Problem Statement

A mayoral election is being held in AtCoder City. The candidates are Takahashi and Aoki.

There are N valid votes cast for either of the two candidates, and the counting is currently underway. Here, N is an odd number.

The current vote count is T votes for Takahashi and A votes for Aoki.

Determine if the outcome of the election is already decided at this point.

Constraints

• 1 \leq N \leq 99
• N is an odd number.
• 0 \leq T, A \leq N
• T + A \leq N
• All input values are integers.

Sample Input 1

7 4 2

Sample Output 1

Yes

Even if the remaining one vote goes to Aoki, Takahashi will still win. That is, his victory is decided, so print Yes.

Sample Input 2

99 12 48

Sample Output 2

No

Although Aoki currently has more votes, Takahashi would win if he receives the remaining 39 votes. Therefore, print No.

Sample Input 3

1 0 0

Sample Output 3

No

Problem Statement

Assuming that the horizon seen from a place x meters above the ground is \sqrt{x(12800000+x)} meters away, find how many meters away the horizon seen from a place H meters above the ground is.

Constraints

• 1 \leq H \leq 10^5
• H is an integer.

Sample Input 1

333

Sample Output 1

65287.907678222

We have \sqrt{333(12800000+333)} = 65287.9076782\ldots. Outputs such as 65287.91 would also be accepted.

Sample Input 2

634

Sample Output 2

90086.635834623

We have \sqrt{634(12800000+634)} = 90086.6358346\ldots.

Problem Statement

At the entrance of AtCoder Land, there is a single ticket booth where visitors line up to purchase tickets one by one. The purchasing process takes A seconds per person. Once the person at the front of the line finishes purchasing their ticket, the next person (if any) immediately starts their purchasing process.

Currently, there is no one in line at the ticket booth, and N people will come to buy tickets one after another. Specifically, the i-th person will arrive at the ticket booth T_i seconds from now. If there is already a line, they will join the end of it; if not, they will start the purchasing process immediately. Here, T_1 < T_2 < \dots < T_N.

For each i\ (1 \leq i \leq N), determine how many seconds from now the i-th person will finish purchasing their ticket.

Constraints

• 1 \leq N \leq 100
• 0 \leq T_1 < T_2 < \dots < T_N \leq 10^6
• 1 \leq A \leq 10^6
• All input values are integers.

Sample Input 1

3 4
0 2 10

Sample Output 1

4
8
14

The events proceed in the following order:

• At 0 seconds: The 1st person arrives at the ticket booth and starts the purchasing process.
• At 2 seconds: The 2nd person arrives at the ticket booth and joins the line behind the 1st person.
• At 4 seconds: The 1st person finishes purchasing their ticket, and the 2nd person starts the purchasing process.
• At 8 seconds: The 2nd person finishes purchasing their ticket.
• At 10 seconds: The 3rd person arrives at the ticket booth and starts the purchasing process.
• At 14 seconds: The 3rd person finishes purchasing their ticket.

Sample Input 2

3 3
1 4 7

Sample Output 2

4
7
10

The events proceed in the following order:

• At 1 second: The 1st person arrives at the ticket booth and starts the purchasing process.
• At 4 seconds: The 1st person finishes purchasing their ticket, and the 2nd person arrives at the ticket booth and starts the purchasing process.
• At 7 seconds: The 2nd person finishes purchasing their ticket, and the 3rd person arrives at the ticket booth and starts the purchasing process.
• At 10 seconds: The 3rd person finishes purchasing their ticket.

Sample Input 3

10 50000
120190 165111 196897 456895 540000 552614 561627 743796 757613 991216

Sample Output 3

170190
220190
270190
506895
590000
640000
690000
793796
843796
1041216

Problem Statement

There are N people numbered 1,2,\ldots,N.

M parties were held. k_i people attended the i-th (1\leq i \leq M) party, and they were People x_{i,1},x_{i,2},\ldots,x_{i,k_i}.

Determine if every two people attended the same party at least once.

Constraints

• 2\leq N \leq 100
• 1\leq M \leq 100
• 2\leq k_i \leq N
• 1\leq x_{i,1}<x_{i,2}<\ldots < x_{i,k_i}\leq N
• All values in the input are integers.

Sample Input 1

3 3
2 1 2
2 2 3
2 1 3

Sample Output 1

Yes

Both Person 1 and Person 2 attended the 1-st party.

Both Person 2 and Person 3 attended the 2-nd party.

Both Person 1 and Person 3 attended the 3-rd party.

Therefore, every two people attended the same party at least once, so the answer is Yes.

Sample Input 2

4 2
3 1 2 4
3 2 3 4

Sample Output 2

No

Person 1 and Person 3 did not attend the same party, so the answer is No.

Problem Statement

Takahashi has two sheets A and B, each composed of black squares and transparent squares, and an infinitely large sheet C composed of transparent squares.
There is also an ideal sheet X for Takahashi composed of black squares and transparent squares.

