Problem Statement

You are given a string S of length 16 consisting of 0 and 1.

If the i-th character of S is 0 for every even number i from 2 through 16, print Yes; otherwise, print No.

Constraints

• S is a string of length 16 consisting of 0 and 1.

Sample Input 1

1001000000001010

Sample Output 1

No

The 4-th character of S= 1001000000001010 is 1, so you should print No.

Sample Input 2

1010100000101000

Sample Output 2

Yes

Every even-positioned character in S= 1010100000101000 is 0, so you should print Yes.

Sample Input 3

1111111111111111

Sample Output 3

No

Every even-positioned character in S is 1. Particularly, they are not all 0, so you should print No.

Problem Statement

There are N players numbered 1 to N, who have played a round-robin tournament. For every match in this tournament, one player won and the other lost.

The results of the matches are given as N strings S_1,S_2,\ldots,S_N of length N each, in the following format:

• If i\neq j, the j-th character of S_i is o or x. o means that player i won against player j, and x means that player i lost to player j.

• If i=j, the j-th character of S_i is -.

The player with more wins ranks higher. If two players have the same number of wins, the player with the smaller player number ranks higher. Report the player numbers of the N players in descending order of rank.

Constraints

• 2\leq N\leq 100
• N is an integer.
• S_i is a string of length N consisting of o, x, and -.
• S_1,\ldots,S_N conform to the format described in the problem statement.

Sample Input 1

3
-xx
o-x
oo-

Sample Output 1

3 2 1

Player 1 has 0 wins, player 2 has 1 win, and player 3 has 2 wins. Thus, the player numbers in descending order of rank are 3,2,1.

Sample Input 2

7
-oxoxox
x-xxxox
oo-xoox
xoo-ooo
ooxx-ox
xxxxx-x
oooxoo-

Sample Output 2

4 7 3 1 5 2 6

Both players 4 and 7 have 5 wins, but player 4 ranks higher because their player number is smaller.

Problem Statement

The programming contest World Tour Finals is underway, where N players are participating, and half of the competition time has passed. There are M problems in this contest, and the score A_i of problem i is a multiple of 100 between 500 and 2500, inclusive.

For each i = 1, \ldots, N, you are given a string S_i that indicates which problems player i has already solved. S_i is a string of length M consisting of o and x, where the j-th character of S_i is o if player i has already solved problem j, and x if they have not yet solved it. Here, none of the players have solved all the problems yet.

The total score of player i is calculated as the sum of the scores of the problems they have solved, plus a bonus score of i points.
For each i = 1, \ldots, N, answer the following question.

• At least how many of the problems that player i has not yet solved must player i solve to exceed all other players' current total scores?

Note that under the conditions in this statement and the constraints, it can be proved that player i can exceed all other players' current total scores by solving all the problems, so the answer is always defined.

Constraints

• 2\leq N\leq 100
• 1\leq M\leq 100
• 500\leq A_i\leq 2500
• A_i is a multiple of 100.
• S_i is a string of length M consisting of o and x.
• S_i contains at least one x.
• All numeric values in the input are integers.

Sample Input 1

3 4
1000 500 700 2000
xxxo
ooxx
oxox

Sample Output 1

0
1
1

The players' total scores at the halfway point of the competition time are 2001 points for player 1, 1502 points for player 2, and 1703 points for player 3.

Player 1 is already ahead of all other players' total scores without solving any more problems.

Player 2 can, for example, solve problem 4 to have a total score of 3502 points, which would exceed all other players' total scores.

Player 3 can also, for example, solve problem 4 to have a total score of 3703 points, which would exceed all other players' total scores.

Sample Input 2

5 5
1000 1500 2000 2000 2500
xxxxx
oxxxx
xxxxx
oxxxx
oxxxx

Sample Output 2

1
1
1
1
0

Sample Input 3

7 8
500 500 500 500 500 500 500 500
xxxxxxxx
oxxxxxxx
ooxxxxxx
oooxxxxx
ooooxxxx
oooooxxx
ooooooxx

Sample Output 3

7
6
5
4
3
2
0

Problem Statement

Initially, there are N sizes of slimes.
Specifically, for each 1\leq i\leq N, there are C_i slimes of size S_i.

Takahashi can repeat slime synthesis any number of times (possibly zero) in any order.
Slime synthesis is performed as follows.

• Choose two slimes of the same size. Let this size be X, and a new slime of size 2X appears. Then, the two original slimes disappear.

Takahashi wants to minimize the number of slimes. What is the minimum number of slimes he can end up with by an optimal sequence of syntheses?

Constraints

• 1\leq N\leq 10^5
• 1\leq S_i\leq 10^9
• 1\leq C_i\leq 10^9
• S_1,S_2,\ldots,S_N are all different.
• All input values are integers.

