Problem Statement

You are given a sequence A of length N.
Freely choose exactly K elements from A and remove them, then concatenate the remaining elements in their original order to form a new sequence B.
Find the minimum possible value of this: the maximum value of B minus the minimum value of B.

Constraints

• All inputs are integers.
• 1 \le K < N \le 2 \times 10^5
• 1 \le A_i \le 10^9

Sample Input 1

5 2
3 1 5 4 9

Sample Output 1

2

Consider removing exactly two elements from A=(3,1,5,4,9).

• For example, if you remove the 2nd element 1 and the 5th element 9, the resulting sequence is B=(3,5,4).
  • In this case, the maximum value of B is 5 and the minimum value is 3, so (maximum value of B) - (minimum value of B) =2, which is the minimum possible value.

Sample Input 2

6 5
1 1 1 1 1 1

Sample Output 2

0

Sample Input 3

8 3
31 43 26 6 18 36 22 13

Sample Output 3

18

Problem Statement

You are given a string S of length N consisting of lowercase English letters, ,, and ". It is guaranteed that S contains an even number of ".

Let 2K be the number of " in S. For each i=1,2,\ldots,K, the characters from the (2i-1)-th " through the (2i)-th " are said to be enclosed.

Your task is to replace each , in S that is not an enclosed character with . and print the resulting string.

Constraints

• N is an integer between 1 and 2\times 10^5, inclusive.
• S is a string of length N consisting of lowercase English letters, ,, and ".
• S contains an even number of ".

Sample Input 1

8
"a,b"c,d

Sample Output 1

"a,b"c.d

In S, "a,b" are enclosed characters, and c,d are not.

The , in S that is not an enclosed character is the seventh character from the left in S, so replace that character with . to get the answer.

Sample Input 2

5
,,,,,

Sample Output 2

.....

Sample Input 3

20
a,"t,"c,"o,"d,"e,"r,

Sample Output 3

a."t,"c."o,"d."e,"r.

Problem Statement

A superman, Takahashi, is about to jump off the roof of a building to help a person in trouble on the ground. Takahashi's planet has a constant value g that represents the strength of gravity, and the time it takes for him to reach the ground after starting to fall is \frac{A}{\sqrt{g}}.

It is now time 0, and g = 1. Takahashi will perform the following operation as many times as he wants (possibly zero).

• Use a superpower to increase the value of g by 1. This takes a time of B.

Then, he will jump off the building. After starting to fall, he cannot change the value of g. Additionally, we only consider the time it takes to perform the operation and fall.

Find the earliest time Takahashi can reach the ground.

Constraints

• 1 \leq A \leq 10^{18}
• 1 \leq B \leq 10^{18}
• All values in the input are integers.

Sample Input 1

10 1

Sample Output 1

7.7735026919

• If he performs the operation zero times, he will reach the ground at time 1\times 0+\frac{10}{\sqrt{1}} = 10.
• If he performs the operation once, he will reach the ground at time 1\times 1+\frac{10}{\sqrt{2}} \fallingdotseq 8.07.
• If he performs the operation twice, he will reach the ground at time 1\times 2+\frac{10}{\sqrt{3}} \fallingdotseq 7.77.
• If he performs the operation three times, he will reach the ground at time 1\times 3+\frac{10}{\sqrt{4}} = 8.

Performing the operation four or more times will only delay the time to reach the ground. Therefore, it is optimal to perform the operation twice before jumping off, and the answer is 2+\frac{10}{\sqrt{3}}.

Sample Input 2

5 10

Sample Output 2

5.0000000000

It is optimal not to perform the operation at all.

Sample Input 3

1000000000000000000 100

Sample Output 3

8772053214538.5976562500

Problem Statement

Takahashi and Aoki are playing a game using N cards. The front side of the i-th card has A_i written on it, and the back side has B_i written on it. Initially, the N cards are laid out on the table. With Takahashi going first, the two players take turns performing the following operation:

• Choose a pair of cards from the table such that either the numbers on their front sides are the same or the numbers on their back sides are the same, and remove these two cards from the table. If no such pair of cards exists, the player cannot perform the operation.

The player who is first to be unable to perform the operation loses, and the other player wins. Determine who wins if both players play optimally.

Constraints

• 1 \leq N \leq 18
• 1 \leq A_i, B_i \leq 10^9
• All input values are integers.

Sample Input 1

5
1 9
2 5
4 9
1 4
2 5

Sample Output 1

Aoki

If Takahashi first removes

• the first and third cards: Aoki can win by removing the second and fifth cards.

• the first and fourth cards: Aoki can win by removing the second and fifth cards.

• the second and fifth cards: Aoki can win by removing the first and third cards.

These are the only three pairs of cards Takahashi can remove in his first move, and Aoki can win in all cases. Therefore, the answer is Aoki.

Sample Input 2

9
3 2
1 7
4 1
1 8
5 2
9 8
2 1
6 8
5 2

Sample Output 2

Takahashi

Problem Statement

Snuke is planning to build a maze as a new attraction in AtCoder Land. The maze is represented as a grid with N rows and M columns, with the top edge of the top-right cell being the entrance and the bottom edge of the bottom-right cell being the exit. He will create the maze by appropriately placing walls between adjacent cells.

He loves simple mazes, so he wants the path from the entrance to the exit to pass through exactly K cells without any branches. Determine if it is possible to create such a maze, and if possible, construct one.

For example, in the following figure, N=3 and M=3, and walls are placed at the solid lines (walls are always placed around the outer perimeter except for the entrance and exit). In this case, the path from the entrance to the exit passes through exactly 7 cells without any branches.

Below is a formal statement.

There is a grid with N rows and M columns. Let (i, j) denote the cell at the i-th row from the top and j-th column from the left. For each pair of side-adjacent cells, you can decide whether to place a wall between them. Determine whether it is possible to place walls to satisfy the following condition, and if it is possible, construct one such placement.

Consider an undirected graph G with NM vertices. Each vertex of G is uniquely labeled by a pair of integers (i,j)\ (1\leq i\leq N, 1\leq j\leq M). Two distinct vertices (i_1,j_1) and (i_2,j_2) are connected by an edge if and only if |i_1-i_2|+|j_1-j_2|=1 and there is no wall between the corresponding cells (i_1,j_1) and (i_2,j_2) on the grid.

Condition: there exists a simple path with K vertices that connects the two vertices (1,M) and (N,M), and the connected component containing the vertices (1,M) and (N,M) consists only of this path.

Constraints

• 2\leq N \leq 100
• 1\leq M \leq 100
• 1\leq K\leq NM
• All input values are integers.

Sample Input 1

3 3 7

Sample Output 1

Yes
+++++S+
+o.o.o+
+.+-+-+
+o.o.o+
+-+-+.+
+o.o|o+
+++++G+

This is the same placement of walls as in the figure in the problem statement.

Sample Input 2

3 3 2

Sample Output 2

No

Sample Input 3

4 1 4

Sample Output 3

Yes
+S+
+o+
+.+
+o+
+.+
+o+
+.+
+o+
+G+