Problem Statement

Given a string S of length N consisting of lowercase English alphabets, print the last character of S.

Constraints

• N is an integer.
• 1 ≤ N ≤ 1000
• S is a string of length N consisting of lowercase English alphabets.

Sample Input 1

5
abcde

Sample Output 1

e

The last character of S = {}abcde is e, so e should be printed.

Sample Input 2

1
a

Sample Output 2

a

Problem Statement

Consider an xy-plane. The positive direction of the x-axis is in the direction of east, and the positive direction of the y-axis is in the direction of north.
Takahashi is initially at point (x, y) = (0, 0) and facing east (in the positive direction of the x-axis).

You are given a string T = t_1t_2\ldots t_N of length N consisting of S and R. Takahashi will do the following move for each i = 1, 2, \ldots, N in this order.

• If t_i = S, Takahashi advances in the current direction by distance 1.
• If t_i = R, Takahashi turns 90 degrees clockwise without changing his position. As a result, Takahashi's direction changes as follows.
  • If he is facing east (in the positive direction of the x-axis) before he turns, he will face south (in the negative direction of the y-axis) after he turns.
  • If he is facing south (in the negative direction of the y-axis) before he turns, he will face west (in the negative direction of the x-axis) after he turns.
  • If he is facing west (in the negative direction of the x-axis) before he turns, he will face north (in the positive direction of the y-axis) after he turns.
  • If he is facing north (in the positive direction of the y-axis) before he turns, he will face east (in the positive direction of the x-axis) after he turns.

Print the coordinates Takahashi is at after all the steps above have been done.

Constraints

• 1 \leq N \leq 10^5
• N is an integer.
• T is a string of length N consisting of S and R.

Sample Input 1

4
SSRS

Sample Output 1

2 -1

Takahashi is initially at (0, 0) facing east. Then, he moves as follows.

1. t_1 = S, so he advances in the direction of east by distance 1, arriving at (1, 0).
2. t_2 = S, so he advances in the direction of east by distance 1, arriving at (2, 0).
3. t_3 = R, so he turns 90 degrees clockwise, resulting in facing south.
4. t_4 = S, so he advances in the direction of south by distance 1, arriving at (2, -1).

Thus, Takahashi's final position, (x, y) = (2, -1), should be printed.

Sample Input 2

20
SRSRSSRSSSRSRRRRRSRR

Sample Output 2

0 1

Problem Statement

Takahashi and Aoki will play the following game against each other.

Starting from Takahashi, the two alternatingly declare an integer between 1 and 2N+1 (inclusive) until the game ends. Any integer declared by either player cannot be declared by either player again. The player who is no longer able to declare an integer loses; the player who didn't lose wins.

In this game, Takahashi will always win. Your task is to actually play the game on behalf of Takahashi and win the game.

Constraints

• 1 \leq N \leq 1000
• N is an integer.

Input and Output

This task is an interactive task (in which your program and the judge program interact with each other via inputs and outputs).
Your program plays the game on behalf of Takahashi, and the judge program plays the game on behalf of Aoki.

First, your program is given a positive integer N from Standard Input. Then, the following procedures are repeated until the game ends.

1. Your program outputs an integer between 1 and 2N+1 (inclusive) to Standard Output, which defines the integer that Takahashi declares. (You cannot output an integer that is already declared by either player.)
2. The integer that Aoki declares is given by the judge program to your program from Standard Input. (No integer that is already declared by either player will be given.) If Aoki has no more integer to declare, 0 is given instead, which means that the game ended and Takahashi won.

Notes

• After each output, you must flush Standard Output. Otherwise, you may get TLE.
• After the game ended and Takahashi won, the program must be terminated immediately. Otherwise, the judge does not necessarily give AC.
• If your program outputs something that violates the rules of the game (such as an integer that has already been declared by either player), your answer is considered incorrect. In such case, the verdict is indeterminate. It does not necessarily give WA.

Sample Input and Output

Problem Statement

There are three Takahashis numbered 1, 2 and 3, and three hats colored red, green, and blue. Each Takahashi is wearing one hat. The color of the hat that Takahashi i is currently wearing is represented by a character S_i. Here, R corresponds to red, G to green, and B to blue. Now, they will do the following operation exactly 10^{18} times.

Operation

• Choose two out of the three Takahashis. The two exchange the hats they are wearing.

Is it possible to make Takahashi i wearing the hat of color corresponding to character T_i after the 10^{18} repetitions?