The sizes of sheets A, B, and X are H_A rows \times W_A columns, H_B rows \times W_B columns, and H_X rows \times W_X columns, respectively.
The squares of sheet A are represented by H_A strings of length W_A, A_1, A_2, \ldots, A_{H_A} consisting of . and #.
If the j-th character (1\leq j\leq W_A) of A_i (1\leq i\leq H_A) is ., the square at the i-th row from the top and j-th column from the left is transparent; if it is #, that square is black.
Similarly, the squares of sheets B and X are represented by H_B strings of length W_B, B_1, B_2, \ldots, B_{H_B}, and H_X strings of length W_X, X_1, X_2, \ldots, X_{H_X}, respectively.

Takahashi's goal is to create sheet X using all black squares in sheets A and B by following the steps below with sheets A, B, and C.

1. Paste sheets A and B onto sheet C along the grid. Each sheet can be pasted anywhere by translating it, but it cannot be cut or rotated.
2. Cut out an H_X\times W_X area from sheet C along the grid. Here, a square of the cut-out sheet will be black if a black square of sheet A or B is pasted there, and transparent otherwise.

Determine whether Takahashi can achieve his goal by appropriately choosing the positions where the sheets are pasted and the area to cut out, that is, whether he can satisfy both of the following conditions.

• The cut-out sheet includes all black squares of sheets A and B. The black squares of sheets A and B may overlap on the cut-out sheet.
• The cut-out sheet coincides sheet X without rotating or flipping.

Constraints

• 1\leq H_A, W_A, H_B, W_B, H_X, W_X\leq 10
• H_A, W_A, H_B, W_B, H_X, W_X are integers.
• A_i is a string of length W_A consisting of . and #.
• B_i is a string of length W_B consisting of . and #.
• X_i is a string of length W_X consisting of . and #.
• Sheets A, B, and X each contain at least one black square.

Sample Input 1

3 5
#.#..
.....
.#...
2 2
#.
.#
5 3
...
#.#
.#.
.#.
...

Sample Output 1

Yes

First, paste sheet A onto sheet C, as shown in the figure below.

     \vdots
  .......  
  .#.#...  
\cdots.......\cdots
  ..#....  
  .......  
     \vdots

Next, paste sheet B so that its top-left corner aligns with that of sheet A, as shown in the figure below.

     \vdots
  .......  
  .#.#...  
\cdots..#....\cdots
  ..#....  
  .......  
     \vdots

Now, cut out a 5\times 3 area with the square in the first row and second column of the range illustrated above as the top-left corner, as shown in the figure below.

...
#.#
.#.
.#.
...

This includes all black squares of sheets A and B and matches sheet X, satisfying the conditions.

Therefore, print Yes.

Sample Input 2

2 2
#.
.#
2 2
#.
.#
2 2
##
##

Sample Output 2

No

Note that sheets A and B may not be rotated or flipped when pasting them.

Sample Input 3

1 1
#
1 2
##
1 1
#

Sample Output 3

No

No matter how you paste or cut, you cannot cut out a sheet that includes all black squares of sheet B, so you cannot satisfy the first condition. Therefore, print No.

Sample Input 4

3 3
###
...
...
3 3
#..
#..
#..
3 3
..#
..#
###

Sample Output 4

Yes

Problem Statement

You are given a grid of H rows and W columns.
Let (i,j) denote the cell at row i (1 \leq i \leq H) from the top and column j (1 \leq j \leq W) from the left.
The state of the grid is represented by H strings S_1, S_2, \ldots, S_H, each of length W, as follows:

• If the j-th character of S_i is #, cell (i,j) is painted black.
• If the j-th character of S_i is ., cell (i,j) is painted white.
• If the j-th character of S_i is ?, cell (i,j) is not yet painted.

Takahashi wants to paint each not-yet-painted cell white or black so that all the black cells form a rectangle.
More precisely, he wants there to exist a quadruple of integers (a,b,c,d) (1 \leq a \leq b \leq H, 1 \leq c \leq d \leq W) such that:

For each cell (i,j) (1 \leq i \leq H, 1 \leq j \leq W), if a \leq i \leq b and c \leq j \leq d, the cell is black;
otherwise, the cell is white.

Determine whether this is possible.

Constraints

• 1 \leq H, W \leq 1000
• H and W are integers.
• Each S_i is a string of length W consisting of #, ., ?.
• There is at least one cell that is already painted black.

Sample Input 1

3 5
.#?#.
.?#?.
?...?

Sample Output 1

Yes

The grid is in the following state. ? indicates a cell that are not yet painted.

By painting cells (1,3), (2,2), and (2,4) black and cells (3,1) and (3,5) white, the black cells can form a rectangle as follows:

Therefore, print Yes.

Sample Input 2

3 3
?##
#.#
##?

Sample Output 2

No

To form a rectangle with all black cells, you would need to paint cell (2,2) black, but it is already painted white.
Therefore, it is impossible to make all black cells form a rectangle, so print No.

