Problem Statement

You are given an integer N between 0 and 9999 (inclusive).

Print it as a four-digit string after appending to it the necessary number of leading zeros.

Constraints

• 0 \leq N \leq 9999
• N is an integer.

Sample Input 1

321

Sample Output 1

0321

321 has three digits, so we need to add one leading zero to it to make it have four digits.

Sample Input 2

7777

Sample Output 2

7777

Sample Input 3

1

Sample Output 3

0001

Problem Statement

N students took an exam. The students are labeled as Student 1, Student 2, \dots, Student N, and Student i scored a_i points.

A student who scored less than P points are considered to have failed the exam and cannot earn the credit. Find the number of students who failed the exam.

Constraints

• 1 \leq N \leq 10^5
• 1 \leq P \leq 100
• 0 \leq a_i \leq 100 (1 \leq i \leq N)
• All values in input are integers.

Sample Input 1

4 50
80 60 40 0

Sample Output 1

2

Students 1 and 2, who scored 80 and 60 points, respectively, succeeded in scoring at least 50 points to earn the credit.
On the other hand, Students 3 and 4, who scored 40 and 0 points, respectively, fell below 50 points and failed the exam. Thus, the answer is 2.

Sample Input 2

3 90
89 89 89

Sample Output 2

3

Sample Input 3

2 22
6 37

Sample Output 3

1

Problem Statement

2N players, with ID numbers 1 through 2N, will participate in a rock-scissors-paper contest.

The contest has M rounds. Each round has N one-on-one matches, where each player plays in one of them.

For each i=0, 1, \ldots, M, the players' ranks at the end of the i-th round are determined as follows.

• A player with more wins in the first i rounds ranks higher.
• Ties are broken by ID numbers: a player with a smaller ID number ranks higher.

Additionally, for each i=1, \ldots, M, the matches in the i-th round are arranged as follows.

• For each k=1, 2, \ldots, N, a match is played between the players who rank (2k-1)-th and 2k-th at the end of the (i-1)-th round.

In each match, the two players play a hand just once, resulting in one player's win and the other's loss, or a draw.

Takahashi, who can foresee the future, knows that Player i will play A_{i, j} in their match in the j-th round, where A_{i,j} is G, C, or P.
Here, G stands for rock, C stands for scissors, and P stands for paper. (These derive from Japanese.)

Find the players' ranks at the end of the M-th round.

Constraints

• 1 \leq N \leq 50
• 1 \leq M \leq 100
• A_{i,j} is G, C, or P.

Sample Input 1

2 3
GCP
PPP
CCC
PPC

Sample Output 1

3
1
2
4

In the first round, matches are played between Players 1 and 2, and between Players 3 and 4. Player 2 wins the former, and Player 3 wins the latter.
In the second round, matches are played between Players 2 and 3, and between Players 1 and 4. Player 3 wins the former, and Player 1 wins the latter.
In the third round, matches are played between Players 3 and 1, and between Players 2 and 4. Player 3 wins the former, and Player 4 wins the latter.
Therefore, in the end, the players are ranked in the following order: 3,1,2,4, from highest to lowest.

Sample Input 2

2 2
GC
PG
CG
PP

Sample Output 2

1
2
3
4

In the first round, matches are played between Players 1 and 2, and between Players 3 and 4. Player 2 wins the former, and Player 3 wins the latter.
In the second round, matches are played between Players 2 and 3, and between Players 1 and 4. The former is drawn, and Player 1 wins the latter.
Therefore, in the end, the players are ranked in the following order: 1,2,3,4, from highest to lowest.

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.

Problem Statement

Given are a tree with N vertices, a sequence of M numbers A=(A_1,\ldots,A_M), and an integer K.
The vertices are numbered 1 through N, and the i-th edge connects Vertices U_i and V_i.

We will paint each of the N-1 edges of this tree red or blue. Among the 2^{N-1} such ways, find the number of ones that satisfies the following condition, modulo 998244353.

Condition:
Let us put a piece on Vertex A_1, and for each i=1,\ldots,M-1 in this order, move it from Vertex A_i to Vertex A_{i+1} along the edges in the shortest path. After all of these movements, R-B=K holds, where R and B are the numbers of times the piece traverses a red edge and a blue edge, respectively.

Constraints

• 2 \leq N \leq 1000
• 2 \leq M \leq 100
• |K| \leq 10^5
• 1 \leq A_i \leq N
• 1\leq U_i,V_i\leq N
• The given graph is a tree.
• All values in input are integers.

Sample Input 1

4 5 0
2 3 2 1 4
1 2
2 3
3 4

Sample Output 1

2

If we paint the 1-st and 3-rd edges red and the 2-nd edge blue, the piece will traverse the following numbers of red and blue edges:

• 0 red edges and 1 blue edge when moving from Vertex 2 to 3,
• 0 red edges and 1 blue edge when moving from Vertex 3 to 2,
• 1 red edge and 0 blue edges when moving from Vertex 2 to 1,
• 2 red edges and 1 blue edge when moving from Vertex 1 to 4,

for a total of 3 red edges and 3 blue edges, satisfying the condition.