Constraints

• S_1, S_2, S_3 are a permutation of R, G, B.
• T_1, T_2, T_3 are a permutation of R, G, B.

Sample Input 1

R G B
R G B

Sample Output 1

Yes

For example, the objective can be achieved by repeating 10^{18} times the operation of swapping the hats of Takahashi 1 and Takahashi 2.

Problem Statement

You are given a simple undirected graph with N vertices and M edges. The vertices are numbered from 1 through N, and the edges are numbered from 1 through M. Edge i connects Vertex U_i and Vertex V_i.

You are given integers K, S, T, and X. How many sequences A = (A_0, A_1, \dots, A_K) are there satisfying the following conditions?

• A_i is an integer between 1 and N (inclusive).
• A_0 = S
• A_K = T
• There is an edge that directly connects Vertex A_i and Vertex A_{i+1}.
• Integer X\ (X≠S,X≠T) appears even number of times (possibly zero) in sequence A.

Since the answer can be very large, find the answer modulo 998244353.

Constraints

• All values in input are integers.
• 2≤N≤2000
• 1≤M≤2000
• 1≤K≤2000
• 1≤S,T,X≤N
• X≠S
• X≠T
• 1≤U_i<V_i≤N
• If i ≠ j, then (U_i, V_i) ≠ (U_j, V_j).

Sample Input 1

4 4 4 1 3 2
1 2
2 3
3 4
1 4

Sample Output 1

4

The following 4 sequences satisfy the conditions:

• (1, 2, 1, 2, 3)
• (1, 2, 3, 2, 3)
• (1, 4, 1, 4, 3)
• (1, 4, 3, 4, 3)

On the other hand, (1, 2, 3, 4, 3) and (1, 4, 1, 2, 3) do not, since there are odd number of occurrences of 2.

Sample Input 2

6 5 10 1 2 3
2 3
2 4
4 6
3 6
1 5

Sample Output 2

0

The graph is not necessarily connected.

Sample Input 3

10 15 20 4 4 6
2 6
2 7
5 7
4 5
2 4
3 7
1 7
1 4
2 9
5 10
1 3
7 8
7 9
1 6
1 2

Sample Output 3

952504739

Find the answer modulo 998244353.

Problem Statement

You are given a simple connected undirected graph with N vertices and M edges. (A graph is said to be simple if it has no multi-edges and no self-loops.)
For i = 1, 2, \ldots, M, the i-th edge connects Vertex u_i and Vertex v_i.

A sequence (A_1, A_2, \ldots, A_k) is said to be a path of length k if both of the following two conditions are satisfied:

• For all i = 1, 2, \dots, k, it holds that 1 \leq A_i \leq N.
• For all i = 1, 2, \ldots, k-1, Vertex A_i and Vertex A_{i+1} are directly connected by an edge.

An empty sequence is regarded as a path of length 0.

Let S = s_1s_2\ldots s_N be a string of length N consisting of 0 and 1. A path A = (A_1, A_2, \ldots, A_k) is said to be a good path with respect to S if the following conditions are satisfied:

• For all i = 1, 2, \ldots, N, it holds that:
  • if s_i = 0, then A has even number of i's.
  • if s_i = 1, then A has odd number of i's.

There are 2^N possible S (in other words, there are 2^N strings of length N consisting of 0 and 1). Find the sum of "the length of the shortest good path with respect to S" over all those S.

Under the Constraints of this problem, it can be proved that, for any string S of length N consisting of 0 and 1, there is at least one good path with respect to S.

Constraints

• 2 \leq N \leq 17
• N-1 \leq M \leq \frac{N(N-1)}{2}
• 1 \leq u_i, v_i \leq N
• The given graph is simple and connected.
• All values in input are integers.

Sample Input 1

3 2
1 2
2 3

Sample Output 1

14

• For S = 000, the empty sequence () is the shortest good path with respect to S, whose length is 0.
• For S = 100, (1) is the shortest good path with respect to S, whose length is 1.
• For S = 010, (2) is the shortest good path with respect to S, whose length is 1.
• For S = 110, (1, 2) is the shortest good path with respect to S, whose length is 2.
• For S = 001, (3) is the shortest good path with respect to S, whose length is 1.
• For S = 101, (1, 2, 3, 2) is the shortest good path with respect to S, whose length is 4.
• For S = 011, (2, 3) is the shortest good path with respect to S, whose length is 2.
• For S = 111, (1, 2, 3) is the shortest good path with respect to S, whose length is 3.

