Problem Statement

There are zero or more sensors placed on a grid of H rows and W columns. Let (i, j) denote the square in the i-th row from the top and the j-th column from the left.
Whether each square contains a sensor is given by the strings S_1, S_2, \ldots, S_H, each of length W. (i, j) contains a sensor if and only if the j-th character of S_i is #.
These sensors interact with other sensors in the squares horizontally, vertically, or diagonally adjacent to them and operate as one sensor. Here, a cell (x, y) and a cell (x', y') are said to be horizontally, vertically, or diagonally adjacent if and only if \max(|x-x'|,|y-y'|) = 1.
Note that if sensor A interacts with sensor B and sensor A interacts with sensor C, then sensor B and sensor C also interact.

Considering the interacting sensors as one sensor, find the number of sensors on this grid.

Constraints

• 1 \leq H, W \leq 1000
• H and W are integers.
• S_i is a string of length W where each character is # or ..

Sample Input 1

5 6
.##...
...#..
....##
#.#...
..#...

Sample Output 1

3

When considering the interacting sensors as one sensor, the following three sensors exist:

• The interacting sensors at (1,2),(1,3),(2,4),(3,5),(3,6)
• The sensor at (4,1)
• The interacting sensors at (4,3),(5,3)

Sample Input 2

3 3
#.#
.#.
#.#

Sample Output 2

1

Sample Input 3

4 2
..
..
..
..

Sample Output 3

0

Sample Input 4

5 47
.#..#..#####..#...#..#####..#...#...###...#####
.#.#...#.......#.#...#......##..#..#...#..#....
.##....#####....#....#####..#.#.#..#......#####
.#.#...#........#....#......#..##..#...#..#....
.#..#..#####....#....#####..#...#...###...#####

Sample Output 4

7

Problem Statement

You are given a string S of length N. Let S_i\ (1\leq i \leq N) be the i-th character of S from the left.

You may perform the following two kinds of operations zero or more times in any order:

• Pay A yen (the currency in Japan). Move the leftmost character of S to the right end. In other words, change S_1S_2\ldots S_N to S_2\ldots S_NS_1.

• Pay B yen. Choose an integer i between 1 and N, and replace S_i with any lowercase English letter.

How many yen do you need to pay to make S a palindrome?

Constraints

• 1\leq N \leq 5000
• 1\leq A,B\leq 10^9
• S is a string of length N consisting of lowercase English letters.
• All values in the input except for S are integers.

Sample Input 1

5 1 2
rrefa

Sample Output 1

3

First, pay 2 yen to perform the operation of the second kind once: let i=5 to replace S_5 with e. S is now rrefe.

Then, pay 1 yen to perform the operation of the first kind once. S is now refer, which is a palindrome.

Thus, you can make S a palindrome for 3 yen. Since you cannot make S a palindrome for 2 yen or less, 3 is the answer.

Sample Input 2

8 1000000000 1000000000
bcdfcgaa

Sample Output 2

4000000000

Note that the answer may not fit into a 32-bit integer type.

Problem Statement

You are given N circles on the xy-coordinate plane. For each i = 1, 2, \ldots, N, the i-th circle is centered at (x_i, y_i) and has a radius of r_i.

Determine whether it is possible to get from (s_x, s_y) to (t_x, t_y) by only passing through points that lie on the circumference of at least one of the N circles.

Constraints

• 1 \leq N \leq 3000
• -10^9 \leq x_i, y_i \leq 10^9
• 1 \leq r_i \leq 10^9
• (s_x, s_y) lies on the circumference of at least one of the N circles.
• (t_x, t_y) lies on the circumference of at least one of the N circles.
• All values in input are integers.

Sample Input 1

4
0 -2 3 3
0 0 2
2 0 2
2 3 1
-3 3 3

Sample Output 1

Yes

Here is one way to get from (0, -2) to (3, 3).