Sample Input 1

3
3 3
5 1
6 1

Sample Output 1

3

Initially, there are three slimes of size 3, one of size 5, and one of size 6.
Takahashi can perform the synthesis twice as follows:

• First, perform the synthesis by choosing two slimes of size 3. There will be one slime of size 3, one of size 5, and two of size 6.
• Next, perform the synthesis by choosing two slimes of size 6. There will be one slime of size 3, one of size 5, and one of size 12.

No matter how he repeats the synthesis from the initial state, he cannot reduce the number of slimes to 2 or less, so you should print 3.

Sample Input 2

3
1 1
2 1
3 1

Sample Output 2

3

He cannot perform the synthesis.

Sample Input 3

1
1000000000 1000000000

Sample Output 3

13

Problem Statement

Takahashi has a playlist with N songs. Song i (1 \leq i \leq N) lasts T_i seconds.
Takahashi has started random play of the playlist at time 0.

Random play repeats the following: choose one song from the N songs with equal probability and play that song to the end. Here, songs are played continuously: once a song ends, the next chosen song starts immediately. The same song can be chosen consecutively.

Find the probability that song 1 is being played (X + 0.5) seconds after time 0, modulo 998244353.

Constraints

• 2 \leq N\leq 10^3
• 0 \leq X\leq 10^4
• 1 \leq T_i\leq 10^4
• All input values are integers.

Sample Input 1

3 6
3 5 6

Sample Output 1

369720131

Song 1 will be playing 6.5 seconds after time 0 if songs are played in one of the following orders.

• Song 1 \to Song 1 \to Song 1
• Song 2 \to Song 1
• Song 3 \to Song 1

The probability that one of these occurs is \frac{7}{27}.
We have 369720131\times 27\equiv 7 \pmod{998244353}, so you should print 369720131.

Sample Input 2

5 0
1 2 1 2 1

Sample Output 2

598946612

0.5 seconds after time 0, the first song to be played is still playing, so the sought probability is \frac{1}{5}.
Note that different songs may have the same length.

Sample Input 3

5 10000
1 2 3 4 5

Sample Output 3

586965467

Problem Statement

Takahashi and a cargo are on a coordinate plane.

Takahashi is currently at (X_A,Y_A), and the cargo is at (X_B,Y_B). He wants to move the cargo to (X_C,Y_C).

When he is at (x,y), he can make one of the following moves in a single action.

• Move to (x+1,y). If the cargo is at (x+1,y) before the move, move it to (x+2,y).
• Move to (x-1,y). If the cargo is at (x-1,y) before the move, move it to (x-2,y).
• Move to (x,y+1). If the cargo is at (x,y+1) before the move, move it to (x,y+2).
• Move to (x,y-1). If the cargo is at (x,y-1) before the move, move it to (x,y-2).

Find the minimum number of actions required to move the cargo to (X_C,Y_C).

Constraints

• -10^{17}\leq X_A,Y_A,X_B,Y_B,X_C,Y_C\leq 10^{17}
• (X_A,Y_A)\neq (X_B,Y_B)
• (X_B,Y_B)\neq (X_C,Y_C)
• All input values are integers.

Sample Input 1

1 2 3 3 0 5

Sample Output 1

9

Takahashi can move the cargo to (0,5) in nine actions as follows.

• Move to (2,2).
• Move to (3,2).
• Move to (3,3). The cargo moves to (3,4).
• Move to (3,4). The cargo moves to (3,5).
• Move to (4,4).
• Move to (4,5).
• Move to (3,5). The cargo moves to (2,5).
• Move to (2,5). The cargo moves to (1,5).
• Move to (1,5). The cargo moves to (0,5).

It is impossible to move the cargo to (0,5) in eight or fewer actions, so you should print 9.

Sample Input 2

0 0 1 0 -1 0

Sample Output 2

6

Sample Input 3

-100000000000000000 -100000000000000000 100000000000000000 100000000000000000 -100000000000000000 -100000000000000000

Sample Output 3

800000000000000003

Problem Statement

You are given a permutation P=(P_1,P_2,\ldots,P_N) of (1,2,\ldots,N).

For each K=0,1,\ldots,N-1, find the number, modulo 998244353, of trees with N vertices numbered 1 to N that satisfy the following condition.

• Among the pairs of vertices (u_i,v_i)\ (u_i < v_i) that are directly connected by an edge in the tree, exactly K pairs satisfy P_{u_i}>P_{v_i}.

Constraints

• 2\leq N\leq 500
• P is a permutation of (1,2,\ldots,N).

Sample Input 1

3
1 3 2

Sample Output 1

1 2 0

The answer for K=0 is 1: the tree with edges connecting vertices 1,2 and 1,3. Indeed, P_1 < P_2 and P_1<P_3.

The answer for K=1 is 2: the tree with edges connecting vertices 1,2 and 2,3, and another with edges connecting vertices 1,3 and 2,3. Indeed, in the tree with edges connecting vertices 1,2 and 2,3, we have P_1 < P_2, P_2>P_3.

Sample Input 2

10
3 1 4 10 8 6 9 2 7 5

Sample Output 2

294448 2989776 12112684 25422152 30002820 20184912 7484084 1397576 108908 2640