Problem Statement

There is a grid with H horizontal rows and W vertical columns, in which two distinct squares have a piece.

The state of the squares is represented by H strings S_1, \dots, S_H of length W. S_{i, j} = o means that there is a piece in the square at the i-th row from the top and j-th column from the left; S_{i, j} = - means that the square does not have a piece. Here, S_{i, j} denotes the j-th character of the string S_i.

Consider repeatedly moving one of the pieces to one of the four adjacent squares. It is not allowed to move the piece outside the grid. How many moves are required at minimum for the piece to reach the square with the other piece?

Constraints

• 2 \leq H, W \leq 100
• H and W are integers.
• S_i \, (1 \leq i \leq H) is a string of length W consisting of o and -.
• There exist exactly two pairs of integers 1 \leq i \leq H, 1 \leq j \leq W such that S_{i, j} = o.

Sample Input 1

2 3
--o
o--

Sample Output 1

3

The piece at the 1-st row from the top and 3-rd column from the left can reach the square with the other piece in 3 moves: down, left, left. Since it is impossible to do so in two or less moves, 3 should be printed.

Sample Input 2

5 4
-o--
----
----
----
-o--

Sample Output 2

4

Problem Statement

You are given a tree with N vertices and N-1 edges.
The vertices are numbered 1,2,\ldots,N. The i-th edge connects Vertex a_i and Vertex b_i.

Determine whether this tree is a star.

Here, a star is a tree where there is a vertex directly connected to all other vertices.

Notes

For the definition of a tree, see Tree (graph theory) - Wikipedia.

Constraints

• 3 \leq N \leq 10^5
• 1 \leq a_i \lt b_i \leq N
• The given graph is a tree.

Sample Input 1

5
1 4
2 4
3 4
4 5

Sample Output 1

Yes

The given graph is a star.

Sample Input 2

4
2 4
1 4
2 3

Sample Output 2

No

The given graph is not a star.

Sample Input 3

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

Sample Output 3

Yes

Problem Statement

We have an empty box.
Takahashi can cast the following two spells any number of times in any order.

• Spell A: puts one new ball into the box.
• Spell B: doubles the number of balls in the box.

Tell us a way to have exactly N balls in the box with at most \mathbf{120} casts of spells.
It can be proved that there always exists such a way under the Constraints given.

There is no way other than spells to alter the number of balls in the box.

Constraints

• 1 \leq N \leq 10^{18}
• All values in input are integers.

Sample Input 1

5

Sample Output 1

AABA

This changes the number of balls as follows: 0 \xrightarrow{A} 1\xrightarrow{A} 2 \xrightarrow{B}4\xrightarrow{A} 5.
There are also other acceptable outputs, such as AAAAA.

Sample Input 2

14

Sample Output 2

BBABBAAAB

This changes the number of balls as follows: 0 \xrightarrow{B} 0 \xrightarrow{B} 0 \xrightarrow{A}1 \xrightarrow{B} 2 \xrightarrow{B} 4 \xrightarrow{A}5 \xrightarrow{A}6 \xrightarrow{A} 7 \xrightarrow{B}14.
It is not required to minimize the length of S.

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

A sequence of n numbers, S = (s_1, s_2, \dots, s_n), is said to be non-decreasing if and only if s_i \leq s_{i+1} holds for every i (1 \leq i \leq n - 1).

Given are non-decreasing sequences of N integers each: A = (a_1, a_2, \dots, a_N) and B = (b_1, b_2, \dots, b_N).
Consider a non-decreasing sequence of N integers, C = (c_1, c_2, \dots, c_N), that satisfies the following condition:

• a_i \leq c_i \leq b_i for every i (1 \leq i \leq N).

Find the number, modulo 998244353, of sequences that can be C.

Constraints

• 1 \leq N \leq 3000
• 0 \leq a_i \leq b_i \leq 3000 (1 \leq i \leq N)
• The sequences A and B are non-decreasing.
• All values in input are integers.

Sample Input 1

2
1 1
2 3

Sample Output 1

5

There are five sequences that can be C, as follows.

• (1, 1)
• (1, 2)
• (1, 3)
• (2, 2)
• (2, 3)

Note that (2, 1) does not satisfy the condition since it is not non-decreasing.

Sample Input 2

3
2 2 2
2 2 2

Sample Output 2

1

There is one sequence that can be C, as follows.

• (2, 2, 2)

Sample Input 3

10
1 2 3 4 5 6 7 8 9 10
1 4 9 16 25 36 49 64 81 100

Sample Output 3

978222082

Be sure to find the count modulo 998244353.