Sample Input 3

1 1
#

Sample Output 3

Yes

Problem Statement

There is a directed graph with N vertices and M edges, where vertices are numbered from 1 to N and edges are numbered from 1 to M. Edge i is a directed edge from vertex A_i to vertex B_i with weight W_i.

Find the minimum value of the bitwise \mathrm{XOR} of the weights of edges included in a walk from vertex 1 to vertex N.

Constraints

• 2 \leq N \leq 1000
• 0 \leq M \leq 1000
• 1 \leq A_i,B_i \leq N
• 0 \leq W_i < 2^{10}
• All input values are integers.

Sample Input 1

3 3
1 2 4
2 3 5
1 3 2

Sample Output 1

1

The bitwise \mathrm{XOR} of the weights of edges included in the walk (edge 1, edge 2) is 1.

Sample Input 2

4 4
1 4 7
4 2 2
2 3 4
3 4 1

Sample Output 2

0

The bitwise \mathrm{XOR} of the weights of edges included in the walk (edge 1, edge 2, edge 3, edge 4) is 0.

Note that the walk may include vertex N in the middle.

Sample Input 3

999 4
1 2 9
2 1 8
1 2 7
1 1 6

Sample Output 3

-1

If there is no walk from vertex 1 to vertex N, output -1.

Problem Statement

Aoki, an employee at AtCoder Inc., has his salary for this month determined by an integer N and a sequence A of length N as follows.
First, he is given an N-sided die (dice) that shows the integers from 1 to N with equal probability, and a variable x=0.

Then, the following steps are repeated until terminated.

• Roll the die once and let y be the result.
  • If x<y, pay him A_y yen and let x=y.
  • Otherwise, terminate the process.

Aoki's salary for this month is the total amount paid through this process.
Find the expected value of Aoki's salary this month, modulo 998244353.

Constraints

• All inputs are integers.
• 1 \le N \le 3 \times 10^5
• 0 \le A_i < 998244353

Sample Input 1

3
3 2 6

Sample Output 1

776412280

Here is an example of how the process goes.

• Initially, x=0.
• Roll the die once, and it shows 1. Since 0<1, pay him A_1 = 3 yen and let x=1.
• Roll the die once, and it shows 3. Since 1<3, pay him A_3 = 6 yen and let x=3.
• Roll the die once, and it shows 1. Since 3 \ge 1, terminate the process.

In this case, his salary for this month is 9 yen.

It can be calculated that the expected value of his salary this month is \frac{49}{9} yen, whose representation modulo 998244353 is 776412280.

Sample Input 2

1
998244352

Sample Output 2

998244352

Sample Input 3

9
3 14 159 2653 58979 323846 2643383 27950288 419716939

Sample Output 3

545252774

Problem Statement

There are N Christmas trees on a two-dimensional plane. The i-th (1\le i\le N) Christmas tree is located at coordinates (X_i,Y_i).

You are given Q queries. Process the queries in order. Each query is one of the following:

• Type 1 : Given in the form 1 i x y. Change the coordinates of the i-th Christmas tree to (x,y).
• Type 2 : Given in the form 2 L R x y. Output the Manhattan distance from the coordinates (x,y) to the farthest Christmas tree among the L,L+1,\ldots,R-th Christmas trees.

Here, the Manhattan distance between coordinates (x_1,y_1) and coordinates (x_2,y_2) is defined as |x_1-x_2|+|y_1-y_2|.

Constraints

• 1\le N,Q\le 2\times 10^5
• -10^9\le X_i,Y_i\le 10^9
• 1\le i\le N
• 1\le L\le R\le N
• -10^9\le x,y\le 10^9
• All input values are integers.

Sample Input 1

3 4
-1 -1
1 2
-2 1
2 1 2 0 0
2 1 3 -1 2
1 1 0 1
2 1 3 -1 2

Sample Output 1

3
3
2

Initially, the 1st, 2nd, 3rd Christmas trees are located at coordinates (-1,-1), (1,2), (-2,1), respectively.

Each query is processed as follows:

• The Manhattan distances from the 1st and 2nd Christmas trees to coordinates (0,0) are 2 and 3, respectively. Thus, output 3, which is the maximum value among 2,3.
• The Manhattan distances from the 1st, 2nd, 3rd Christmas trees to coordinates (-1,2) are 3, 2, 2, respectively. Thus, output 3, which is the maximum value among 3,2,2.
• Change the coordinates of the 1st Christmas tree to (0,1). The coordinates of the 1st, 2nd, 3rd Christmas trees become (0,1),(1,2),(-2,1), respectively.
• The Manhattan distances from the 1st, 2nd, 3rd Christmas trees to coordinates (-1,2) are 2, 2, 2, respectively. Thus, output 2, which is the maximum value among 2,2,2.

Sample Input 2

5 7
-9 5
-2 -9
10 -6
9 8
2 9
1 3 -9 -6
2 3 4 2 7
1 4 -2 -10
2 1 2 0 -10
2 3 4 10 -9
2 3 4 8 7
2 5 5 0 2

Sample Output 2

24
24
22
30
9