Therefore, the sought answer is 0 + 1 + 1 + 2 + 1 + 4 + 2 + 3 = 14.

Sample Input 2

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

Sample Output 2

108

Problem Statement

You are given a simple connected undirected graph with N vertices and M edges. (A graph is said to be simple if it has no multi-edges and no self-loops.)
For i = 1, 2, \ldots, M, the i-th edge connects Vertex u_i and Vertex v_i.

A sequence (A_1, A_2, \ldots, A_k) is said to be a path of length k if both of the following two conditions are satisfied:

• For all i = 1, 2, \dots, k, it holds that 1 \leq A_i \leq N.
• For all i = 1, 2, \ldots, k-1, Vertex A_i and Vertex A_{i+1} are directly connected with an edge.

An empty sequence is regarded as a path of length 0.

You are given a sting S = s_1s_2\ldots s_N of length N consisting of 0 and 1. A path A = (A_1, A_2, \ldots, A_k) is said to be a good path with respect to S if the following conditions are satisfied:

• For all i = 1, 2, \ldots, N, it holds that:
  • if s_i = 0, then A has even number of i's.
  • if s_i = 1, then A has odd number of i's.

Under the Constraints of this problem, it can be proved that there is at least one good path with respect to S of length at most 4N. Print a good path with respect to S of length at most 4N.

Constraints

• 2 \leq N \leq 10^5
• N-1 \leq M \leq \min\lbrace 2 \times 10^5, \frac{N(N-1)}{2}\rbrace
• 1 \leq u_i, v_i \leq N
• The given graph is simple and connected.
• N, M, u_i, and v_i are integers.
• S is a string of length N consisting of 0 and 1.

Sample Input 1

6 6
6 3
2 5
4 2
1 3
6 5
3 2
110001

Sample Output 1

9
2 5 6 5 6 3 1 3 6

The path (2, 5, 6, 5, 6, 3, 1, 3, 6) has a length no greater than 4N, and

• it has odd number (1) of 1
• it has odd number (1) of 2
• it has even number (2) of 3
• it has even number (0) of 4
• it has even number (2) of 5
• it has odd number (3) of 6

so it is a good path with respect to S = 110001.

Sample Input 2

3 3
3 1
3 2
1 2
000

Sample Output 2

0

An empty path () is a good path with respect to S = 000000. Alternatively, paths like (1, 2, 3, 1, 2, 3) are also accepted.

Problem Statement

There is a set S of points on a two-dimensional plane. S is initially empty.

For each i = 1, 2, \dots, Q in this order, process the following query.

• You are given integers X_i, Y_i, A_i, and B_i. Add point (X_i, Y_i) to S, and then find \displaystyle \max_{(x,y) \in S}\left\{A_ix + B_iy\right\}.

Constraints

• All values in input are integers.
• 1≤Q≤2 \times 10^5
• |X_i|, |Y_i|, |A_i|, |B_i| ≤10^9
• If i ≠ j, then (X_i, Y_i) ≠ (X_j, Y_j).

Sample Input 1

4
1 0 -1 -1
0 1 2 0
-1 0 1 1
0 -1 1 -2

Sample Output 1

-1
2
1
2

• When i = 1: add point (1, 0) to S, then it will become S = \{(1, 0)\}. For (x, y) = (1, 0), we have -x - y = -1, which is the maximum.
• When i = 2: add point (0, 1) to S, then it will become S = \{(0, 1), (1, 0)\}. For (x, y) = (1, 0), we have 2x = 2, which is the maximum.
• When i = 3: add point (-1, 0) to S, then it will become S = \{(-1, 0), (0, 1), (1, 0)\}. For (x, y) = (1, 0) or (x, y) = (0, 1), we have x + y = 1, which is the maximum.
• When i = 4: add point (0, -1) to S, then it will become S = \{(-1, 0), (0, -1), (0, 1), (1, 0)\}. For (x, y) = (0, -1), we have x - 2y = 2, which is the maximum.

Sample Input 2

9
-1 4 -8 -2
9 -9 -7 7
4 1 6 7
-4 -1 -4 -5
-9 3 -2 -6
-1 0 -8 5
-8 -5 0 0
8 3 0 -4
2 -5 2 5

Sample Output 2

0
35
31
21
36
87
0
36
31