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

You are given two strings S and T. Determine whether it is possible to make S equal T by performing the following operation some number of times (possibly zero).

Between two consecutive equal characters in S, insert a character equal to these characters. That is, take the following three steps.

1. Let N be the current length of S, and S = S_1S_2\ldots S_N.
2. Choose an integer i between 1 and N-1 (inclusive) such that S_i = S_{i+1}. (If there is no such i, do nothing and terminate the operation now, skipping step 3.)
3. Insert a single copy of the character S_i(= S_{i+1}) between the i-th and (i+1)-th characters of S. Now, S is a string of length N+1: S_1S_2\ldots S_i S_i S_{i+1} \ldots S_N.

Constraints

• Each of S and T is a string of length between 2 and 2 \times 10^5 (inclusive) consisting of lowercase English letters.

Sample Input 1

abbaac
abbbbaaac

Sample Output 1

Yes

You can make S = abbaac equal T = abbbbaaac by the following three operations.

• First, insert b between the 2-nd and 3-rd characters of S. Now, S = abbbaac.
• Next, insert b again between the 2-nd and 3-rd characters of S. Now, S = abbbbaac.
• Lastly, insert a between the 6-th and 7-th characters of S. Now, S = abbbbaaac.

Thus, Yes should be printed.

Sample Input 2

xyzz
xyyzz

Sample Output 2

No

No sequence of operations makes S = xyzz equal T = xyyzz. Thus, No should be printed.

Problem Statement

We have a polynomial of degree N, A(x)=A_Nx^N+A_{N-1}x^{N-1}+\cdots +A_1x+A_0,
and another of degree M, B(x)=B_Mx^M+B_{M-1}x^{M-1}+\cdots +B_1x+B_0.
Here, each coefficient of A(x) and B(x) is an integer whose absolute value is at most 100, and the leading coefficients are not 0.

Also, let the product of them be C(x)=A(x)B(x)=C_{N+M}x^{N+M}+C_{N+M-1}x^{N+M-1}+\cdots +C_1x+C_0.

Given A_0,A_1,\ldots, A_N and C_0,C_1,\ldots, C_{N+M}, find B_0,B_1,\ldots, B_M.
Here, the given inputs guarantee that there is a unique sequence B_0, B_1, \ldots, B_M that satisfies the given conditions.

Constraints

• 1 \leq N < 100
• 1 \leq M < 100
• |A_i| \leq 100
• |C_i| \leq 10^6
• A_N \neq 0
• C_{N+M} \neq 0
• There is a unique sequence B_0, B_1, \ldots, B_M that satisfies the conditions given in the statement.

Sample Input 1

1 2
2 1
12 14 8 2

Sample Output 1

6 4 2

For A(x)=x+2 and B(x)=2x^2+4x+6, we have C(x)=A(x)B(x)=(x+2)(2x^2+4x+6)=2x^3+8x^2+14x+12, so B(x)=2x^2+4x+6 satisfies the given conditions. Thus, B_0=6, B_1=4, B_2=2 should be printed in this order, with spaces in between.

Sample Input 2

1 1
100 1
10000 0 -1

Sample Output 2

100 -1

We have A(x)=x+100, C(x)=-x^2+10000, for which B(x)=-x+100 satisfies the given conditions. Thus, 100, -1 should be printed in this order, with spaces in between.

Problem Statement

There is a directed graph with N vertices numbered 1 to N and N edges.
The out-degree of every vertex is 1, and the edge from vertex i points to vertex a_i.
Count the number of pairs of vertices (u, v) such that vertex v is reachable from vertex u.

Here, vertex v is reachable from vertex u if there exists a sequence of vertices w_0, w_1, \dots, w_K of length K+1 that satisfies the following conditions. In particular, if u = v, it is always reachable.

• w_0 = u.
• w_K = v.
• For every 0 \leq i \lt K, there is an edge from vertex w_i to vertex w_{i+1}.

Constraints

• 1 \leq N \leq 2 \times 10^5
• 1 \leq a_i \leq N
• All input values are integers.

Sample Input 1

4
2 1 1 4

Sample Output 1

8

The vertices reachable from vertex 1 are vertices 1, 2.
The vertices reachable from vertex 2 are vertices 1, 2.
The vertices reachable from vertex 3 are vertices 1, 2, 3.
The vertex reachable from vertex 4 is vertex 4.
Therefore, the number of pairs of vertices (u, v) such that vertex v is reachable from vertex u is 8.
Note that the edge from vertex 4 is a self-loop, that is, it points to vertex 4 itself.

Sample Input 2

5
2 4 3 1 2

Sample Output 2

14

Sample Input 3

10
6 10 4 1 5 9 8 6 5 1

Sample Output 3

41

Problem Statement

There are integer sequences A, B, C of length M each.

C is represented by integers x_1, \dots, x_N, y_1, \dots, y_N. The first y_1 terms of C are x_1, the subsequent y_2 terms are x_2, \ldots, the last y_N terms are x_N.

B is defined by B_i = \sum_{k = 1}^i C_k \, (1 \leq i \leq M).

A is defined by A_i = \sum_{k = 1}^i B_k \, (1 \leq i \leq M).

Find the maximum value among A_1, \dots, A_M.

You will be given T test cases to solve.

Constraints

• 1 \leq T \leq 2 \times 10^5
• 1 \leq N \leq 2 \times 10^5
• The sum of N in a single file is at most 2 \times 10^5.
• 1 \leq M \leq 10^9
• |x_i| \leq 4 \, (1 \leq i \leq N)
• y_i \gt 0 \, (1 \leq i \leq N)
• \sum_{k = 1}^N y_k = M
• All values in input are integers.

Sample Input 1

3
3 7
-1 2
2 3
-3 2
10 472
-4 12
1 29
2 77
-1 86
0 51
3 81
3 17
-2 31
-4 65
4 23
1 1000000000
4 1000000000

Sample Output 1

4
53910
2000000002000000000

In the first test case, we have:

• C = (-1, -1, 2, 2, 2, -3, -3)
• B = (-1, -2, 0, 2, 4, 1, -2)
• A = (-1, -3, -3, -1, 3, 4, 2)

Thus, the maximum value among A_1, \dots, A_M is 4.