• From (0, -2), pass through the circumference of the 1-st circle counterclockwise to reach (1, -\sqrt{3}).
• From (1, -\sqrt{3}), pass through the circumference of the 2-nd circle clockwise to reach (2, 2).
• From (2, 2), pass through the circumference of the 3-rd circle counterclockwise to reach (3, 3).

Thus, Yes should be printed.

Sample Input 2

3
0 1 0 3
0 0 1
0 0 2
0 0 3

Sample Output 2

No

It is impossible to get from (0, 1) to (0, 3) by only passing through points on the circumference of at least one of the circles, so No should be printed.

Problem Statement

There is a H \times W-square grid with H horizontal rows and W vertical columns. Let (i, j) denote the square at the i-th row from the top and j-th column from the left.

The grid has a rook, initially on (x_1, y_1). Takahashi will do the following operation K times.

• Move the rook to a square that shares the row or column with the square currently occupied by the rook. Here, it must move to a square different from the current one.

How many ways are there to do the K operations so that the rook will be on (x_2, y_2) in the end? Since the answer can be enormous, find it modulo 998244353.

Constraints

• 2 \leq H, W \leq 10^9
• 1 \leq K \leq 10^6
• 1 \leq x_1, x_2 \leq H
• 1 \leq y_1, y_2 \leq W

Sample Input 1

2 2 2
1 2 2 1

Sample Output 1

2

We have the following two ways.

• First, move the rook from (1, 2) to (1, 1). Second, move it from (1, 1) to (2, 1).
• First, move the rook from (1, 2) to (2, 2). Second, move it from (2, 2) to (2, 1).

Sample Input 2

1000000000 1000000000 1000000
1000000000 1000000000 1000000000 1000000000

Sample Output 2

24922282

Be sure to find the count modulo 998244353.

Sample Input 3

3 3 3
1 3 3 3

Sample Output 3

9

Problem Statement

There are N cards called card 1, card 2, \ldots, card N. On card i (1\leq i\leq N), an integer A_i is written.

For K=1, 2, \ldots, N, solve the following problem.

We have a bag that contains K cards: card 1, card 2, \ldots, card K.
Let us perform the following operation twice, and let x and y be the numbers recorded, in the recorded order.

Draw a card from the bag uniformly at random, and record the number written on that card. Then, return the card to the bag.

Print the expected value of \max(x,y), modulo 998244353 (see Notes).
Here, \max(x,y) denotes the value of the greater of x and y (or x if they are equal).

Notes

It can be proved that the sought expected value is always finite and rational. Additionally, under the Constraints of this problem, when that value is represented as \frac{P}{Q} with two coprime integers P and Q, it can be proved that there is a unique integer R such that R \times Q \equiv P\pmod{998244353} and 0 \leq R \lt 998244353. Print this R.

Constraints

• 1 \leq N \leq 2\times 10^5
• 1 \leq A_i \leq 2\times 10^5
• All values in the input are integers.

Sample Input 1

3
5 7 5

Sample Output 1

5
499122183
443664163

For instance, the answer for K=2 is found as follows.

The bag contains card 1 and card 2, with A_1=5 and A_2=7 written on them, respectively.

• If you draw card 1 in the first draw and card 1 again in the second draw, we have x=y=5, so \max(x,y)=5.
• If you draw card 1 in the first draw and card 2 in the second draw, we have x=5 and y=7, so \max(x,y)=7.
• If you draw card 2 in the first draw and card 1 in the second draw, we have x=7 and y=5, so \max(x,y)=7.
• If you draw card 2 in the first draw and card 2 again in the second draw, we have x=y=7, so \max(x,y)=7.

These events happen with the same probability, so the sought expected value is \frac{5+7+7+7}{4}=\frac{13}{2}. We have 499122183\times 2\equiv 13 \pmod{998244353}, so 499122183 should be printed.

Sample Input 2

7
22 75 26 45 72 81 47

Sample Output 2

22
249561150
110916092
873463862
279508479
360477194
529680742