14 (Clang), C++17 (Clang), C++20 (Clang)

## 채점 및 기타 정보

• 100점 이상을 획득해야 를 받는다.