Problem Statement

There are N people numbered Person 1, Person 2, \dots, and Person N. Person i has a family name s_i and a given name t_i.

Consider giving a nickname to each of the N people. Person i's nickname a_i should satisfy all the conditions below.

• a_i coincides with Person i's family name or given name. In other words, a_i = s_i and/or a_i = t_i holds.
• a_i does not coincide with the family name and the given name of any other person. In other words, for all integer j such that 1 \leq j \leq N and i \neq j, it holds that a_i \neq s_j and a_i \neq t_j.

Is it possible to give nicknames to all the N people? If it is possible, print Yes; otherwise, print No.

Constraints

• 2 \leq N \leq 100
• N is an integer.
• s_i and t_i are strings of lengths between 1 and 10 (inclusive) consisting of lowercase English alphabets.

Sample Input 1

3
tanaka taro
tanaka jiro
suzuki hanako

Sample Output 1

Yes

The following assignment satisfies the conditions of nicknames described in the Problem Statement: a_1 = taro, a_2 = jiro, a_3 = hanako. (a_3 may be suzuki, too.)
However, note that we cannot let a_1 = tanaka, which violates the second condition of nicknames, since Person 2's family name s_2 is tanaka too.

Sample Input 2

3
aaa bbb
xxx aaa
bbb yyy

Sample Output 2

No

There is no way to give nicknames satisfying the conditions in the Problem Statement.

Sample Input 3

2
tanaka taro
tanaka taro

Sample Output 3

No

There may be a pair of people with the same family name and the same given name.

Sample Input 4

3
takahashi chokudai
aoki kensho
snu ke

Sample Output 4

Yes

We can let a_1 = chokudai, a_2 = kensho, and a_3 = ke.

Problem Statement

You are given a string S of length N consisting of lowercase English letters. The x-th (1 \le x \le N) character of S is S_x.

For each i=1,2,\dots,N-1, find the maximum non-negative integer l that satisfies all of the following conditions:

• l+i \le N, and
• for all integers k such that 1 \le k \le l, it holds that S_{k} \neq S_{k+i}.

Note that l=0 always satisfies the conditions.

Constraints

• N is an integer such that 2 \le N \le 5000.
• S is a string of length N consisting of lowercase English letters.

Sample Input 1

6
abcbac

Sample Output 1

5
1
2
0
1

In this input, S= abcbac.

• When i=1, we have S_1 \neq S_2, S_2 \neq S_3, \dots, and S_5 \neq S_6, so the maximum value is l=5.
• When i=2, we have S_1 \neq S_3 but S_2 = S_4, so the maximum value is l=1.
• When i=3, we have S_1 \neq S_4 and S_2 \neq S_5 but S_3 = S_6, so the maximum value is l=2.
• When i=4, we have S_1 = S_5, so the maximum value is l=0.
• When i=5, we have S_1 \neq S_6, so the maximum value is l=1.

Problem Statement

Takahashi and Aoki will play the following game against each other.

Starting from Takahashi, the two alternatingly declare an integer between 1 and 2N+1 (inclusive) until the game ends. Any integer declared by either player cannot be declared by either player again. The player who is no longer able to declare an integer loses; the player who didn't lose wins.

In this game, Takahashi will always win. Your task is to actually play the game on behalf of Takahashi and win the game.

Constraints

• 1 \leq N \leq 1000
• N is an integer.

Input and Output

This task is an interactive task (in which your program and the judge program interact with each other via inputs and outputs).
Your program plays the game on behalf of Takahashi, and the judge program plays the game on behalf of Aoki.

First, your program is given a positive integer N from Standard Input. Then, the following procedures are repeated until the game ends.

1. Your program outputs an integer between 1 and 2N+1 (inclusive) to Standard Output, which defines the integer that Takahashi declares. (You cannot output an integer that is already declared by either player.)
2. The integer that Aoki declares is given by the judge program to your program from Standard Input. (No integer that is already declared by either player will be given.) If Aoki has no more integer to declare, 0 is given instead, which means that the game ended and Takahashi won.

Notes

• After each output, you must flush Standard Output. Otherwise, you may get TLE.
• After the game ended and Takahashi won, the program must be terminated immediately. Otherwise, the judge does not necessarily give AC.
• If your program outputs something that violates the rules of the game (such as an integer that has already been declared by either player), your answer is considered incorrect. In such case, the verdict is indeterminate. It does not necessarily give WA.

Sample Input and Output

Problem Statement

You are given an integer X. The following action on this integer is called an operation.

• Choose and do one of the following.
  • Add 1 to X.
  • Subtract 1 from X.

The terms in the arithmetic progression S with N terms whose initial term is A and whose common difference is D are called good numbers.
Consider performing zero or more operations to make X a good number. Find the minimum number of operations required to do so.

Constraints

• All values in input are integers.
• -10^{18} \le X,A \le 10^{18}
• -10^6 \le D \le 10^6
• 1 \le N \le 10^{12}

Sample Input 1

6 2 3 3

Sample Output 1

1

Since A=2,D=3,N=3, we have S=(2,5,8).
You can subtract 1 from X once to make X=6 a good number.
It is impossible to make X good in zero operations.

Sample Input 2

0 0 0 1

Sample Output 2

0

We might have D=0. Additionally, no operation might be required.

Sample Input 3

998244353 -10 -20 30

Sample Output 3

998244363

Sample Input 4

-555555555555555555 -1000000000000000000 1000000 1000000000000

Sample Output 4

444445

Problem Statement

This is an interactive task, where your program and the judge interact via Standard Input and Output.

The judge has a string of length N consisting of 0 and 1: S = S_1S_2\ldots S_N. Here, S_1 = 0 and S_N = 1.

You are given the length N of S, but not the contents of S. Instead, you can ask the judge at most 20 questions as follows.

• Choose an integer i such that 1 \leq i \leq N and ask the value of S_i.

Print an integer p such that 1 \leq p \leq N-1 and S_p \neq S_{p+1}.
It can be shown that such p always exists under the settings of this problem.

Constraints

• 2 \leq N \leq 2 \times 10^5

Input and Output

First, receive the length N of the string S from Standard Input:

N

Then, you get to ask the judge at most 20 questions as described in the problem statement.

Print each question to Standard Output in the following format, where i is an integer satisfying 1 \leq i \leq N:

? i

In response to this, the value of S_i will be given from Standard Input in the following format:

S_i

Here, S_i is 0 or 1.

When you find an integer p satisfying the condition in the problem statement, print it in the following format, and immediately quit the program:

! p

If multiple solutions exist, you may print any of them.

Notes

• Print a newline and flush Standard Output at the end of each message. Otherwise, you may get a TLE verdict.
• If there is malformed output during the interaction or your program quits prematurely, the verdict will be indeterminate.
• After printing the answer, immediately quit the program. Otherwise, the verdict will be indeterminate.
• The string S will be fixed at the start of the interaction and will not be changed according to your questions or other factors.

Sample Input and Output

In the following interaction, N = 7 and S = 0010011.

For the presented p = 5, we have 1 \leq p \leq N-1 and S_p \neq S_{p+1}. Thus, if the program immediately quits here, this case will be judged as correctly solved.