Problem Statement

Takahashi eats three plates for breakfast: rice, miso soup, and salad.

His table is long and narrow, so he arranged the three plates in a row. The arrangement is given by a string S, where the i-th plate from the left is rice if S_i is R, miso soup if S_i is M, and salad if S_i is S.

Determine whether the plate of rice is to the left of the plate of miso soup.

Constraints

• |S| = 3
• S contains one R, one M, and one S.

Sample Input 1

RSM

Sample Output 1

Yes

The plate of rice is at the 1st position from the left, and the plate of miso soup is at the 3rd position from the left. Since the plate of rice is to the left, print Yes.

Sample Input 2

SMR

Sample Output 2

No

The plates are arranged as salad, miso soup, and rice from left to right.

Problem Statement

You are given two strings S and T consisting of lowercase English letters.

Determine if there exists a pair of integers c and w such that 1 \leq c \leq w < |S| and the following condition is satisfied. Here, |S| denotes the length of the string S. Note that w must be less than |S|.

• If S is split at every w characters from the beginning, the concatenation of the c-th characters of the substrings of length at least c in order equals T.

Constraints

• S and T are strings consisting of lowercase English letters.
• 1 \leq |T| \leq |S| \leq 100

Sample Input 1

atcoder toe

Sample Output 1

Yes

If S is split at every two characters, it looks like this:

at
co
de
r

Then, the concatenation of the 2nd characters of the substrings of length at least 2 is toe, which equals T. Thus, print Yes.

Sample Input 2

beginner r

Sample Output 2

No

w=|S| is not allowed, and no pair of integers 1 \leq c \leq w < |S| satisfies the condition. Thus, print No.

Sample Input 3

verticalreading agh

Sample Output 3

No

Problem Statement

There are N boxes numbered 1 to N and N items numbered 1 to N. Item i (1 \leq i \leq N) is in box A_i and has a weight of W_i.

You can repeatedly perform the operation of choosing an item and moving it to another box zero or more times. If the weight of the item being moved is w, the cost of the operation is w.

Find the minimum total cost required to make each box contain exactly one item.

Constraints

• 1 \leq N \leq 10^{5}
• 1 \leq A_i \leq N (1 \leq i \leq N)
• 1 \leq W_i \leq 10^{4} (1 \leq i \leq N)
• All input values are integers.

Sample Input 1

5
2 2 3 3 5
33 40 2 12 16

Sample Output 1

35

With the following two moves, you can make each box contain exactly one item:

• Move item 1 from box 2 to box 1. The cost is 33.
• Move item 3 from box 3 to box 4. The cost is 2.

The total cost of these two moves is 35. It is impossible to make each box contain exactly one item with a cost less than 35, so print 35.

Sample Input 2

12
3 6 7 4 12 4 8 11 11 1 8 11
3925 9785 9752 3587 4013 1117 3937 7045 6437 6208 3391 6309

Sample Output 2

17254

Problem Statement

There are N ants on a number line, labeled 1 to N. Ant i (1 \leq i \leq N) starts at coordinate X_i and faces either a positive or negative direction. Initially, all ants are at distinct coordinates. The direction each ant is facing is represented by a binary string S of length N, where ant i is facing the negative direction if S_i is 0 and the positive direction if S_i is 1.

Let the current time be 0, and the ants move in their respective directions at a speed of 1 unit per unit time for (T+0.1) units of time until time (T+0.1). If multiple ants reach the same coordinate, they pass through each other without changing direction or speed. After (T+0.1) units of time, all ants stop.

Find the number of pairs (i, j) such that 1 \leq i < j \leq N and ants i and j pass each other from now before time (T+0.1).

Constraints

• 2 \leq N \leq 2 \times 10^{5}
• 1 \leq T \leq 10^{9}
• S is a string of length N consisting of 0 and 1.
• -10^{9} \leq X_i \leq 10^{9} (1 \leq i \leq N)
• X_i \neq X_j (1 \leq i < j \leq N)
• N, T, and X_i (1 \leq i \leq N) are integers.

