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

There are N people, each of whom is either a child or an adult. The i-th person has a weight of W_i.
Whether each person is a child or an adult is specified by a string S of length N consisting of 0 and 1.
If the i-th character of S is 0, then the i-th person is a child; if it is 1, then the i-th person is an adult.

When Takahashi the robot is given a real number X, Takahashi judges a person with a weight less than X to be a child and a person with a weight more than or equal to X to be an adult.
For a real value X, let f(X) be the number of people whom Takahashi correctly judges whether they are children or adults.

Find the maximum value of f(X) for all real values of X.

Constraints

• 1\leq N\leq 2\times 10^5
• S is a string of length N consisting of 0 and 1.
• 1\leq W_i\leq 10^9
• N and W_i are integers.

Sample Input 1

5
10101
60 45 30 40 80

Sample Output 1

4

When Takahashi is given X=50, it judges the 2-nd, 3-rd, and 4-th people to be children and the 1-st and 5-th to be adults.
In reality, the 2-nd and 4-th are children, and the 1-st, 3-rd, and 5-th are adults, so the 1-st, 2-nd, 4-th, and 5-th people are correctly judged. Thus, f(50)=4.

This is the maximum since there is no X that judges correctly for all 5 people. Thus, 4 should be printed.

Sample Input 2

3
000
1 2 3

Sample Output 2

3

For example, X=10 achieves the maximum value f(10)=3.
Note that the people may be all children or all adults.

Sample Input 3

5
10101
60 50 50 50 60

Sample Output 3

4

For example, X=55 achieves the maximum value f(55)=4.
Note that there may be multiple people with the same weight.

Problem Statement

You are given a tree with N vertices: vertex 1, vertex 2, \ldots, vertex N. The i-th edge (1\leq i\lt N) connects vertex u _ i and vertex v _ i.

Consider repeating the following operation some number of times:

• Choose one leaf vertex v and delete it along with all incident edges.

Find the minimum number of operations required to delete vertex 1.

Constraints

• 2\leq N\leq3\times10^5
• 1\leq u _ i\lt v _ i\leq N\ (1\leq i\lt N)
• The given graph is a tree.
• All input values are integers.

Sample Input 1

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

Sample Output 1

5

The given graph looks like this:

For example, you can choose vertices 9,8,7,6,1 in this order to delete vertex 1 in five operations.

Vertex 1 cannot be deleted in four or fewer operations, so print 5.

Sample Input 2

6
1 2
2 3
2 4
3 5
3 6

Sample Output 2

1

In the given graph, vertex 1 is a leaf. Hence, you can choose and delete vertex 1 in the first operation.

Sample Input 3

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

Sample Output 3

12

Problem Statement

There is an island of size H \times W, surrounded by the sea.
The island is divided into H rows and W columns of 1 \times 1 sections, and the elevation of the section at the i-th row from the top and the j-th column from the left (relative to the current sea level) is A_{i,j}.

Starting from now, the sea level rises by 1 each year.
Here, a section that is vertically or horizontally adjacent to the sea or a section sunk into the sea and has an elevation not greater than the sea level will sink into the sea.
Here, when a section newly sinks into the sea, any vertically or horizontally adjacent section with an elevation not greater than the sea level will also sink into the sea simultaneously, and this process repeats for the newly sunk sections.

For each i=1,2,\ldots, Y, find the area of the island that remains above sea level i years from now.

Constraints

• 1 \leq H, W \leq 1000
• 1 \leq Y \leq 10^5
• 1 \leq A_{i,j} \leq 10^5
• All input values are integers.

Sample Input 1

3 3 5
10 2 10
3 1 4
10 5 10

Sample Output 1

9
7
6
5
4

Let (i,j) denote the section at the i-th row from the top and the j-th column from the left. Then, the following happens:

• After 1 year, the sea level is higher than now by 1, but there are no sections with an elevation of 1 that are adjacent to the sea, so no sections sink. Thus, the first line should contain 9.
• After 2 years, the sea level is higher than now by 2, and (1,2) sinks into the sea. This makes (2,2) adjacent to a sunken section, and its elevation is not greater than 2, so it also sinks. No other sections sink at this point. Thus, two sections sink, and the second line should contain 9-2=7.
• After 3 years, the sea level is higher than now by 3, and (2,1) sinks into the sea. No other sections sink. Thus, the third line should contain 6.
• After 4 years, the sea level is higher than now by 4, and (2,3) sinks into the sea. No other sections sink. Thus, the fourth line should contain 5.
• After 5 years, the sea level is higher than now by 5, and (3,2) sinks into the sea. No other sections sink. Thus, the fifth line should contain 4.

Therefore, print 9, 7, 6, 5, 4 in this order, each on a new line.

Sample Input 2

3 5 3
2 2 3 3 3
2 1 2 1 3
2 2 3 3 3

Sample Output 2

15
7
0

Problem Statement

We have a grid with H rows and W columns.

We choose K cells in this grid uniformly at random. The score is the number of cells in the minimum rectangle (whose edges are parallel to the axes of the grid) that contains all of the chosen cells.

Find the expected score modulo 998244353.

Constraints

• 1\leq H,W \leq 1000
• 1\leq K \leq HW
• All values in the input are integers.

Sample Input 1

2 2 2

Sample Output 1

665496238

The score equals 4 in the following two cases: if cells (1,1) and (2,2) are chosen, or cells (1,2) and (2,1) are chosen. The other four cases yield a score of 2.

Thus, the expected score equals \frac{4 \times 2 + 2 \times 4} {6} = \frac{8}{3}. Since 665496238 \times 3 \equiv 8\pmod{998244353}, you should print 665496238.

Sample Input 2

10 10 1

Sample Output 2

1

Sample Input 3

314 159 2653

Sample Output 3

639716353