Problem Statement

You are given a string D representing one of the eight directions (north, east, west, south, northeast, northwest, southeast, southwest). The correspondence between the directions and their representing strings is as follows.

• North: N
• East: E
• West: W
• South: S
• Northeast: NE
• Northwest: NW
• Southeast: SE
• Southwest: SW

Print the string representing the direction opposite to the direction denoted by D.

Constraints

• D is one of N, E, W, S, NE, NW, SE, SW.

Sample Input 1

N

Sample Output 1

S

Print S, which represents south, the direction opposite to north.

Sample Input 2

SE

Sample Output 2

NW

Print NW, which represents northwest, the direction opposite to southeast.

Problem Statement

There is a stack of N cards, and the i-th card from the top has an integer A_i written on it.

You take K cards from the bottom of the stack and place them on top of the stack, maintaining their order.

Print the integers written on the cards from top to bottom after the operation.

Constraints

• 1 \leq K < N \leq 100
• 1 \leq A_i \leq 100
• All input values are integers.

Sample Input 1

5 3
1 2 3 4 5

Sample Output 1

3 4 5 1 2

Initially, the integers written on the cards are 1,2,3,4,5 from top to bottom.

After taking three cards from the bottom of the stack and placing them on top, the integers written on the cards become 3,4,5,1,2 from top to bottom.

Sample Input 2

6 2
1 2 1 2 1 2

Sample Output 2

1 2 1 2 1 2

The integers written on the cards are not necessarily distinct.

Problem Statement

You are given N-by-N matrices A and B where each element is 0 or 1.
Let A_{i,j} and B_{i,j} denote the element at the i-th row and j-th column of A and B, respectively.
Determine whether it is possible to rotate A so that B_{i,j} = 1 for every pair of integers (i, j) such that A_{i,j} = 1.
Here, to rotate A is to perform the following operation zero or more times:

• for every pair of integers (i, j) such that 1 \leq i, j \leq N, simultaneously replace A_{i,j} with A_{N + 1 - j,i}.

Constraints

• 1 \leq N \leq 100
• Each element of A and B is 0 or 1.
• All values in the input are integers.

Sample Input 1

3
0 1 1
1 0 0
0 1 0
1 1 0
0 0 1
1 1 1

Sample Output 1

Yes

Initially, A is :

0 1 1
1 0 0
0 1 0

After performing the operation once, A is :

0 1 0
1 0 1 
0 0 1

After performing the operation once again, A is :

0 1 0
0 0 1
1 1 0

Here, B_{i,j} = 1 for every pair of integers (i, j) such that A_{i,j} = 1, so you should print Yes.

Sample Input 2

2
0 0
0 0
1 1
1 1

Sample Output 2

Yes

Sample Input 3

5
0 0 1 1 0
1 0 0 1 0
0 0 1 0 1
0 1 0 1 0
0 1 0 0 1
1 1 0 0 1
0 1 1 1 0
0 0 1 1 1
1 0 1 0 1
1 1 0 1 0

Sample Output 3

No

Problem Statement

You are given an odd number N that is at least 3.

There is a grid with N rows and N columns, where all cells are initially empty. Now, you will write integers in each cell of this grid according to the following procedure. Let (i,j) denote the cell at the (i+1)-th row from the top and (j+1)-th column from the left (0\leq i<N, 0\leq j<N).

1. Write 1 in cell (0,\frac{N-1}{2}).
2. Repeat the following operation N^2-1 times:
   • Let (r,c) be the cell where an integer was written last time, and k be the integer written. If cell ((r-1) \bmod N, (c+1) \bmod N) is empty, write k+1 in that cell; otherwise, write k+1 in cell ((r+1) \bmod N,c). Here, x \bmod N denotes the remainder when x is divided by N.

Find the integer that will be written in each cell in this procedure. It can be proved that each cell will have an integer written in it exactly once.

Constraints

• 3\leq N \leq 99
• N is an odd number.

Sample Input 1

3

Sample Output 1

8 1 6
3 5 7
4 9 2

Integers are written in each cell as follows:

1. Write 1 in cell (0,\frac{3-1}{2})=(0,1).
2. Cell ((0-1) \bmod 3, (1+1) \bmod 3)=(2,2) is empty, so write 2 there.
3. Cell ((2-1) \bmod 3, (2+1) \bmod 3)=(1,0) is empty, so write 3 there.
4. Cell ((1-1) \bmod 3, (0+1) \bmod 3)=(0,1) is not empty, so write 4 in cell ((1+1) \bmod 3,0)=(2,0).
5. \vdots

Sample Input 2

5

Sample Output 2

17 24 1 8 15
23 5 7 14 16
4 6 13 20 22
10 12 19 21 3
11 18 25 2 9

Problem Statement

For positive integers x and y, define f(x, y) as the remainder of (x + y) divided by 10^8.

You are given a sequence of positive integers A = (A_1, \ldots, A_N) of length N. Find the value of the following expression:

Constraints

• 2 \leq N \leq 3\times 10^5
• 1 \leq A_i < 10^8
• All input values are integers.

Sample Input 1

3
3 50000001 50000002

Sample Output 1

100000012

• f(A_1,A_2)=50000004
• f(A_1,A_3)=50000005
• f(A_2,A_3)=3

Thus, the answer is f(A_1,A_2) + f(A_1,A_3) + f(A_2,A_3) = 100000012.

Note that you are not asked to compute the remainder of the sum divided by 10^8.

Sample Input 2

5
1 3 99999999 99999994 1000000

Sample Output 2

303999988

Problem Statement

We have N fuses connected in series. The i-th fuse from the left has a length of A_i centimeters and burns at a constant speed of B_i centimeters per second.