Sample Input 1

6 3
101010
-5 -1 0 1 2 4

Sample Output 1

5

The following five pairs of ants pass each other:

• Ant 3 and ant 4 pass each other at time 0.5.
• Ant 5 and ant 6 pass each other at time 1.
• Ant 1 and ant 2 pass each other at time 2.
• Ant 3 and ant 6 pass each other at time 2.
• Ant 1 and ant 4 pass each other at time 3.

No other pairs of ants pass each other, so print 5.

Sample Input 2

13 656320850
0100110011101
-900549713 -713494784 -713078652 -687818593 -517374932 -498415009 -472742091 -390030458 -379340552 -237481538 -44636942 352721061 695864366

Sample Output 2

14

Problem Statement

There are N - 1 white balls and one black ball. These N balls are arranged in a row, with the black ball initially at the leftmost position.

Takahashi will perform the following operation exactly K times.

• Choose an integer uniformly at random between 1 and N, inclusive, twice. Let a and b the chosen integers. If a \neq b, swap the a-th and b-th balls from the left.

After K operations, let the black ball be at the x-th position from the left. Find the expected value of x, modulo 998244353.

Constraints

• 1 \leq N \leq 998244352
• 1 \leq K \leq 10^5

Sample Input 1

2 1

Sample Output 1

499122178

After one operation, the probabilities that the black ball is at the 1st position and the 2nd position from the left are both \displaystyle \frac{1}{2}. Thus, the expected value is \displaystyle \frac{3}{2}.

Sample Input 2

3 2

Sample Output 2

554580198

Sample Input 3

4 4

Sample Output 3

592707587

Problem Statement

You are given N intervals numbered 1 to N. Interval i is [L_i, R_i].

Two intervals [l_a, r_a] and [l_b, r_b] are said to intersect if and only if they satisfy either (l_a < l_b < r_a < r_b) or (l_b < l_a < r_b < r_a).

Define f(l, r) as the number of intervals i (1 \leq i \leq N) that intersect with the interval [l, r].

Among all pairs of integers (l, r) satisfying 0 \leq l < r \leq 10^{9}, find the pair (l, r) that maximizes f(l, r). If there are multiple such pairs, choose the one with the smallest l. If there are still multiple pairs, choose the one with the smallest r among them. (Since 0 \leq l < r, the pair (l, r) to be answered is uniquely determined.)

Constraints

• 1 \leq N \leq 10^{5}
• 0 \leq L_i < R_i \leq 10^{9} (1 \leq i \leq N)
• All input values are integers.

Sample Input 1

5
1 7
3 9
7 18
10 14
15 20

Sample Output 1

4 11

The maximum value of f(l, r) is 4, and among the pairs (l, r) that achieve f(l, r) = 4, the smallest l is 4. The pairs (l, r) that satisfy f(l, r) = 4 and l = 4 are the following five:

• (l, r) = (4, 11)
• (l, r) = (4, 12)
• (l, r) = (4, 13)
• (l, r) = (4, 16)
• (l, r) = (4, 17)

Among these, the smallest r is 11, so print 4 and 11.

Sample Input 2

11
856977192 996441446
298251737 935869360
396653206 658841528
710569907 929136831
325371222 425309117
379628374 697340458
835681913 939343451
140179224 887672320
375607390 611397526
93530028 581033295
249611310 775998537

Sample Output 2

396653207 887672321

Problem Statement

You are given an integer sequence A of length N. Takahashi will perform the following operation exactly once:

• Choose an integer x between 1 and N, inclusive, and an arbitrary integer y. Replace A_x with y.

Find the maximum possible length of a longest increasing subsequence (LIS) of A after performing the operation.

Constraints

• 1 \leq N \leq 2 \times 10^5
• 1 \leq A_i \leq 10^9

Sample Input 1

4
3 2 2 4

Sample Output 1

3

The length of a LIS of the given sequence A is 2. For example, if you replace A_1 with 1, the length of a LIS of A becomes 3, which is the maximum.

Sample Input 2

5
4 5 3 6 7

Sample Output 2

4