Another way to satisfy the condition is to paint the 1-st and 3-rd edges blue and the 2-nd edge red. There is no other way to satisfy it, so the answer is 2.

Sample Input 2

3 10 10000
1 2 1 2 1 2 2 1 1 2
1 2
1 3

Sample Output 2

0

There may be no way to paint the tree to satisfy the condition.

Sample Input 3

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

Sample Output 3

126

Sample Input 4

5 8 -1
1 4 1 4 2 1 3 5
1 2
4 1
3 1
1 5

Sample Output 4

2

Problem Statement

The Kingdom of AtCoder is composed of N towns and N-1 roads.
The towns are labeled as Town 1, Town 2, \dots, Town N. Similarly, the roads are labeled as Road 1, Road 2, \dots, Road N-1. Road i connects Towns A_i and B_i bidirectionally, and you have to pay the toll of C_i to go through it. For every pair of different towns (i, j), it is possible to go from Town A_i to Town B_j via the roads.

You are given a sequence D = (D_1, D_2, \dots, D_N), where D_i is the cost to do sightseeing in Town i. Let us define the travel cost E_{i,j} from Town i to Town j as the total toll incurred when going from Town i to Town j, plus D_{j}.

• More formally, E_{i,j} is defined as D_j + \displaystyle\sum_{l=0}^{k-1} c_l, where the shortest path between i and j is i = p_0, p_1, \dots, p_{k-1}, p_k = j and the toll for the road connecting Towns p_{l} and p_{l+1} is c_l.

For every i, find the maximum possible travel cost when traveling from Town i to another town.

• More formally, for every i, find \max_{1 \leq j \leq N, j \neq i} E_{i,j}.

Constraints

• 2 \leq N \leq 2 \times 10^5
• 1 \leq A_i \leq N (1 \leq i \leq N-1)
• 1 \leq B_i \leq N (1 \leq i \leq N-1)
• 1 \leq C_i \leq 10^9 (1 \leq i \leq N-1)
• 1 \leq D_i \leq 10^9 (1 \leq i \leq N)
• It is possible to travel from Town i to Town j via some number of roads, for a pair of integers (i,j) such that 1 \leq i \lt j \leq N.
• All values in input are integers.

Sample Input 1

3
1 2 2
2 3 3
1 2 3

Sample Output 1

8
6
6

The value of E_{i,j} for every ordered pair of towns (i,j) is as follows.

• E_{1,2} = 2 + 2 = 4
• E_{1,3} = 5 + 3 = 8
• E_{2,1} = 2 + 1 = 3
• E_{2,3} = 3 + 3 = 6
• E_{3,1} = 5 + 1 = 6
• E_{3,2} = 3 + 2 = 5

Sample Input 2

6
1 2 3
1 3 1
1 4 4
1 5 1
1 6 5
9 2 6 5 3 100

Sample Output 2

105
108
106
109
106
14

Sample Input 3

6
1 2 1000000000
2 3 1000000000
3 4 1000000000
4 5 1000000000
5 6 1000000000
1 2 3 4 5 6

Sample Output 3

5000000006
4000000006
3000000006
3000000001
4000000001
5000000001

Problem Statement

We have a sequence 2,22,222,2222,\ldots, where the i-th term is an i-digit integer whose digits are all 2.

Where does a multiple of K appear in this sequence for the first time? If the first multiple of K is the x-th term of the sequence, print x; if there is no multiple of K, print -1.

Given T cases, solve each of them.

Constraints

• 1 \leq T \leq 200
• 1 \leq K \leq 10^8
• All values in input are integers.

Sample Input 1

4
1
7
10
999983

Sample Output 1

1
6
-1
999982

We have four cases.

• 2 is a multiple of 1.
• None of 2,22,222,2222,22222 is a multiple of 7, but 222222 is.
• None of 2,22,\ldots is a multiple of 10.

Problem Statement

For a positive integer N, a rooted binary tree that satisfies the following conditions is said to be a beautiful binary tree of degree N.

• Each vertex has 0 or 1 written on it.
• Each vertex that is a leaf has 1 written on it.
• It is possible to do the following operation at most N-1 times so that the root has N written on it and the other vertices have 0 written on them.
  • Choose vertices u and v, where v must be a child of u or a child of "a child of u." Let a_u \gets a_u + a_v, a_v \gets 0, where a_u and a_v are the numbers written on u and v, respectively.

Given N, find the number, modulo 998244353, of beautiful binary trees of degree N.

Constraints

• 1 \leq N \leq 10^7
• All values in input are integers.

Sample Input 1

1

Sample Output 1

1

The only binary tree that satisfies the condition is a tree with one vertex whose root has 1 written on it.

Sample Input 2

2

Sample Output 2

6

The binary trees that satisfy the condition are the six trees shown below.

Sample Input 3

222

Sample Output 3

987355927

Sample Input 4

222222

Sample Output 4

675337738