Consider igniting this object from left and right ends simultaneously. Find the distance between the position where the two flames will meet and the left end of the object.

Constraints

• 1 \leq N \leq 10^5
• 1 \leq A_i,B_i \leq 1000
• All values in input are integers.

Sample Input 1

3
1 1
2 1
3 1

Sample Output 1

3.000000000000000

The two flames will meet at 3 centimeters from the left end of the object.

Sample Input 2

3
1 3
2 2
3 1

Sample Output 2

3.833333333333333

Sample Input 3

5
3 9
1 2
4 6
1 5
5 3

Sample Output 3

8.916666666666668

Problem Statement

The AtCoder Archipelago consists of N islands connected by N bridges. The islands are numbered from 1 to N, and the i-th bridge (1\leq i\leq N-1) connects islands i and i+1 bidirectionally, while the N-th bridge connects islands N and 1 bidirectionally. There is no way to travel between islands other than crossing the bridges.

On the islands, a tour that starts from island X_1 and visits islands X_2, X_3, \dots, X_M in order is regularly conducted. The tour may pass through islands other than those being visited, and the total number of times bridges are crossed during the tour is defined as the length of the tour.

More precisely, a tour is a sequence of l+1 islands a_0, a_1, \dots, a_l that satisfies all the following conditions, and its length is defined as l:

• For all j\ (0\leq j\leq l-1), islands a_j and a_{j+1} are directly connected by a bridge.
• There are some 0 = y_1 < y_2 < \dots < y_M = l such that for all k\ (1\leq k\leq M), a_{y_k} = X_k.

Due to financial difficulties, the islands will close one bridge to reduce maintenance costs. Determine the minimum possible length of the tour when the bridge to be closed is chosen optimally.

Constraints

• 3\leq N \leq 2\times 10^5
• 2\leq M \leq 2\times 10^5
• 1\leq X_k\leq N
• X_k\neq X_{k+1}\ (1\leq k\leq M-1)
• All input values are integers.

Sample Input 1

3 3
1 3 2

Sample Output 1

2

• If the first bridge is closed: By taking the sequence of islands (a_0, a_1, a_2) = (1, 3, 2), it is possible to visit islands 1, 3, 2 in order, and a tour of length 2 can be conducted. There is no shorter tour.
• If the second bridge is closed: By taking the sequence of islands (a_0, a_1, a_2, a_3) = (1, 3, 1, 2), it is possible to visit islands 1, 3, 2 in order, and a tour of length 3 can be conducted. There is no shorter tour.
• If the third bridge is closed: By taking the sequence of islands (a_0, a_1, a_2, a_3) = (1, 2, 3, 2), it is possible to visit islands 1, 3, 2 in order, and a tour of length 3 can be conducted. There is no shorter tour.

Therefore, the minimum possible length of the tour when the bridge to be closed is chosen optimally is 2.

The following figure shows, from left to right, the cases when bridges 1, 2, 3 are closed, respectively. The circles with numbers represent islands, the lines connecting the circles represent bridges, and the blue arrows represent the shortest tour routes.

Sample Input 2

4 5
2 4 2 4 2

Sample Output 2

8

The same island may appear multiple times in X_1, X_2, \dots, X_M.

Sample Input 3

163054 10
62874 19143 77750 111403 29327 56303 6659 18896 64175 26369

Sample Output 3

390009

Problem Statement

You are given positive integers N,A,B. It is guaranteed that both A and B are between 1 and N, inclusive.

There is an N\times N grid. The cell at the i-th row from the top and j-th column from the left is denoted as cell (i,j). Initially, a piece is placed at cell (1,1).

By repeating N^2-2 moves, each of which moves the piece to an adjacent cell (up, down, left, or right), you want to move the piece to cell (N,N) while visiting every cell except cell (A,B). You must not visit the same cell more than once (you must not visit cells (1,1) and (N,N) in the middle, either).

Determine whether such a sequence of moves is possible, and if so, output one such sequence.

You are given T test cases; solve each of them.

Constraints

• 1\le T \le 5000
• 2\le N\le 10^3
• 1\le A,B\le N
• (A,B)\neq (1,1),(N,N)
• The sum of N^2 over all test cases is at most 10^6.
• All input values are integers.

Sample Input 1

3
2 1 2
3 2 2
4 3 2

Sample Output 1

Yes
DR
No
Yes
RRRDLLLDDRRURD

Consider the first test case.

Starting with the piece at cell (1,1), move it twice as follows:

• Move the piece downward. The piece moves to cell (2,1).
• Move the piece rightward. The piece moves to cell (2,2).

This sequence of moves satisfies the conditions.

Problem Statement

We have a typewriter with N rows. The keys in the i-th row from the top can type the characters in a string S_i.

Let us use this keyboard to enter a string, as follows.

• First, choose an integer 1 \le k \le N.
• Then, start with an empty string and only use the keys in the k-th row from the top to enter a string of length exactly L.

How many strings of length L can be entered in this way? Since the answer can be enormous, print it modulo 998244353.

Constraints

• N and L are integers.
• 1 \le N \le 18
• 1 \le L \le 10^9
• S_i is a (not necessarily contiguous) non-empty subsequence of abcdefghijklmnopqrstuvwxyz.

Sample Input 1

2 2
ab
ac

Sample Output 1

7

We can enter seven strings: aa, ab, ac, ba, bb, ca, cc.

Sample Input 2

4 3
abcdefg
hijklmnop
qrstuv
wxyz

Sample Output 2

1352

Sample Input 3

5 1000000000
abc
acde
cefg
abcfh
dghi

Sample Output 3

346462871

Be sure to print the answer modulo 998244353.