Problem Statement

There is a grid with H rows and W columns.

Each cell of the grid is land or sea, which is represented by H strings S_1, S_2, \ldots, S_H of length W. Let (i, j) denote the cell at the i-th row from the top and j-th column from the left, and (i, j) is land if the j-th character of S_i is ., and (i, j) is sea if the character is #.

The constraints guarantee that all cells on the perimeter of the grid (that is, the cells (i, j) that satisfy at least one of i = 1, i = H, j = 1, j = W) are sea.

Takahashi's spaceship has crash-landed on a cell in the grid. Afterward, he moved N times on the grid following the instructions represented by a string T of length N consisting of L, R, U, and D. For i = 1, 2, \ldots, N, the i-th character of T describes the i-th move as follows:

• L indicates a move of one cell to the left. That is, if he is at (i, j) before the move, he will be at (i, j-1) after the move.
• R indicates a move of one cell to the right. That is, if he is at (i, j) before the move, he will be at (i, j+1) after the move.
• U indicates a move of one cell up. That is, if he is at (i, j) before the move, he will be at (i-1, j) after the move.
• D indicates a move of one cell down. That is, if he is at (i, j) before the move, he will be at (i+1, j) after the move.

It is known that all cells along his path (including the cell where he crash-landed and the cell he is currently on) are not sea. Print the number of cells that could be his current position.

Constraints

• H, W, and N are integers.
• 3 \leq H, W \leq 500
• 1 \leq N \leq 500
• T is a string of length N consisting of L, R, U, and D.
• S_i is a string of length W consisting of . and #.
• There is at least one cell that could be Takahashi's current position.
• All cells on the perimeter of the grid are sea.

Sample Input 1

6 7 5
LULDR
#######
#...#.#
##...##
#.#...#
#...#.#
#######

Sample Output 1

2

The following two cases are possible, so there are two cells that could be Takahashi's current position: (3, 4) and (4, 5).

• He crash-landed on cell (3, 5) and moved (3, 5) \rightarrow (3, 4) \rightarrow (2, 4) \rightarrow (2, 3) \rightarrow (3, 3) \rightarrow (3, 4).
• He crash-landed on cell (4, 6) and moved (4, 6) \rightarrow (4, 5) \rightarrow (3, 5) \rightarrow (3, 4) \rightarrow (4, 4) \rightarrow (4, 5).

Sample Input 2

13 16 9
ULURDLURD
################
##..##.#..####.#
###.#..#.....#.#
#..##..#####.###
#...#..#......##
###.##.#..#....#
##.#####....##.#
###.###.#.#.#..#
######.....##..#
#...#.#.######.#
##..###..#..#.##
#...#.#.#...#..#
################

Sample Output 2

6

Problem Statement

There is a tree T with N vertices. The i-th edge (1\leq i\leq N-1) connects vertex U_i and vertex V_i.

You are given two different vertices X and Y in T. List all vertices along the simple path from vertex X to vertex Y in order, including endpoints.

It can be proved that, for any two different vertices a and b in a tree, there is a unique simple path from a to b.

Constraints

• 1\leq N\leq 2\times 10^5
• 1\leq X,Y\leq N
• X\neq Y
• 1\leq U_i,V_i\leq N
• All values in the input are integers.
• The given graph is a tree.

Sample Input 1

5 2 5
1 2
1 3
3 4
3 5

Sample Output 1

2 1 3 5

The tree T is shown below. The simple path from vertex 2 to vertex 5 is 2 \to 1 \to 3 \to 5.
Thus, 2,1,3,5 should be printed in this order, with spaces in between.

Sample Input 2

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

Sample Output 2

1 2

The tree T is shown below.

Problem Statement

You are given a sequence of non-negative integers A=(a_1,a_2,\ldots,a_N).

Let S be the set of non-negative integers that can be the sum of K terms in A (with distinct indices).

Find the greatest multiple of D in S. If there is no multiple of D in S, print -1 instead.

Constraints

• 1 \leq K \leq N \leq 100
• 1 \leq D \leq 100
• 0 \leq a_i \leq 10^9
• All values in the input are integers.

Sample Input 1

4 2 2
1 2 3 4

Sample Output 1

6

Here are all the ways to choose two terms in A.

• Choose a_1 and a_2, whose sum is 1+2=3.
• Choose a_1 and a_3, whose sum is 1+3=4.
• Choose a_1 and a_4, whose sum is 1+4=5.
• Choose a_2 and a_3, whose sum is 2+3=5.
• Choose a_2 and a_4, whose sum is 2+4=6.
• Choose a_3 and a_4, whose sum is 3+4=7.

Thus, we have S=\{3,4,5,6,7\}. The greatest multiple of 2 in S is 6, so you should print 6.

Sample Input 2

3 1 2
1 3 5

Sample Output 2

-1

In this example, we have S=\{1,3,5\}. Nothing in S is a multiple of 2, so you should print -1.

Problem Statement

How many integer sequences A=(A_1,\ldots,A_N) of length N satisfy all the conditions below?

• 1\le A_i \le M (1 \le i \le N)

• |A_i - A_{i+1}| \geq K (1 \le i \le N - 1)

Since the count can be enormous, find it modulo 998244353.

Constraints

• 2 \leq N \leq 1000
• 1 \leq M \leq 5000
• 0 \leq K \leq M-1
• All values in input are integers.

Sample Input 1

2 3 1

Sample Output 1

6

The following 6 sequences satisfy the conditions.

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

Sample Input 2

3 3 2

Sample Output 2

2

The following 2 sequences satisfy the conditions.

• (1,3,1)
• (3,1,3)

Sample Input 3

100 1000 500

Sample Output 3

657064711

Print the count modulo 998244353.

Problem Statement

There are apple trees lined up on a number line, and N apples fall from the trees.
Specifically, for each 1\leq i\leq N, an apple falls at coordinate X_i at time T_i.

Takahashi has a basket with durability D and length W, and he can take the following action exactly once.

Choose positive integers S and L. He sets up the basket to cover the range L-0.5\leq x\leq L+W-0.5 at time S-0.5, and retrieves it at time S+D-0.5. He gets all the apples that fell into the range covered by the basket between the time it was set up and the time it was retrieved.

He cannot move the basket once it has been set up, nor can he set it up again once it has been retrieved.
Find the maximum number of apples that he can get.

Constraints

• 1 \leq N\leq 2\times 10^5
• 1 \leq D\leq 2\times 10^5
• 1 \leq W\leq 2\times 10^5
• 1 \leq T_i\leq 2\times 10^5
• 1 \leq X_i\leq 2\times 10^5
• All pairs (T_i,X_i) are different.
• All input values are integers.

Sample Input 1

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

Sample Output 1

5

If Takahashi chooses S=3 and L=2, he will set up the basket to cover the range 1.5\leq x\leq 4.5 from time 2.5 to 6.5. Then, he gets the following five apples:

• The apple that falls at coordinate X_2=4 at time T_2=3
• The apple that falls at coordinate X_3=4 at time T_3=6
• The apple that falls at coordinate X_4=2 at time T_4=5
• The apple that falls at coordinate X_5=2 at time T_5=4
• The apple that falls at coordinate X_6=3 at time T_6=4

There is no way to get six or more apples, so print 5.